Am 21.03.2025 um 19:48 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
Learn that [...] Cantor [once] has [uttered] that the positive
numbers have more
reality than the even positive numbers. He said that is not in
conflict with the identical cardinality of both
sets. And he was right!
I doubt very much Cantor said such rubbish.
Actually, WM is right here. But the notion of "more reality" clearly
wasn't meant as a technical term (by Cantor). He -Cantor- was just
trying to explain the mathematical fact that 2IN is a PROPER subset of
IN, while both sets still have the same cardinality. (I'd dare to bet
that this was the only time he ever used that phrase in this context.)
Hint: WM is all about words.
Am 21.03.2025 um 20:37 schrieb Moebius:
Am 21.03.2025 um 19:48 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
Learn that [...] Cantor [once] has [uttered] that the positive
numbers have more
reality than the even positive numbers. He said that is not in
conflict with the identical cardinality of both
sets. And he was right!
I doubt very much Cantor said such rubbish.
Actually, WM is right here. But the notion of "more reality" clearly
wasn't meant as a technical term (by Cantor). He -Cantor- was just
trying to explain the mathematical fact that 2IN is a PROPER subset of
IN, while both sets still have the same cardinality. (I'd dare to bet
that this was the only time he ever used that phrase in this context.)
Her's the original quote:
"Sei M die Gesamtheit (nü) aller endlichen Zahlen nü, M' die
Gesamtheit (2nü) aller geraden Zahlen 2nü. Hier ist unbedingt richtig, daß M seiner Entität nach /reicher/ ist, als M'; enthält doch M außer den geraden Zahlen, aus welchen M' besteht, noch außerdem alle ungeraden
Zahlen M''. Andererseits ist ebenso unbedingt richtig, daß den beiden
Mengen M und M' nach Nr. 2 und 3 /dieselbe/ Kardinalzahl zukommt. Beides
ist sicher und keines steht dem andern im Wege, wenn man nur auf die Distinktion von /Realität/ und /Zahl/ achtet. Man muß also sagen: /die Menge M hat mehr Realität wie M', weil sie M' und außerdem M'' als Bestandteile enthält; die den beiden Mengen M und M' zukommenden Kardinalzahlen sind aber gleich/." (G. Cantor)
Google Translator:
"Let M be the totality (nu) of all finite numbers nu, and M' the
totality (2nu) of all even numbers 2nu. Here it is absolutely true that
M is /richer/ than M' in its essence [entity]; after all, M contains, in addition to the even numbers of which M' consists, all the odd numbers
M''. On the other hand, it is equally absolutely true that the two sets
M and M', according to no. 2 and 3, have /the same/ cardinal number.
Both are certain, and neither precludes the other, if one only pays
attention to the distinction between /reality/ and /number/. One must therefore say: /the set M has more reality than M' because it contains
M' and, in addition, M'' as components; but the cardinal numbers
belonging to the two sets M and M' are equal/."
Hint: WM is all about words.
WM <wolfgang.mueckenheim@tha.de> wrote:Actually, WM is right here. But the notion of "more reality" clearly
Learn that [...] Cantor [once] has [uttered] that the positive numbers have more
reality than the even positive numbers.
He said that is not in conflict with the identical cardinality of both
sets. And he was right!
I doubt very much Cantor said such rubbish.
Am 21.03.2025 um 20:46 schrieb Moebius:
Am 21.03.2025 um 20:37 schrieb Moebius:
Am 21.03.2025 um 19:48 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
Learn that [...] Cantor [once] has [uttered] that the positive
numbers have more
reality than the even positive numbers. He said that is not in
conflict with the identical cardinality of both
sets. And he was right!
I doubt very much Cantor said such rubbish.
Actually, WM is right here. But the notion of "more reality" clearly
wasn't meant as a technical term (by Cantor). He -Cantor- was just
trying to explain the mathematical fact that 2IN is a PROPER subset
of IN, while both sets still have the same cardinality. (I'd dare to
bet that this was the only time he ever used that phrase in this
context.)
Her's the original quote:
"Sei M die Gesamtheit (nü) aller endlichen Zahlen nü, M' die
Gesamtheit (2nü) aller geraden Zahlen 2nü. Hier ist unbedingt richtig,
daß
M seiner Entität nach /reicher/ ist, als M'; enthält doch M außer den
geraden Zahlen, aus welchen M' besteht, noch außerdem alle ungeraden
Zahlen M''. Andererseits ist ebenso unbedingt richtig, daß den beiden
Mengen M und M' nach Nr. 2 und 3 /dieselbe/ Kardinalzahl zukommt. Beides
ist sicher und keines steht dem andern im Wege, wenn man nur auf die
Distinktion von /Realität/ und /Zahl/ achtet. Man muß also sagen: /die
Menge M hat mehr Realität wie M', weil sie M' und außerdem M'' als
Bestandteile enthält; die den beiden Mengen M und M' zukommenden
Kardinalzahlen sind aber gleich/." (G. Cantor)
Google Translator:
"Let M be the totality (nu) of all finite numbers nu, and M' the
totality (2nu) of all even numbers 2nu. Here it is absolutely true
that M is /richer/ than M' in its essence [entity]; after all, M
contains, in addition to the even numbers of which M' consists, all
the odd numbers M''. On the other hand, it is equally absolutely true
that the two sets M and M', according to no. 2 and 3, have /the same/
cardinal number. Both are certain, and neither precludes the other, if
one only pays attention to the distinction between /reality/ and /
number/. One must therefore say: /the set M has more reality than M'
because it contains M' and, in addition, M'' as components; but the
cardinal numbers belonging to the two sets M and M' are equal/."
Well, what can we say? Set theory in its infancy.
Hint: WM is all about words.
Am 21.03.2025 um 20:37 schrieb Moebius:
Am 21.03.2025 um 19:48 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
Learn that [...] Cantor [once] has [uttered] that the positive
numbers have more
reality than the even positive numbers. He said that is not in
conflict with the identical cardinality of both
sets. And he was right!
I doubt very much Cantor said such rubbish.
Actually, WM is right here. But the notion of "more reality" clearly
wasn't meant as a technical term (by Cantor). He -Cantor- was just
trying to explain the mathematical fact that 2IN is a PROPER subset of
IN, while both sets still have the same cardinality. (I'd dare to bet
that this was the only time he ever used that phrase in this context.)
Her's the original quote:
"Sei M die Gesamtheit (nü) aller endlichen Zahlen nü, M' die
Gesamtheit (2nü) aller geraden Zahlen 2nü. Hier ist unbedingt richtig, daß M seiner Entität nach /reicher/ ist, als M'; enthält doch M außer den geraden Zahlen, aus welchen M' besteht, noch außerdem alle ungeraden
Zahlen M''. Andererseits ist ebenso unbedingt richtig, daß den beiden
Mengen M und M' nach Nr. 2 und 3 /dieselbe/ Kardinalzahl zukommt. Beides
ist sicher und keines steht dem andern im Wege, wenn man nur auf die Distinktion von /Realität/ und /Zahl/ achtet. Man muß also sagen: /die Menge M hat mehr Realität wie M', weil sie M' und außerdem M'' als Bestandteile enthält; die den beiden Mengen M und M' zukommenden Kardinalzahlen sind aber gleich/." (G. Cantor)
Google Translator:
"Let M be the totality (nu) of all finite numbers nu, and M' the
totality (2nu) of all even numbers 2nu. Here it is absolutely true that
M is /richer/ than M' in its essence [entity]; after all, M contains, in addition to the even numbers of which M' consists, all the odd numbers
M''. On the other hand, it is equally absolutely true that the two sets
M and M', according to no. 2 and 3, have /the same/ cardinal number.
Both are certain, and neither precludes the other, if one only pays attention to the distinction between /reality/ and /number/. One must therefore say: /the set M has more reality than M' because it contains
M' and, in addition, M'' as components; but the cardinal numbers
belonging to the two sets M and M' are equal/."
Hint: WM is all about words.
Moebius <invalid@example.invalid> wrote:
Am 21.03.2025 um 20:37 schrieb Moebius:
Am 21.03.2025 um 19:48 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
Learn that [...] Cantor [once] has [uttered] that the positive
numbers have more
reality than the even positive numbers. He said that is not in
conflict with the identical cardinality of both
sets. And he was right!
I doubt very much Cantor said such rubbish.
Actually, WM is right here. But the notion of "more reality" clearly
wasn't meant as a technical term (by Cantor). He -Cantor- was just
trying to explain the mathematical fact that 2IN is a PROPER subset of
IN, while both sets still have the same cardinality. (I'd dare to bet
that this was the only time he ever used that phrase in this context.)
Her's the original quote:
"Sei M die Gesamtheit (nü) aller endlichen Zahlen nü, M' die
Gesamtheit (2nü) aller geraden Zahlen 2nü. Hier ist unbedingt richtig, daß
M seiner Entität nach /reicher/ ist, als M'; enthält doch M außer den
geraden Zahlen, aus welchen M' besteht, noch außerdem alle ungeraden
Zahlen M''. Andererseits ist ebenso unbedingt richtig, daß den beiden
Mengen M und M' nach Nr. 2 und 3 /dieselbe/ Kardinalzahl zukommt. Beides
ist sicher und keines steht dem andern im Wege, wenn man nur auf die
Distinktion von /Realität/ und /Zahl/ achtet. Man muß also sagen: /die
Menge M hat mehr Realität wie M', weil sie M' und außerdem M'' als
Bestandteile enthält; die den beiden Mengen M und M' zukommenden
Kardinalzahlen sind aber gleich/." (G. Cantor)
What was the context of this quote? Was it a letter to a fellow mathematician, or a in a published work, or what?
It seems Cantor was fumbling around, trying to get a handle on some
concept, and called it Realität for want of a better word.
With the benefit of over a century of development, we can see there is no need for
this concept, which might not even be consistent.
OK, Cantor said/wrote this. It is still rubbish from a modern point of
view. It is only to be expected that the pioneers of a new field, along
with valid developments, also make mistakes.
Am 21.03.2025 um 19:48 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
Actually, WM is right here. But the notion of "more reality" clearlyLearn that [...] Cantor [once] has [uttered] that the positive
numbers have more
reality than the even positive numbers. He said that is not in
conflict with the identical cardinality of both
sets. And he was right!
I doubt very much Cantor said such rubbish.
wasn't meant as a technical term (by Cantor). He -Cantor- was just
trying to explain the mathematical fact that 2IN is a PROPER subset of
IN, while both sets still have the same cardinality.
(I'd dare to bet
that this was the only time he ever used that phrase in this context.)
Hint: WM is all about words.
"Substance" might have worked too, imho. :-P
BUT: The notion of "potential infinity" was not invented by Cantor or
his contemporaries, actually it goes back to _Aristotle_. Cantor just
voted for the (existence of the) actual infinite as represented by his infinite sets
What was the context of this quote? Was it a letter to a fellow mathematician, or a in a published work, or what?
It seems Cantor was fumbling around, trying to get a handle on some
concept, and called it Realität for want of a better word. With the
benefit of over a century of development, we can see there is no need for this concept, which might not even be consistent.
OK, Cantor said/wrote this. It is still rubbish from a modern point of
view.
WM <wolfgang.mueckenheim@tha.de> wrote:
On 21.03.2025 18:39, Jim Burns wrote:
On 3/21/2025 3:50 AM, WM wrote:
On 20.03.2025 23:25, Jim Burns wrote:
For sets not.having a WM.size,
Bob vanishing isn't a size.change.
Only if reducing isn't reducing.
What you (WM) think is reducing
isn't reducing.
You confuse the clear fact that in the reality of sets vanishing means
reducing with the foolish claim that cardinality was a meaningful notion.
Learn that even Cantor has accepted that the positive numbers have more
reality than the even positive numbers.
You mean something like positive numbers have a reality score of 5, and
the even positive numbers only have a reality score of 3?
He said that is not in conflict with the identical cardinality of both
sets. And he was right!
I doubt very much Cantor said such rubbish.
He was a mathematician.
"Coun[t]able" is simply another name for potential infinity.
Not even close.
Therefore the sentence "What you (WM) think is reducing isn't
reducing" exhibits you as a snooty dilettante who cannot distinguish
between cardinality and reality.
Hah! He's got to you, has he?
Am 21.03.2025 um 22:25 schrieb Alan Mackenzie:
What was the context of this quote? Was it a letter to a fellow
mathematician, or a in a published work, or what?
I really don't know (can't tell).
On 21.03.2025 20:37, Moebius wrote:Yeah, one can just talk about subsets, and nobody disputes that.
Am 21.03.2025 um 19:48 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:Actually, WM is right here. But the notion of "more reality" clearly
Learn that [...] Cantor [once] has [uttered] that the positiveI doubt very much Cantor said such rubbish.
numbers have more reality than the even positive numbers. He said
that is not in conflict with the identical cardinality of both sets.
And he was right!
wasn't meant as a technical term (by Cantor). He -Cantor- was just
trying to explain the mathematical fact that 2IN is a PROPER subset of
IN, while both sets still have the same cardinality.
This proves that cardinality does not concern quantities but only judgesNo, there are also finite cardinalities, and "actually infinite" sets
the behaviour of potentially infinite sets: They go on and on. Every
pair of a bijection has infinitely may successors. A final
quantification is impossible.
Then you shouldn't abuse "reality" like that.(I'd dare to bet that this was the only time he ever used that phraseYou would win.
in this context.)
*facepalm*Hint: WM is all about words.Reality is more than words.
On 21.03.2025 19:48, Alan Mackenzie wrote:Aaand those numbers are equally infinite. You just never learned that
WM <wolfgang.mueckenheim@tha.de> wrote:No, The number of positive numbers is |ℕ|. The number of even natural numbers is |ℕ|/2. It needs really years of brainwashing to honestly
On 21.03.2025 18:39, Jim Burns wrote:You mean something like positive numbers have a reality score of 5, and
On 3/21/2025 3:50 AM, WM wrote:You confuse the clear fact that in the reality of sets vanishing means
On 20.03.2025 23:25, Jim Burns wrote:What you (WM) think is reducing isn't reducing.
For sets not.having a WM.size, Bob vanishing isn't a size.change.Only if reducing isn't reducing.
reducing with the foolish claim that cardinality was a meaningful
notion.
Learn that even Cantor has accepted that the positive numbers have
more reality than the even positive numbers.
the even positive numbers only have a reality score of 3?
believe that addition of a number or subset leaves the number of
elements unchanged. It leaves the cardinality unchanged because this
notion is tantamount to potential infinity.
"Countable" means finite or bijective to N.You are simply unable to follow reasonable ideas."Coun[t]able" is simply another name for potential infinity.Not even close.
On 21.03.2025 19:48, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
On 21.03.2025 18:39, Jim Burns wrote:
On 3/21/2025 3:50 AM, WM wrote:
On 20.03.2025 23:25, Jim Burns wrote:
For sets not.having a WM.size,
Bob vanishing isn't a size.change.
Only if reducing isn't reducing.
What you (WM) think is reducing
isn't reducing.
You confuse the clear fact that in the reality of sets vanishing
means reducing with the foolish claim that cardinality was a
meaningful notion.
Learn that even Cantor has accepted that the positive numbers have
more reality than the even positive numbers.
You mean something like positive numbers have a reality score of 5,
and the even positive numbers only have a reality score of 3?
No, The number of positive numbers is |ℕ|. The number of even natural numbers is |ℕ|/2.
It needs really years of brainwashing to honestly believe that addition
of a number or subset leaves the number of elements unchanged.
It leaves the cardinality unchanged because this notion is tantamount
to potential infinity.
He said that is not in conflict with the identical cardinality of both
sets. And he was right!
I doubt very much Cantor said such rubbish.
You have pronounced your own sentence: Your opinions are rubbish.
He was a mathematician.
And you are not at all educated in this field.
"Coun[t]able" is simply another name for potential infinity.
Not even close.
You are simply unable to follow reasonable ideas.
Therefore the sentence "What you (WM) think is reducing isn't
reducing" exhibits you as a snooty dilettante who cannot distinguish
between cardinality and reality.
Hah! He's got to you, has he?
No, that is my judgement on JB.
Regards, WM
Am 22.03.2025 um 15:04 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
It leaves the cardinality unchanged because this notion is tantamount
to potential infinity.
You're stuck in the 1880s.
Nope. Actually, his claim ist absurd nonsense. Es handelt sich bei WM um einen aufmerksamkeitsheischenden, geisteskranker Spinner.
WM <wolfgang.mueckenheim@tha.de> wrote:
It leaves the cardinality unchanged because this notion is tantamount
to potential infinity.
You're stuck in the 1880s.
Am 22.03.2025 um 15:44 schrieb Moebius:
Am 22.03.2025 um 15:04 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
It leaves the cardinality unchanged because this notion is tantamount
to potential infinity.
You're stuck in the 1880s.
Nope. Actually, his claim ist absurd nonsense. Es handelt sich bei WM um
einen aufmerksamkeitsheischenden, geisteskranker Spinner.
Hint: (a) Sets have /cardinality/. (b) There are no "potentially
infinite" sets. Either a set is infinite or not (i.e. finite).
On 03/21/2025 11:15 PM, Moebius wrote:
Am 21.03.2025 um 22:25 schrieb Alan Mackenzie:
What was the context of this quote? Was it a letter to a fellow
mathematician, or a in a published work, or what?
I really don't know (can't tell).
It is from
4. Mitteilungen zur Lehre vom Transfiniten. VIII Nr. 7/8. 417
(the header of page 417 in his Gesammelte Abhandlungen)
Where did you take this quote from?
Am 22.03.2025 um 09:34 schrieb Ralf Bader:
On 03/21/2025 11:15 PM, Moebius wrote:
Am 21.03.2025 um 22:25 schrieb Alan Mackenzie:
What was the context of this quote? Was it a letter to a fellow
mathematician, or a in a published work, or what?
I really don't know (can't tell).
It is from
4. Mitteilungen zur Lehre vom Transfiniten. VIII Nr. 7/8. 417
(the header of page 417 in his Gesammelte Abhandlungen)
Where did you take this quote from?
IIRC, I've once seen it in one of WM's posts.
I guess, one of these is the original source:
Mitteilungen zur Lehre vom Transfiniten. 1. Zeitschrift für Philosophie
und philosophische Kritik 91 (1887), S. 81 − 125 und 252 − 270.
Mitteilungen zur Lehre vom Transfiniten. 2. Zeitschrift für Philosophie
und philosophische Kritik 92 (1888), S. 240 − 265.
Nuff said (concerning the context). :-P
Am 22.03.2025 um 21:07 schrieb Moebius:
Am 22.03.2025 um 09:34 schrieb Ralf Bader:
On 03/21/2025 11:15 PM, Moebius wrote:
Am 21.03.2025 um 22:25 schrieb Alan Mackenzie:
What was the context of this quote? Was it a letter to a fellow
mathematician, or a in a published work, or what?
I really don't know (can't tell).
It is from
4. Mitteilungen zur Lehre vom Transfiniten. VIII Nr. 7/8. 417
(the header of page 417 in his Gesammelte Abhandlungen)
Where did you take this quote from?
You can find it here:
https://gdz.sub.uni-goettingen.de/id/PPN237853094? tify=%7B%22pages%22%3A%5B429%5D%2C%22pan%22%3A%7B%22x%22%3A0.491%2C%22y%22%3A0.708%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.454%7D
IIRC, I've once seen it in one of WM's posts.
I guess, one of these is the original source:
Mitteilungen zur Lehre vom Transfiniten. 1. Zeitschrift für
Philosophie und philosophische Kritik 91 (1887), S. 81 − 125 und
252 − 270.
Mitteilungen zur Lehre vom Transfiniten. 2. Zeitschrift für
Philosophie und philosophische Kritik 92 (1888), S. 240 − 265.
Nuff said (concerning the context). :-P
WM <wolfgang.mueckenheim@tha.de> wrote:
Tell me, which of these infinite sets is bigger: {0, 4, 8, 12, 16, ....}
and {1, 3, 5, 7, 9, ....}?
The mathematically correct answer is that they are both the same size (cardinality) because there is a bijection between them.
I doubt very much Cantor said such rubbish.
You have pronounced your own sentence: Your opinions are rubbish.
On 03/22/2025 01:07 PM, Moebius wrote:
Nuff said (concerning the context). :-P
Does it live in
a mathematical platonist's real universe
of all the mathematical objects?
Or rather, mathematical platonism's?
Am 22.03.2025 um 15:44 schrieb Moebius:
Am 22.03.2025 um 15:04 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
It leaves the cardinality unchanged because this notion is tantamount
to potential infinity.
You're stuck in the 1880s.
Nope. Actually, his claim ist absurd nonsense. Es handelt sich bei WM
um einen aufmerksamkeitsheischenden, geisteskranker Spinner.
Hint: (a) Sets have /cardinality/. (b) There are no "potentially
infinity" sets. Either a set is infinite or not (i.e. finite).
The term "potentially infinite" has no use in
mathematics. Philosophers, etc., might cling to it, though.
You should be ashamed to be so misinformed and nevertheless a bigmouth.
On 22.03.2025 16:28, Alan Mackenzie wrote:
The term "potentially infinite" has no use in
mathematics. Philosophers, etc., might cling to it, though.
For your education.:
"On account of the matter I would like to add that in conventional mathematics, in particular in differential- and integral calculus, you
can gain little or no information about the transfinite because here the potential infinite plays the important role, I don't say the only role
but the role emerging to surface (which most mathematicians are readily satisfied with). Even Leibniz with whom I don't harmonize in many other respects too, has [...] fallen into most spectacular contradictions with respect to the actual infinite." [G. Cantor, letter to A. Schmid (26 Mar 1887)]
Will you show gratitude to be educated in great detail?
Regards, WM
for infinite sets "reality" and cardinality don't have to correspond.
What is the "reality" (in this sense) of N?
Le 23/03/2025 à 18:18, WM a écrit :
You should be ashamed to be so misinformed and nevertheless a bigmouth.
On 22.03.2025 12:31, joes wrote:
for infinite sets "reality" and cardinality don't have to correspond.
So it is! Substance and cardinality have nothing in common because sets
of very different substance have same cardinality.
Regards, WM
WM <wolfgang.mueckenheim@tha.de> wrote:
sets of very different substance [can] have [the] same cardinality.
Am 23.03.2025 um 22:28 schrieb Alan Mackenzie:
What is the "reality" (in this sense) of N?
The "reality" (or rather "substance") of N are its elements
1, 2, 3, ...
Clearly, IN\{1} has less "substance" than IN. Actually, the element 1 is missing in IN\{1} (in comparison to IN).
Moebius <invalid@example.invalid> wrote:
Am 23.03.2025 um 22:28 schrieb Alan Mackenzie:
What is the "reality" (in this sense) of N?
The "reality" (or rather "substance") of N are its elements
1, 2, 3, ...
Clearly, IN\{1} has less "substance" than IN. Actually, the element 1 is
missing in IN\{1} (in comparison to IN).
Are you sure?
You seem to be implying that ...
WM seemed to be saying that the "reality"/"substance" of any two sets
could be ranked, with one greater than the other
I doubt very much that Cantor intended "Realität" to have a mathematicaldefinition.
He was merely using the term in an effort to get others tounderstand how two sets, one a subset of the other, could have the same cardinality.
definition. He was merely using the term in an effort to get others to understand how two sets, one a subset of the other, could have the same cardinality.
On 22.03.2025 15:58, Moebius wrote:lolno. An infinite set has a cardinality. Bijections do that. AFAIU,
Am 22.03.2025 um 15:44 schrieb Moebius:That is a self-contradiction. Cardinality concerns only the potentially infinite subset of an actually infinite set. Bijections of infinite sets would have to cover all elements with no remainder.
Am 22.03.2025 um 15:04 schrieb Alan Mackenzie:Hint: (a) Sets have /cardinality/. (b) There are no "potentially
WM <wolfgang.mueckenheim@tha.de> wrote:Nope. Actually, his claim ist absurd nonsense. Es handelt sich bei WM
It leaves the cardinality unchanged because this notion isYou're stuck in the 1880s.
tantamount to potential infinity.
um einen aufmerksamkeitsheischenden, geisteskranker Spinner.
infinity" sets. Either a set is infinite or not (i.e. finite).
On 22.03.2025 15:04, Alan Mackenzie wrote:They both diverge to exactly omega.
WM <wolfgang.mueckenheim@tha.de> wrote:The second, of course. You need only consider finite sections and take
Tell me, which of these infinite sets is bigger: {0, 4, 8, 12, 16,
....} and {1, 3, 5, 7, 9, ....}?
the limit. Great mathematicians have devised this method.
Which are also bijected.The mathematically correct answer is that they are both the same sizeNonsense. The "bijection" is invalid because there are always infinitely
(cardinality) because there is a bijection between them.
many elements following after every defined pair.
Nothing to add.You should be ashamed to be so misinformed and nevertheless a bigmouth.You have pronounced your own sentence: Your opinions are rubbish.
WM <wolfgang.mueckenheim@tha.de> wrote:
On 22.03.2025 12:31, joes wrote:
for infinite sets "reality" and cardinality don't have to correspond.
So it is! Substance and cardinality have nothing in common because sets
of very different substance have same cardinality.
What is this "reality" or "substance" of which you speak?
The cardinality of N is aleph-0.
What is the "reality" (in this sense) of N?
You did not, in your
voluminous post, cite any indication of a _use_ of "potentially
infinite", only some philosophising about it.
Am Sun, 23 Mar 2025 18:18:15 +0100 schrieb WM:
The "bijection" is invalid because there are always infinitelyWhich are also bijected.
many elements following after every defined pair.
WM seemed to be saying that the "reality"/"substance" of any two sets
could be ranked, with one greater than the other (unless they were,
somehow, the same).
On 23.03.2025 20:39, Alan Mackenzie wrote:
You did not, in your
voluminous post, cite any indication of a _use_ of "potentially
infinite", only some philosophising about it.
Have you not read Hilbert and Cantor? In analysis potential infinity and
only it is used.
In modern mathematics there are the notions finite and infinite. They
are useful. I challenge you to produce a theorem which cannot be proven
with those notions, yet can be proven with, additionally, "potentially
infinite".
If neither you nor anybody else can do this, then we must conclude that
"potentially infinite" has no use in mathematics.
In fact all meaningful and correct applications of infinity in
mathematics concern potential infinity, because actual infinity either
is a chimera only or it is dark and therefore cannot be manipulated and applied in mathematics.
Regards, WM
On 22.03.2025 15:04, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
Tell me, which of these infinite sets is bigger: {0, 4, 8, 12, 16, ....}
and {1, 3, 5, 7, 9, ....}?
The second, of course.
You need only consider finite sections and take the limit. Great mathematicians have devised this method.
The mathematically correct answer is that they are both the same size
(cardinality) because there is a bijection between them.
Nonsense. The "bijection" is invalid because there are always infinitely
many elements following after every defined pair.
I doubt very much Cantor said such rubbish.
You have pronounced your own sentence: Your opinions are rubbish.
You should be ashamed to be so misinformed and nevertheless a bigmouth.
Regards, WM
On 23.03.2025 23:19, Alan Mackenzie wrote:
WM seemed to be saying that the "reality"/"substance" of any two sets
could be ranked, with one greater than the other (unless they were,
somehow, the same).
We should say: Sets with different elements like even and odd integers
have different substance but same amount of substance (number of elements).
Regards, WM
WM <wolfgang.mueckenheim@tha.de> wrote:
Have you not read Hilbert and Cantor? In analysis potential infinity and
only it is used.
What is used is the infinite. It needs no redundant qualifier
"potential".
"Potential infinity", I repeat, is unnecessary, given the well defined notions of "finite" and "infinite". [...]Indeed! Cantor already came to this conclusion...
WM <wolfgang.mueckenheim@tha.de> wrote:
On 22.03.2025 15:04, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
Tell me, which of these infinite sets is bigger: {0, 4, 8, 12, 16, ....} >>> and {1, 3, 5, 7, 9, ....}?
The second, of course.
Prove it.
You need only consider finite sections and take the limit. [...]
What on Earth do you mean by "finite sections", if anything? And take
what limit? [...]
On 23.03.2025 22:28, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
On 22.03.2025 12:31, joes wrote:
for infinite sets "reality" and cardinality don't have to correspond.
So it is! Substance and cardinality have nothing in common because sets
of very different substance have same cardinality.
What is this "reality" or "substance" of which you speak?
Substance is by far the better word. It denotes the number of elements.
The set {1, 2, 3} has more substance than the set {7, 14}.
For many sets the relative substance cannot be determined. But this
drawback is less disastrous than to lump every countable set together.
The cardinality of N is aleph-0.
What is the "reality" (in this sense) of N?
The substance of ℕ is |ℕ|. It is larger than every finite set. The substance of the set of prime numbers is far less than |ℕ| ....
.... but larger than every finite set. These are useful mathematical findings.
Regards, WM
WM <wolfgang.mueckenheim@tha.de> wrote:
On 23.03.2025 23:19, Alan Mackenzie wrote:
WM seemed to be saying that the "reality"/"substance" of any two sets
could be ranked, with one greater than the other (unless they were,
somehow, the same).
We [might] say: Sets with different elements like even and odd integers
have different substance but same [number of elements].*)
Am 24.03.2025 um 21:28 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
On 22.03.2025 15:04, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
Tell me, which of these infinite sets is bigger: {0, 4, 8, 12, 16, ....} >>>> and {1, 3, 5, 7, 9, ....}?
The second, of course.
Prove it.
You need only consider finite sections and take the limit. [...]
What on Earth do you mean by "finite sections", if anything? And take
what limit? [...]
Guess he's referring to this notion: https://en.wikipedia.org/wiki/Natural_density
Am 24.03.2025 um 21:40 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
On 23.03.2025 23:19, Alan Mackenzie wrote:
WM seemed to be saying that the "reality"/"substance" of any two sets
where one is a subset of the other
could be ranked, with one greater than the other (unless they were,
somehow, the same).
We [might] say: Sets with different elements like even and odd integers
have different substance but same [number of elements].*)
Indeed! :-P
With "number of elements" == "cardinality"
Nuff said. :-P
__________________________________________________________________________
*) And what would be the arguments for this claim? I mean, that {2n : n
e IN} and {2n + 1 : n e IN} have the same "number of elements" in
WMmath? Just because HE said so?
You have dishonestly snipped the core of my last post.
IF we measure
"the size" of a set using the notion of cardinality, clearly "the set of positive integers is not [...] larger than the set of perfect squares". :-)
Moebius <invalid@example.invalid> wrote:
Guess he's referring to this notion:
https://en.wikipedia.org/wiki/Natural_density
OK, thanks. That natural density is not a property of sets [in general]
The Wikipedia article describes [that natural density] it as a branch of number theory, not set theory.
"Die gesamte Mathematik inklusive der klassischen Zahlenbereiche läßt
sich als Teilgebiet der Mengenlehre auffassen." ["All of mathematics, including classical number domains, can be regarded as a subfield of
set theory."] (A. Oberschelp, Aufbau des Zahhlensystems, 1968)
Lit.: https://en.wikipedia.org/wiki/Arnold_Oberschelp
Am 24.03.2025 um 22:44 schrieb Moebius:
"Die gesamte Mathematik inklusive der klassischen Zahlenbereiche läßt
sich als Teilgebiet der Mengenlehre auffassen." ["All of mathematics,
including classical number domains, can be regarded as a subfield of
set theory."] (A. Oberschelp, Aufbau des Zahhlensystems, 1968)
Too polemic?
How about: "Die meisten Teile der Mathematik inklusive der klassischen Zahlenbereiche lassen sich als Teilgebiet der Mengenlehre
auffassen." ["Most parts of mathematics, including classical number
domains, can be regarded as a subfield of set theory."]
Lit.: https://en.wikipedia.org/wiki/Arnold_Oberschelp
Am 24.03.2025 um 22:44 schrieb Moebius:
"Die gesamte Mathematik inklusive der klassischen Zahlenbereiche läßt
sich als Teilgebiet der Mengenlehre auffassen." ["All of mathematics,
including classical number domains, can be regarded as a subfield of
set theory."] (A. Oberschelp, Aufbau des Zahhlensystems, 1968)
Too polemic?
How about: "Die meisten Teile der Mathematik inklusive der klassischen Zahlenbereiche lassen sich als Teilgebiet der Mengenlehre
auffassen." ["Most parts of mathematics, including classical number
domains, can be regarded as a subfield of set theory."]
Lit.: https://en.wikipedia.org/wiki/Arnold_Oberschelp
On 24.03.2025 02:11, joes wrote:How can you disprove it?
Am Sun, 23 Mar 2025 18:18:15 +0100 schrieb WM:
How can you prove that?The "bijection" is invalid because there are always infinitely manyWhich are also bijected.
elements following after every defined pair.
On 23.03.2025 22:28, Alan Mackenzie wrote:You can just say subsets.
WM <wolfgang.mueckenheim@tha.de> wrote:Substance is by far the better word. It denotes the number of elements.
On 22.03.2025 12:31, joes wrote:What is this "reality" or "substance" of which you speak?
for infinite sets "reality" and cardinality don't have to correspond.So it is! Substance and cardinality have nothing in common because
sets of very different substance have same cardinality.
The set {1, 2, 3} has more substance than the set {7, 14}. For many sets
the relative substance cannot be determined. But this drawback is less disastrous than to lump every countable set together.
What's the use?The cardinality of N is aleph-0.The substance of ℕ is |ℕ|. It is larger than every finite set. The substance of the set of prime numbers is far less than |ℕ| but larger
What is the "reality" (in this sense) of N?
than every finite set. These are useful mathematical findings.
Moebius <invalid@example.invalid> wrote:
Guess he's referring to this notion:
https://en.wikipedia.org/wiki/Natural_density
OK, thanks. That natural density is not a property of sets, it's a
property of the natural numbers with additional (arithmetic) structure applied to them.
WM <wolfgang.mueckenheim@tha.de> wrote:
On 23.03.2025 20:39, Alan Mackenzie wrote:
You did not, in your
voluminous post, cite any indication of a _use_ of "potentially
infinite", only some philosophising about it.
Have you not read Hilbert and Cantor? In analysis potential infinity and
only it is used.
What is used is the infinite. It needs no redundant qualifier
"potential".
You have dishonestly snipped the core of my last post. Here it is
again:
In modern mathematics there are the notions finite and infinite. They
are useful. I challenge you to produce a theorem which cannot be proven >>> with those notions, yet can be proven with, additionally, "potentially
infinite".
If neither you nor anybody else can do this, then we must conclude that
"potentially infinite" has no use in mathematics.
In fact all meaningful and correct applications of infinity in
mathematics concern potential infinity, because actual infinity either
is a chimera only or it is dark and therefore cannot be manipulated and
applied in mathematics.
"Potential infinity", I repeat, is unnecessary,
WM <wolfgang.mueckenheim@tha.de> wrote:
On 22.03.2025 15:04, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
Tell me, which of these infinite sets is bigger: {0, 4, 8, 12, 16, ....} >>> and {1, 3, 5, 7, 9, ....}?
The second, of course.
Prove it.
You need only consider finite sections and take the limit. Great
mathematicians have devised this method.
What on Earth do you mean by "finite sections", if anything?
And take
what limit? Which great mathematician(s) were supposedly involved in
this method.
The mathematically correct answer is that they are both the same size
(cardinality) because there is a bijection between them.
Nonsense. The "bijection" is invalid because there are always infinitely
many elements following after every defined pair.
You are (?deliberately) ignorant of the definition of bijection.
"Following after every defined pair", if it's not meaningless, is only
the empty set.
The bijection between these two sets exists and is
uncontroversial.
As I've said more than once, I have a degree in maths.
You do not.
Which one of us is more likely to be misinformed about mathematics?
Why don't you just use the word cardinality, like everybody else does.
It is defined mathematically, and generally understood.
WM <wolfgang.mueckenheim@tha.de> wrote:
How is it disastrous to "lump every [infinite] countable set together"?
Does it lead to a mathematical contradiction? It doesn't that I'm aware
of.
The cardinality of N is aleph-0.
What is the "reality" (in this sense) of N?
The substance of ℕ is |ℕ|. It is larger than every finite set. The
substance of the set of prime numbers is far less than |ℕ| ....
By how much is its "substance" supposedly smaller? Quantify it!
.... but larger than every finite set. These are useful mathematical
findings.
Are they? What use are they?
What mathematical theorems do they enable
the proof of?
On 24.03.2025 21:12, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
On 23.03.2025 20:39, Alan Mackenzie wrote:
You did not, in your
voluminous post, cite any indication of a _use_ of "potentially
infinite", only some philosophising about it.
Have you not read Hilbert and Cantor? In analysis potential infinity and >>> only it is used.
What is used is the infinite. It needs no redundant qualifier
"potential".
You don't grasp it.
You have dishonestly snipped the core of my last post. Here it is
again:
In modern mathematics there are the notions finite and infinite.
They are useful. I challenge you to produce a theorem which cannot
be proven with those notions, yet can be proven with, additionally,
"potentially infinite".
I told you that all mathematics is based on potential infinity.
If neither you nor anybody else can do this, then we must conclude
that "potentially infinite" has no use in mathematics.
Actual infinity has no use in any kind of applied mathematics.
In fact all meaningful and correct applications of infinity in
mathematics concern potential infinity, because actual infinity
either is a chimera only or it is dark and therefore cannot be
manipulated and applied in mathematics.
"Potential infinity", I repeat, is unnecessary,
Try to understand the scholars I quoted.
Regards, WM
Or can you refute my assertions of nonsense by defining this "substance"/"Realität" in a mathematical fashion?
On 24.03.2025 21:40, Alan Mackenzie wrote:
Or can you refute my assertions of nonsense by defining this
"substance"/"Realität" in a mathematical fashion?
The relative amount of substance of two infinite sets of numbers is the limit for n --> oo of the quotient of the numbers of elements within [0, n].
Regards, WM
WM <wolfgang.mueckenheim@tha.de> wrote:
[...] It is simply a property of potentially infinite initial
segments of [an] actually infinite set.
On 24.03.2025 21:52, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
How is it disastrous to "lump every [infinite] countable set together"?
Does it lead to a mathematical contradiction? It doesn't that I'm aware
of.
It doesn't. It is simply a property of potentially infinite initial
segments of actually infinite set. Disastrous is that some naive minds
are lead to believe that the actually infinite sets have "in fact" same substance. Assisted imbecility.
The cardinality of N is aleph-0.
What is the "reality" (in this sense) of N?
The substance of ℕ is |ℕ|. It is larger than every finite set. The
substance of the set of prime numbers is far less than |ℕ| ....
By how much is its "substance" supposedly smaller? Quantify it!
It cannot be quantified yet. That would be a rewarding subject of future research.
.... but larger than every finite set. These are useful mathematical
findings.
Are they? What use are they?
Some researchers may be interested.
What mathematical theorems do they enable the proof of?
Mathematical theorems can only be proved by use of potential infinity.
Regards, WM
WM <wolfgang.mueckenheim@tha.de> wrote:
On 24.03.2025 21:40, Alan Mackenzie wrote:
Or can you refute my assertions of nonsense by defining this
"substance"/"Realität" in a mathematical fashion?
The relative amount of substance of two infinite sets of numbers is the
limit for n --> oo of the quotient of the numbers of elements within [0, n].
Thank you!
So "substance"/"Realität" is not a propery of sets in general, it is a property only of subsets of N.
Am Mon, 24 Mar 2025 20:40:07 +0100 schrieb WM:
On 24.03.2025 02:11, joes wrote:How can you disprove it?
Am Sun, 23 Mar 2025 18:18:15 +0100 schrieb WM:How can you prove that?
The "bijection" is invalid because there are always infinitely manyWhich are also bijected.
elements following after every defined pair.
WM <wolfgang.mueckenheim@tha.de> wrote:
On 24.03.2025 21:52, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
How is it disastrous to "lump every [infinite] countable set together"?
Does it lead to a mathematical contradiction? It doesn't that I'm aware >>> of.
It doesn't. It is simply a property of potentially infinite initial
segments of actually infinite set. Disastrous is that some naive minds
are lead to believe that the actually infinite sets have "in fact" same
substance. Assisted imbecility.
According to one of your other posts today, this "substance" is a
property only of subsets of N.
Countably infinite sets all have the same cardinality.
The cardinality of N is aleph-0.
What is the "reality" (in this sense) of N?
The substance of ℕ is |ℕ|. It is larger than every finite set. The >>>> substance of the set of prime numbers is far less than |ℕ| ....
By how much is its "substance" supposedly smaller? Quantify it!
It cannot be quantified yet. That would be a rewarding subject of future
research.
It can indeed by quantified. The assymptotic distribution of prime
numbers is known: the probability of a number near n being prime is
1/log(n). So the proportion of numbers in {1, ..., n} which are prime
will tend to zero as n tends to infinity.
.... but larger than every finite set. These are useful mathematical
findings.
Are they? What use are they?
Some researchers may be interested.
Maybe. On the other hand, maybe not.
What mathematical theorems do they enable the proof of?
Mathematical theorems can only be proved by use of potential infinity.
That's a very bold statement. Many theorems can be proven without regard
to the infinite.
Many others do in fact use the infinite.
But theorems which require the concept of "potentially infinite", over
and above plain infinite, for their proof? I've asked you before for an example, and you've yet to come up with one.
Am 25.03.2025 um 19:27 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
Just want to point out this idiotic nonsense:
[...] It is simply a property of potentially infinite initial
segments of [an] actually infinite set.
What the hell are "potentially infinite initial segments of <whatever>". Beware of Mückenheim's delusions!
THIS SILLY ASSHOLE FULL OF SHIT seems to refer to FINITE initial
segments of <whatever> here. (FISONs in the case of IN.)
U FIS = IN .
On 25.03.2025 09:18, joes wrote:Of course it does.
Am Mon, 24 Mar 2025 20:40:07 +0100 schrieb WM:Knowing that every pair belongs to a finite initial segment.
On 24.03.2025 02:11, joes wrote:How can you disprove it?
Am Sun, 23 Mar 2025 18:18:15 +0100 schrieb WM:How can you prove that?
The "bijection" is invalid because there are always infinitely manyWhich are also bijected.
elements following after every defined pair.
Upon itNeither can they be disproven.
follow infinitely many elements which cannot be proven to have partners
in the other set.
If you have a degree in maths you should be able to say what you mean
clearly and concisely. Maybe you've lost this skill over the decades
since your graduation.
On 25.03.2025 19:27, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
On 24.03.2025 21:52, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
How is it disastrous to "lump every [infinite] countable set together"? >>>> Does it lead to a mathematical contradiction? It doesn't that I'm aware >>>> of.
It doesn't. It is simply a property of potentially infinite initial
segments of actually infinite set. Disastrous is that some naive minds
are lead to believe that the actually infinite sets have "in fact" same
substance. Assisted imbecility.
According to one of your other posts today, this "substance" is a
property only of subsets of N.
They supply the simplest explanation. But substance is in every
non-empty set.
Countably infinite sets all have the same cardinality.
That proves that cardinality is rather uninteresting.
The cardinality of N is aleph-0.
What is the "reality" (in this sense) of N?
The substance of ℕ is |ℕ|. It is larger than every finite set. The >>>>> substance of the set of prime numbers is far less than |ℕ| ....
By how much is its "substance" supposedly smaller? Quantify it!
It cannot be quantified yet. That would be a rewarding subject of future >>> research.
It can indeed by quantified. The assymptotic distribution of prime
numbers is known: the probability of a number near n being prime is
1/log(n). So the proportion of numbers in {1, ..., n} which are prime
will tend to zero as n tends to infinity.
Tend to yes, but not reaching it.
.... but larger than every finite set. These are useful mathematical >>>>> findings.
Are they? What use are they?
Some researchers may be interested.
Maybe. On the other hand, maybe not.
What mathematical theorems do they enable the proof of?
Mathematical theorems can only be proved by use of potential infinity.
That's a very bold statement. Many theorems can be proven without regard
to the infinite.
Of course I meant theorems using the infinite.
Many others do in fact use the infinite.
But theorems which require the concept of "potentially infinite", over
and above plain infinite, for their proof? I've asked you before for an
example, and you've yet to come up with one.
Every theorem in analysis. This has not much changed since Cantor and Hilbert.
Regards, WM
WM <wolfgang.mueckenheim@tha.de> wrote:
What everybody else refers to as infinte, you seem to want to call "potentially infinite".
Am 26.03.2025 um 00:39 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
What everybody else refers to as infinte, you seem to want to call
"potentially infinite".
Actually, it's rather the other way round. WM has a tendency to call
things "potentially infinite" which everyone else would call "finite".
Am 26.03.2025 um 01:14 schrieb Moebius:
Am 26.03.2025 um 01:06 schrieb Moebius:
Am 26.03.2025 um 00:39 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
What everybody else refers to as infinte, you seem to want to call
"potentially infinite".
Actually, it's rather the other way round. WM has a tendency to call
things "potentially infinite" which everyone else would call "finite".
It seems that he recently recognised this himself, leading to the absurd
phrase
potentially (in-)finite <so-and-so> (WM)
<facepalm>
(You won't find such a the notion ANYWHERE ... else.)
"The amount of energy needed to refute bullshit is an order of magnitude bigger than that needed to produce it." (Brandolini's law, bullshit
asymmetry principle)
Am 26.03.2025 um 01:06 schrieb Moebius:
Am 26.03.2025 um 00:39 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
What everybody else refers to as infinte, you seem to want to call
"potentially infinite".
Actually, it's rather the other way round. WM has a tendency to call
things "potentially infinite" which everyone else would call "finite".
It seems that he recently recognised this himself, leading to the absurd phrase
potentially (in-)finite <so-and-so> (WM)
<facepalm>
(You won't find such a the notion ANYWHERE ... else.)
Am Tue, 25 Mar 2025 20:16:10 +0100 schrieb WM:
On 25.03.2025 09:18, joes wrote:Of course it does.
Am Mon, 24 Mar 2025 20:40:07 +0100 schrieb WM:Knowing that every pair belongs to a finite initial segment.
On 24.03.2025 02:11, joes wrote:How can you disprove it?
Am Sun, 23 Mar 2025 18:18:15 +0100 schrieb WM:How can you prove that?
The "bijection" is invalid because there are always infinitely many >>>>>> elements following after every defined pair.Which are also bijected.
Upon itNeither can they be disproven.
follow infinitely many elements which cannot be proven to have partners
in the other set.
Am 26.03.2025 um 01:06 schrieb Moebius:
Am 26.03.2025 um 00:39 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
What everybody else refers to as infinte, you seem to want to call
"potentially infinite".
Actually, it's rather the other way round. WM has a tendency to call
things "potentially infinite" which everyone else would call "finite".
It seems that he recently recognised this himself, leading to the absurd phrase
potentially (in-)finite <so-and-so> (WM)
(You won't find such a the notion ANYWHERE ... else.)
WM <wolfgang.mueckenheim@tha.de> wrote:
substance is in every non-empty set.
Seems doubtful. What you seem to be saying is that every set has a
superset, you embed the set in that superset, then a portion of the
superset is the original set. That portion, a number between 0 and 1,
then becomes the "substance".
You're saying that the "substance" isn't a property of a set as such,
it's a property of a relationship between a superset and a subset.
For example, to get the "substance" of N with respect to Q, you could embed it in the superset Q: You'd get something like: {0, 1, 1/2, 2, 1/3, 3,
1/4, 2/3, 3/2, 4, 1/5, 5, ....}. Then this "substance" would come out as zero.
So, to come back to my original example, the "substance" of {0, 4, 8, 12,
16, ...} wrt N is 1/4.
The substance of {1, 3, 5, 7, 9, ...} wrt {0,
1/2, 1, 3/2, 2, 5/2, 3, ....} is also 1/4.
Their "subtances" are thus
the same.
I haven't come across this notion of "substance"/"Realität" before, and
it doesn't feel like solid maths. It all feels as though you are making
it up as you go along.
Countably infinite sets all have the same cardinality.
That proves that cardinality is rather uninteresting.
On the contrary, it is fascinating.
Tend to yes, but not reaching it.
I thought you just said you had a degree in maths. But you don't seem to understand the process of limits (a bit like John Gabriel didn't when he
was still around).
Every theorem in analysis. This has not much changed since Cantor and
Hilbert.
Theroems in analysis require the infinite yes. They don't require the confusing notion of "potentially infinite".
In my undergraduate studies,
the term "potentially infinite" wasn't used a single time. The first
time I came across it was in this newsgroup just a few years ago.
What everybody else refers to as infinte, you seem to want to call "potentially infinite".
On 26.03.2025 00:39, Alan Mackenzie wrote:That makes all of them >=omega.
WM <wolfgang.mueckenheim@tha.de> wrote:
No, that is not what I meant. Substance is simply the elements of thesubstance is in every non-empty set.Seems doubtful. What you seem to be saying is that every set has a
superset, you embed the set in that superset, then a portion of the
superset is the original set. That portion, a number between 0 and 1,
then becomes the "substance".
set. The amount of substance is the number of elements. This number
exists also for actually infinity sets but cannot be expressed by
natural numbers.
We only know that ∀k,n ∈ ℕ_def: |ℕ|/k > n.
Try it with infinite sets.You're saying that the "substance" isn't a property of a set as such,The relative amount of substance can be determined. The set {1, 2, 3}
it's a property of a relationship between a superset and a subset.
has more substance than the set {father, mother}.
Infinitely so!For example, to get the "substance" of N with respect to Q, you couldNearly. It is smaller than any definable fraction.
embed it in the superset Q: You'd get something like: {0, 1, 1/2, 2,
1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5, ....}. Then this "substance" would
come out as zero.
Except to you. For finite sets you can just use cardinality.So, to come back to my original example, the "substance" of {0, 4, 8,Yes.
12, 16, ...} wrt N is 1/4.
The substance of {1, 3, 5, 7, 9, ...} wrt {0,Yes.
1/2, 1, 3/2, 2, 5/2, 3, ....} is also 1/4.
Their "subtances" are thus the same.Yes. Their amounts of substance, to be precise.
I haven't come across this notion of "substance"/"Realität" before, andReality is Cantor's expression, Substance is Fritsche's (better)
it doesn't feel like solid maths. It all feels as though you are
making it up as you go along.
expression. For all finite sets, it is solid maths. Limits are
well-known from analysis.
Yes, every natural number has a FIS. "Undefined numbers" aren't naturals.If you consider it with cool blood, then you will recognize that allOn the contrary, it is fascinating.Countably infinite sets all have the same cardinality.That proves that cardinality is rather uninteresting.
pairs of a bijection with ℕ are defined within a finite initial segment
[0, n]. That is true for every n. But the infinity lies in the
successors which are undefined.
No, I asked him for the title.Tend to yes, but not reaching it.I thought you just said you had a degree in maths.
There is no real number other than 0.But you don't seem0/oo = 0. 1/oo is smaller than every definable fraction.
to understand the process of limits (a bit like John Gabriel didn't
when he was still around).
Yes, nobody refers to "actual infinity".They have been created using only this notion. And also Cantor'sEvery theorem in analysis. This has not much changed since Cantor andTheroems in analysis require the infinite yes. They don't require the
Hilbert.
confusing notion of "potentially infinite".
"bijections" are based upon potential infinity.
What we refer to as infinite isn't variable.What everybody else refers to as infinte, you seem to want to callThe potential infinite is a variable finite. Cantor's actual infinity is
"potentially infinite".
not variable but fixed. (Therefore Hilbert's hotel is potential
infinity.)
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:
The potential infinite is a variable finite. Cantor's actual infinity isWhat we refer to as infinite isn't variable.
not variable but fixed. (Therefore Hilbert's hotel is potential
infinity.)
On 26.03.2025 21:06, joes wrote:
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:
The potential infinite is a variable finite. Cantor's actual infinity is >>> not variable but fixed. (Therefore Hilbert's hotel is potentialWhat we refer to as infinite isn't variable.
infinity.)
The number of guests/rooms in Hilbert's hotel is infinite but can grow.
That is variable infinity.
Regards, WM
WM <wolfgang.mueckenheim@tha.de> wrote:
On 26.03.2025 21:06, joes wrote:
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:
The potential infinite is a variable finite. Cantor's actual infinity is >>>> not variable but fixed. (Therefore Hilbert's hotel [concerns]
The number of [...] rooms in Hilbert's hotel is infinite but [...] grow[s].
On 26.03.2025 00:39, Alan Mackenzie wrote:
WM <wolfgang.mueckenheim@tha.de> wrote:
substance is in every non-empty set.
Seems doubtful. What you seem to be saying is that every set has a
superset, you embed the set in that superset, then a portion of the
superset is the original set. That portion, a number between 0 and 1,
then becomes the "substance".
No, that is not what I meant. Substance is simply the elements of the
set. The amount of substance is the number of elements.
This number exists also for actually infinite sets but cannot be
expressed by natural numbers.
We only know that ∀k,n ∈ ℕ_def: |ℕ|/k > n.
You're saying that the "substance" isn't a property of a set as such,
it's a property of a relationship between a superset and a subset.
The relative amount of substance can be determined. The set {1, 2, 3}
has more substance than the set {father, mother}.
For example, to get the "substance" of N with respect to Q, you could
embed it in the superset Q: You'd get something like: {0, 1, 1/2, 2,
1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5, ....}. Then this "substance" would
come out as zero.
Nearly. It is smaller than any definable fraction.
So, to come back to my original example, the "substance" of {0, 4, 8,
12, 16, ...} wrt N is 1/4.
Yes.
The substance of {1, 3, 5, 7, 9, ...} wrt {0, 1/2, 1, 3/2, 2, 5/2, 3,
....} is also 1/4.
Yes.
Their "subtances" are thus the same.
Yes. Their amounts of substance, to be precise.
I haven't come across this notion of "substance"/"Realität" before,
and it doesn't feel like solid maths. It all feels as though you are
making it up as you go along.
Reality is Cantor's expression, Substance is Fritsche's (better)
expression. For all finite sets, it is solid maths. Limits are
well-known from analysis.
Countably infinite sets all have the same cardinality.
That proves that cardinality is rather uninteresting.
On the contrary, it is fascinating.
If you consider it with cool blood, then you will recognize that all
pairs of a bijection with ℕ are defined within a finite initial segment [0, n]. That is true for every n. But the infinity lies in the
successors which are undefined.
Tend to yes, but not reaching it.
I thought you just said you had a degree in maths. But you don't seem to
understand the process of limits (a bit like John Gabriel didn't when he
was still around).
0/oo = 0. 1/oo is smaller than every definable fraction.
Every theorem in analysis. This has not much changed since Cantor and
Hilbert.
Theroems in analysis require the infinite yes. They don't require the
confusing notion of "potentially infinite".
They have been created using only this notion. And also Cantor's "bijections" bare based upon potential infinity.
In my undergraduate studies, the term "potentially infinite" wasn't
used a single time. The first time I came across it was in this
newsgroup just a few years ago.
The Bourbakis have tried to exorcize the potential infinite from mathematics. Your teachers have been taught by them or their pupils.
What everybody else refers to as infinte, you seem to want to call
"potentially infinite".
The potential infinite is a variable finite. Cantor's actual infinity
is not variable but fixed. (Therefore Hilbert's hotel is potential
infinity.)
Regards, WM
On 26.03.2025 21:06, joes wrote:The number of rooms is fixed, otherwise it weren't interesting.
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:
The number of guests/rooms in Hilbert's hotel is infinite but can grow.The potential infinite is a variable finite. Cantor's actual infinityWhat we refer to as infinite isn't variable.
is not variable but fixed. (Therefore Hilbert's hotel is potential
infinity.)
That is variable infinity.
WM <wolfgang.mueckenheim@tha.de> wrote:
Crank talk. You don't understand limits, as I've already said. Have you really got a degree in mathematics? It seems unlikely.
Cantor's "bijections" bare based upon potential infinity.
Am 27.03.2025 um 12:18 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
Crank talk. You don't understand limits, as I've already said. Have you
really got a degree in mathematics? It seems unlikely.
WM doesn't have a degree in mathematics.
Cantor's "bijections" are based upon potential infinity.
Complete nonsense.
Am Wed, 26 Mar 2025 21:28:52 +0100 schrieb WM:
On 26.03.2025 21:06, joes wrote:The number of rooms is fixed,
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:The number of guests/rooms in Hilbert's hotel is infinite but can grow.
The potential infinite is a variable finite. Cantor's actual infinityWhat we refer to as infinite isn't variable.
is not variable but fixed. (Therefore Hilbert's hotel is potential
infinity.)
That is variable infinity.
otherwise it weren't interesting.
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
The number of [...] rooms in Hilbert's hotel is infinite but [...]
grow[s].
No, it doesn't.
WM <wolfgang.mueckenheim@tha.de> wrote:
Substance is simply the elements of the
set. The amount of substance is the number of elements.
You seem to mean the cardinality of the set.
This number exists also for actually infinite sets but cannot be
expressed by natural numbers.
We only know that ∀k,n ∈ ℕ_def: |ℕ|/k > n.
|N|/k is undefined.
You're saying that the "substance" isn't a property of a set as such,
it's a property of a relationship between a superset and a subset.
The relative amount of substance can be determined. The set {1, 2, 3}
has more substance than the set {father, mother}.
You mean it has a larger cardinality.
For example, to get the "substance" of N with respect to Q, you could
embed it in the superset Q: You'd get something like: {0, 1, 1/2, 2,
1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5, ....}. Then this "substance" would
come out as zero.
Nearly. It is smaller than any definable fraction.
Crank talk.
So, to come back to my original example, the "substance" of {0, 4, 8,
12, 16, ...} wrt N is 1/4.
Yes.
The substance of {1, 3, 5, 7, 9, ...} wrt {0, 1/2, 1, 3/2, 2, 5/2, 3,
....} is also 1/4.
Yes.
Their "subtances" are thus the same.
Yes. Their amounts of substance, to be precise.
Or their cardinality, to be even more precise.
If you consider it with cool blood, then you will recognize that all
pairs of a bijection with ℕ are defined within a finite initial segment
[0, n]. That is true for every n. But the infinity lies in the
successors which are undefined.
That's pure baloney. Every element of a bijection is an ordered pair of
an element of set 1 and an element of set 2. Each element of these sets occurs in exactly one ordered pair. There is no need to obfuscate this definition with considerations of finite initial segments or infinity or
what have you.
Tend to yes, but not reaching it.
I thought you just said you had a degree in maths. But you don't seem to >>> understand the process of limits (a bit like John Gabriel didn't when he >>> was still around).
0/oo = 0. 1/oo is smaller than every definable fraction.
More crank talk. Ordinary arithmetic is not defined on infinity. And "smaller than every definable fraction" is zero.
The Bourbakis have tried to exorcize the potential infinite from
mathematics. Your teachers have been taught by them or their pupils.
"Potentially infinite" doesn't belong in mathematics. It's not of any
use, and causes only obfuscation and confusion, not illumination.
What everybody else refers to as infinte, you seem to want to call
"potentially infinite".
The potential infinite is a variable finite. Cantor's actual infinity
is not variable but fixed. (Therefore Hilbert's hotel is potential
infinity.)
Hilbert's hotel is infinite, not "variably finite".
WM <wolfgang.mueckenheim@tha.de> wrote:
On 26.03.2025 21:06, joes wrote:
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:
The potential infinite is a variable finite. Cantor's actual infinity is >>>> not variable but fixed. (Therefore Hilbert's hotel is potentialWhat we refer to as infinite isn't variable.
infinity.)
The number of guests/rooms in Hilbert's hotel is infinite but can grow.
That is variable infinity.
You are mistaken. (Countable) infinity stays the same when you add
finite and countably infinite numbers to it.
Thus in Hilbert's hotel,
although all the rooms are occupied, one of these rooms can be vacated
to make room for a new guest without expelling an existing guest.
Adding that new guest doesn't change the number of guests in the hotel,
or the number of rooms required.
See many of Jim's posts over the last
few days for details.
Moebius <invalid@example.invalid> wrote:
Am 27.03.2025 um 12:18 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
Crank talk.
You don't understand limits, as I've already said.
Have you really got a degree in mathematics?
It seems unlikely.
WM doesn't have a degree in mathematics.
He denied not having one.
Most of the time,
Most of the time,
people don't outright lie on Usenet.
They twist and turn,
answering direct questiongs evasively and inadequately,
and assert half truths.
My working theory at the moment is that
WM has some university degree which had
a small component of mathematics, but
that part of his degree didn't cover
rigorous analysis or rigorous set theory,
or anything much else rigorous.
His outpourings on this newsgroup
(as well as on its German counterpart)
pretty much rule out systematic study of
the foundations of mathematics.
Am 27.03.2025 um 15:59 schrieb joes:*Which is already there*. We are not building new rooms, we are
Am Wed, 26 Mar 2025 21:28:52 +0100 schrieb WM:One more guest requires one more room.
On 26.03.2025 21:06, joes wrote:The number of rooms is fixed,
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:The number of guests/rooms in Hilbert's hotel is infinite but can
The potential infinite is a variable finite. Cantor's actualWhat we refer to as infinite isn't variable.
infinity is not variable but fixed. (Therefore Hilbert's hotel is
potential infinity.)
grow.
That is variable infinity.
Do you believe Hilbert's Hotel cannot accomodate *any* new guests?otherwise it weren't interesting.It is interestimng only insofar as it shows that adherents of set theory
are ready to believe every shit.
Am 27.03.2025 um 12:18 schrieb Alan Mackenzie:That is all that mathematicians talk about.
WM <wolfgang.mueckenheim@tha.de> wrote:
Not at all! Cardinality concerns only definable elements.Substance is simply the elements of the set. The amount of substanceYou seem to mean the cardinality of the set.
is the number of elements.
Like I was saying, they are infinite.It is undefined *as a finite number*. But it is defined by |N|/(k-1) >This number exists also for actually infinite sets but cannot be|N|/k is undefined.
expressed by natural numbers.
We only know that ∀k,n ∈ ℕ_def: |ℕ|/k > n.
|N|/k > |N|/(k+1).
Again, thanks for confirming your "dark numbers" are infinite (or infinitesimal). They are definitely not reals, and we are notFor finite sets cardinality is a meaningful notion expressing the numberYou mean it has a larger cardinality.You're saying that the "substance" isn't a property of a set as such,The relative amount of substance can be determined. The set {1, 2, 3}
it's a property of a relationship between a superset and a subset.
has more substance than the set {father, mother}.
of elements. But I dislike it because it is often extended to cover
infinite sets where it does not describe the number of elements.
Ideas surpassing your knowledge by far.
Crank talk.For example, to get the "substance" of N with respect to Q, you couldNearly. It is smaller than any definable fraction.
embed it in the superset Q: You'd get something like: {0, 1, 1/2, 2,
1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5, ....}. Then this "substance" would
come out as zero.
No, to be imprecise. Their numbers of elements differ but theirOr their cardinality, to be even more precise.So, to come back to my original example, the "substance" of {0, 4, 8,Yes.
12, 16, ...} wrt N is 1/4.
The substance of {1, 3, 5, 7, 9, ...} wrt {0, 1/2, 1, 3/2, 2, 5/2, 3,Yes.
....} is also 1/4.
Their "subtances" are thus the same.Yes. Their amounts of substance, to be precise.
cardinality is the same.
It is correct and shows that cardinality is nonsense.If you consider it with cool blood, then you will recognize that allThat's pure baloney. Every element of a bijection is an ordered pair
pairs of a bijection with ℕ are defined within a finite initial
segment [0, n]. That is true for every n. But the infinity lies in the
successors which are undefined.
of an element of set 1 and an element of set 2. Each element of these
sets occurs in exactly one ordered pair. There is no need to obfuscate
this definition with considerations of finite initial segments or
infinity or what have you.
Study surreal and hyperreal numbers which appear even in modern
More crank talk. Ordinary arithmetic is not defined on infinity. And0/oo = 0. 1/oo is smaller than every definable fraction.Tend to yes, but not reaching it.I thought you just said you had a degree in maths. But you don't
seem to understand the process of limits (a bit like John Gabriel
didn't when he was still around).
"smaller than every definable fraction" is zero.
mathematics.
Since when do you know about them? (That's rhetorical btw.)Look, you don't know much. Not even surreal and hyperreal numbers. WhyThe Bourbakis have tried to exorcize the potential infinite from"Potentially infinite" doesn't belong in mathematics. It's not of any
mathematics. Your teachers have been taught by them or their pupils.
use, and causes only obfuscation and confusion, not illumination.
should I take your word on other topics as fact?
It doesn't.Then its number of guests and of rooms could not change.Hilbert's hotel is infinite, not "variably finite".What everybody else refers to as infinte, you seem to want to callThe potential infinite is a variable finite. Cantor's actual infinity
"potentially infinite".
is not variable but fixed. (Therefore Hilbert's hotel is potential
infinity.)
Am 26.03.2025 um 23:18 schrieb Moebius:Depends on how you look at it. The hotel is always "full" in that all
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
And the number of guests?No, it doesn't.The number of [...] rooms in Hilbert's hotel is infinite but [...]
grow[s].
Real fools like really counterintuitive "results".The whole point of Hilbert's Hotel is illustrating the counterintuitivity
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:It's just counterintuitive, which is excusable, as opposed to your
WM <wolfgang.mueckenheim@tha.de> wrote:That proves that cardinality is nonsense. When a new guest arrives, then
On 26.03.2025 21:06, joes wrote:You are mistaken. (Countable) infinity stays the same when you add
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:
The number of guests/rooms in Hilbert's hotel is infinite but canThe potential infinite is a variable finite. Cantor's actualWhat we refer to as infinite isn't variable.
infinity is not variable but fixed. (Therefore Hilbert's hotel is
potential infinity.)
grow. That is variable infinity.
finite and countably infinite numbers to it.
the number of guests grows by 1.
Natural fools hold their intuition above all.Thus in Hilbert's hotel,Real fools are really delighted by counterintuitive results.
although all the rooms are occupied, one of these rooms can be vacated
to make room for a new guest without expelling an existing guest.
That's what makes math fun.Adding that new guest doesn't change the number of guests in the hotel,Real fools are really delighted by counterintuitive results.
or the number of rooms required.
That's a misrepresentation.See many of Jim's posts over the last few days for details.There is only one important detail, namely that lossless exchanges cause losses. It is sufficient to reject every intelligent being.
Moebius <invalid@example.invalid> wrote:
WM doesn't have a degree in mathematics.
He denied not having one. Most of the time, people don't outright lie
on Usenet. They twist and turn, answering direct questiongs evasively
and inadequately, and assert half truths.
My working theory at the moment is that WM has some university degree
which had a small component of mathematics, but that part of his degree didn't cover rigorous analysis or rigorous set theory, or anything much
else rigorous.
Am 27.03.2025 um 15:59 schrieb joes:
The number of rooms is fixed,
One more guest requires one more room.
For finite sets cardinality is a meaningful notion expressing the numberFor infinite sets it is a "generalisation" of the notion "number of
of elements.
Am 26.03.2025 um 23:18 schrieb Moebius:
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
The number of [...] rooms in Hilbert's hotel is infinite but [...]
grow[s].
No, it doesn't.
And the number of guests?
Am 27.03.2025 um 21:11 schrieb WM:
Am 26.03.2025 um 23:18 schrieb Moebius:
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
The number of [...] rooms in Hilbert's hotel is infinite but [...]
grow[s].
No, it doesn't.
And the number of guests?
Is still the same: aleph_0.
It's just the "reality" (->Cantor) of guests that has grown by one. :-)
Moebius) than {g_1, g_2, g_3, ...}, though both sets have the same cardinality (i.e. cardinal number).
Am 27.03.2025 um 21:11 schrieb WM:
Am 26.03.2025 um 23:18 schrieb Moebius:
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
The number of [...] rooms in Hilbert's hotel is infinite but [...]
grow[s].
No, it doesn't.
And the number of guests?
Is still the same: aleph_0.
It's just the "reality" (->Cantor) of guests that has grown by one. :-)
Am 28.03.2025 um 02:02 schrieb Moebius:
Am 27.03.2025 um 21:11 schrieb WM:Set theory cannot describe reality. :-(
Am 26.03.2025 um 23:18 schrieb Moebius:Is still the same: aleph_0.
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:And the number of guests?
WM <wolfgang.mueckenheim@tha.de> wrote:
No, it doesn't.The number of [...] rooms in Hilbert's hotel is infinite but [...] >>>>>> grow[s].
It's just the "reality" (->Cantor) of guests that has grown by one. :-)
Mathematics is expected to be able to describe reality. :-)
On 3/27/2025 1:50 PM, Alan Mackenzie wrote:
Moebius <invalid@example.invalid> wrote:
Am 27.03.2025 um 12:18 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
Crank talk.
You don't understand limits, as I've already said.
Have you really got a degree in mathematics?
It seems unlikely.
WM doesn't have a degree in mathematics.
He denied not having one.
Most of the time,
Hmmm.
Most of the time,
people don't outright lie on Usenet.
They twist and turn,
answering direct questiongs evasively and inadequately,
and assert half truths.
My working theory at the moment is that
WM has some university degree which had
a small component of mathematics, but
that part of his degree didn't cover
rigorous analysis or rigorous set theory,
or anything much else rigorous.
https://de.wikipedia.org/wiki/Wolfgang_M%C3%BCckenheim
[Google->Eng]
⎛ From 1973 to 1977, Mückenheim studied physics with
⎜ minors in mathematics, astronomy and chemistry ...
⎜
⎜ In 1979, he obtained his doctorate in physics with
⎜ a dissertation on elastic photon scattering on
⎝ the ¹²²uranium, ...
⎛ For the record, my (JB's) own formal education ends with
⎜ a bachelor's degree in physics and
⎝ minors in mathematics and computer science.
For what it's worth,
on paper, Mückenheim is more qualified than I am.
I suspect that that irks him to no end.
His outpourings on this newsgroup (as well as on its German
counterpart) pretty much rule out systematic study of the foundations
of mathematics.
I am honestly mystified by
what I need to explain to Mückenheim,
someone who apparently has taken
upper.level courses in mathematics.
Am I unusual in having had to prove things
in such courses, for homework and for tests?
But I have a theory.
There once was a poster of the crankish persuasion,
convinced that he was going to prove
Fermat's Last Theorem at a highschool level of difficulty.
He seemed to know very little mathematics.
I had some reason to look back over
his much.earlier posts, and
I was surprised to find that
that, much _earlier_ he apparently knew _more_
than the same (?) poster later, _after_ years of
arguing mathematics with mathematicians.
My theory is that
trying to defend bullshit rots the brain.
Set theory cannot describe reality. :-(
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
On 26.03.2025 21:06, joes wrote:
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:
The potential infinite is a variable finite. Cantor's actual infinity is >>>>> not variable but fixed. (Therefore Hilbert's hotel is potentialWhat we refer to as infinite isn't variable.
infinity.)
The number of guests/rooms in Hilbert's hotel is infinite but can
grow. That is variable infinity.
You are mistaken. (Countable) infinity stays the same when you add
finite and countably infinite numbers to it.
That proves that cardinality is nonsense. When a new guest arrives,
then the number of guests grows by 1.
Thus in Hilbert's hotel, although all the rooms are occupied, one of
these rooms can be vacated to make room for a new guest without
expelling an existing guest.
Real fools are really delighted by counterintuitive results.
Adding that new guest doesn't change the number of guests in the
hotel, or the number of rooms required.
Real fools are really delighted by counterintuitive results.
See many of Jim's posts over the last few days for details.
There is only one important detail, namely that lossless exchanges
cause losses. It is sufficient to reject every intelligent being.
Regards, WM
Am 27.03.2025 um 12:18 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
For finite sets cardinality is a meaningful notion expressing the number
of elements. But I dislike it because it is often extended to cover
infinite sets where it does not describe the number of elements.
So, to come back to my original example, the "substance" of {0, 4, 8,
12, 16, ...} wrt N is 1/4.
Yes.
The substance of {1, 3, 5, 7, 9, ...} wrt {0, 1/2, 1, 3/2, 2, 5/2, 3,
....} is also 1/4.
Yes.
Their "subtances" are thus the same.
Yes. Their amounts of substance, to be precise.
Or their cardinality, to be even more precise.
No, to be imprecise. Their numbers of elements differ but their
cardinality is the same.
Tend to yes, but not reaching it.
I thought you just said you had a degree in maths. But you don't seem to >>>> understand the process of limits (a bit like John Gabriel didn't when he >>>> was still around).
0/oo = 0. 1/oo is smaller than every definable fraction.
More crank talk. Ordinary arithmetic is not defined on infinity. And
"smaller than every definable fraction" is zero.
Study surreal and hyperreal numbers which appear even in modern
mathematics.
Look, you don't know much. Not even surreal and hyperreal numbers. Why
should I take your word on other topics as fact?>
The potential infinite is a variable finite. Cantor's actual infinity
is not variable but fixed. (Therefore Hilbert's hotel is potential
infinity.)
Hilbert's hotel is infinite, not "variably finite".
Then its number of guests and of rooms could not change.
Regards, WM
Am 28.03.2025 um 02:02 schrieb Moebius:
Am 27.03.2025 um 21:11 schrieb WM:
Am 26.03.2025 um 23:18 schrieb Moebius:
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
The number of [...] rooms in Hilbert's hotel
is infinite but [...] grow[s].
No, it doesn't.
And the number of guests?
Is still the same: aleph_0.
It's just the "reality" (->Cantor) of guests
that has grown by one. :-)
Set theory cannot describe reality. :-(
Mathematics is expected to
be able to describe reality. :-)
My theory is that
trying to defend bullshit rots the brain.
Am Thu, 27 Mar 2025 20:31:16 +0100 schrieb WM:
Am 27.03.2025 um 15:59 schrieb joes:*Which is already there*. We are not building new rooms, we are
Am Wed, 26 Mar 2025 21:28:52 +0100 schrieb WM:One more guest requires one more room.
On 26.03.2025 21:06, joes wrote:The number of rooms is fixed,
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:The number of guests/rooms in Hilbert's hotel is infinite but can
The potential infinite is a variable finite. Cantor's actualWhat we refer to as infinite isn't variable.
infinity is not variable but fixed. (Therefore Hilbert's hotel is
potential infinity.)
grow.
That is variable infinity.
moving the guests.
Do you believe Hilbert's Hotel cannot accomodate *any* new guests?otherwise it weren't interesting.It is interesting only insofar as it shows that adherents of set theory
are ready to believe every shit.
Am Thu, 27 Mar 2025 21:11:00 +0100 schrieb WM:
Am 26.03.2025 um 23:18 schrieb Moebius:Depends on how you look at it. The hotel is always "full" in that all
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:And the number of guests?
WM <wolfgang.mueckenheim@tha.de> wrote:
No, it doesn't.The number of [...] rooms in Hilbert's hotel is infinite but [...]
grow[s].
rooms are occupied; yet it can always accomodate more guests (but not uncountably many). There are "exactly" as many rooms as guests at any
point, so far as that makes sense for infinities.
Real fools like really counterintuitive "results".The whole point of Hilbert's Hotel is illustrating the counterintuitivity
of infinite cardinalities.
Am 27.03.2025 um 20:31 schrieb WM:
Am 27.03.2025 um 15:59 schrieb joes:
The number of rooms is fixed,
One more guest requires one more room.
Nope.
Am 27.03.2025 um 21:07 schrieb WM:
For finite sets cardinality is a meaningful notion expressing theFor infinite sets it is a "generalisation" of the notion "number of elements".
number of elements.
Hint: "In mathematics, the cardinality of a set is the number of its elements.
It's just the "reality" (->Cantor) of guests that has grown by one. :-)
Am 28.03.2025 um 02:02 schrieb Moebius:
Am 27.03.2025 um 21:11 schrieb WM:
Am 26.03.2025 um 23:18 schrieb Moebius:
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
The number of [...] rooms in Hilbert's hotel is infinite but [...] >>>>>> grow[s].
No, it doesn't.
And the number of guests?
Is still the same: aleph_0.
It's just the "reality" (->Cantor) of guests that has grown by one. :-)
In other words, {g_0, g_1, g_2, g_3, ...} has more "substance"
Moebius) than {g_1, g_2, g_3, ...}, though both sets have the same cardinality (i.e. cardinal number).
Am Fri, 28 Mar 2025 08:34:15 +0100 schrieb WM:
Am 28.03.2025 um 02:02 schrieb Moebius:
Am 27.03.2025 um 21:11 schrieb WM:Set theory cannot describe reality. :-(
Am 26.03.2025 um 23:18 schrieb Moebius:Is still the same: aleph_0.
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:And the number of guests?
WM <wolfgang.mueckenheim@tha.de> wrote:
No, it doesn't.The number of [...] rooms in Hilbert's hotel is infinite but [...] >>>>>>> grow[s].
It's just the "reality" (->Cantor) of guests that has grown by one. :-)
Mathematics is expected to be able to describe reality. :-)
Now you're just confusing the technical and colloquial meaning. You
should have stuck with "substance", but then you could simply talk
about supersets. That's what Cantor referred to off-handedly as
"Realität". No such term is used in mathematics, and still it does
describe the world remarkably well
- not that infinite sets
physically exist.
https://de.wikipedia.org/wiki/Wolfgang_M%C3%BCckenheim
[Google->Eng]
⎛ From 1973 to 1977, Mückenheim studied physics with
⎜ minors in mathematics, astronomy and chemistry ...
⎜
⎜ In 1979, he obtained his doctorate in physics with
⎜ a dissertation on elastic photon scattering on
⎝ the ¹²²uranium, ...
Am 28.03.2025 um 08:34 schrieb WM:
Set theory cannot describe reality. :-(
If you say so. :-)
On the other hand, it suffices that it can be used for _mathematics_. :-)
WM <invalid@no.org> wrote:
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
On 26.03.2025 21:06, joes wrote:
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:
The potential infinite is a variable finite. Cantor's actual infinity is >>>>>> not variable but fixed. (Therefore Hilbert's hotel is potentialWhat we refer to as infinite isn't variable.
infinity.)
The number of guests/rooms in Hilbert's hotel is infinite but can
grow. That is variable infinity.
You are mistaken. (Countable) infinity stays the same when you add
finite and countably infinite numbers to it.
That proves that cardinality is nonsense. When a new guest arrives,
then the number of guests grows by 1.
Yes indeed. There were aleph-0 guests beforehand, the new guest arrives growing that number by 1, giving aleph-0. Why do you find this so
difficult to understand?
There is only one important detail, namely that lossless exchanges
cause losses. It is sufficient to reject every intelligent being.
An infinite process of lossless exchanges can cause loss, as we have
seen.
So tell us, O wise one, how many elements are there in {1, 3, 5, 7, 9,
...}? And how many elements in {0, 4, 8, 12, 16, ...}? Which of these
two numbers is bigger, and why?
There are never empty rooms, all rooms are
always occupied,
that's kinda the point. And yet it can accommodate
a countable number of guests, unlike if it were finite.
On 28.03.2025 17:32, Alan Mackenzie wrote:
So tell us, O wise one, how many elements are there in {1, 3, 5, 7, 9,
...}? And how many elements in {0, 4, 8, 12, 16, ...}? Which of these
two numbers is bigger, and why?
|ℕ|/2 > |ℕ|/4.
Regards, WM
WM <wolfgang.mueckenheim@tha.de> wrote:
On 28.03.2025 17:32, Alan Mackenzie wrote:
So tell us, O wise one, how many elements are there in {1, 3, 5, 7, 9,
...}? And how many elements in {0, 4, 8, 12, 16, ...}? Which of these
two numbers is bigger, and why?
|ℕ|/2 > |ℕ|/4.
Start out with a set of natural numbers. Multiply each member by four, giving a new set. You'd have us believe that the new set contains fewer elements than the original set. Baloney.
Regards, WM
On 28.03.2025 16:45, Alan Mackenzie wrote:
WM <invalid@no.org> wrote:
Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
On 26.03.2025 21:06, joes wrote:
Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:
The potential infinite is a variable finite. Cantor's actualWhat we refer to as infinite isn't variable.
infinity is not variable but fixed. (Therefore Hilbert's hotel
is potential infinity.)
The number of guests/rooms in Hilbert's hotel is infinite but can
grow. That is variable infinity.
You are mistaken. (Countable) infinity stays the same when you add
finite and countably infinite numbers to it.
That proves that cardinality is nonsense. When a new guest arrives,
then the number of guests grows by 1.
Yes indeed. There were aleph-0 guests beforehand, the new guest arrives
growing that number by 1, giving aleph-0. Why do you find this so
difficult to understand?
It is not at all difficult to understand. Difficult to understand is
only why cardinality is used at all.
It is worthless because it cannot describe changes of substance. If
there are |ℕ| natural numbers, then there are |ℕ|^2 positive fractions.
The cardinality is the same because it counts only the first elements.
Potential infinity. Otherwise it could not overlook the big difference.
There is only one important detail, namely that lossless exchanges
cause losses. It is sufficient to reject every intelligent being.
An infinite process of lossless exchanges can cause loss, as we have
seen.
No. You have not seen it.
You are mistaken and try to maintain your mistakes by "limits" which
are not used in Cantor's theory:
"so daß jedes Element der Menge an einer bestimmten Stelle dieser Reihe steht" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts", Springer, Berlin (1932) S.
152]
Regards, WM
WM <wolfgang.mueckenheim@tha.de> wrote:
On 28.03.2025 17:32, Alan Mackenzie wrote:
So tell us, O wise one, how many elements are there in {1, 3, 5, 7, 9,
...}? And how many elements in {0, 4, 8, 12, 16, ...}? Which of these
two numbers is bigger, and why?
|ℕ|/2 > |ℕ|/4.
Start out with a set of natural numbers. Multiply each member by four, giving a new set. You'd have us believe that the new set contains fewer elements than the original set.
It is not at all difficult to understand. Difficult to understand is
only why cardinality is used at all.
Those two sentences contradict eachother. Cardinality is used because it
is a sensible way of comparing the size of sets.
It is worthless because it cannot describe changes of substance. If
there are |ℕ| natural numbers, then there are |ℕ|^2 positive fractions.
Yes, and aleph_0^2 = aleph_0. There are as many positive fractions as natural numbers.
This was proven by Cantor. That you don't understand
the proof is your problem, not ours.
The cardinality is the same because it counts only the first elements.
That's a meaningless concatenation of words.
An infinite process of lossless exchanges can cause loss, as we have
seen.
No. You have not seen it.
What makes you think you know what I have and have not seen?
You are mistaken and try to maintain your mistakes by "limits" which
are not used in Cantor's theory:
I'm quite sure Cantor was enirely familiar with the theory of limits.
"so daß jedes Element der Menge an einer bestimmten Stelle dieser Reihe
steht" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts", Springer, Berlin (1932) S.
152]
"So that each element of the set stands at a definite position of this sequence." That has no relevance to anything at issue here. In
particular, it has no relevance to the loss of your favourite set element caused by an infinite sequence of transpositions.
WM <wolfgang.mueckenheim@tha.de> wrote:WM is mixing up things Cantor expressed with his own muddled "thoughts".
"so daß jedes Element der Menge an einer bestimmten Stelle dieser Reihe
steht" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts", Springer, Berlin (1932) S.
152]
"So that each element of the set stands at a definite position of this sequence." That has no relevance to anything at issue here. [...]
WM <wolfgang.mueckenheim@tha.de> wrote:
"so daß jedes Element der Menge an einer bestimmten Stelle dieser Reihe
steht" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts", Springer, Berlin (1932) S.
152]
"So that each element of the set stands at a definite position of this sequence." That has no relevance to anything at issue here. [...]
Am 03.04.2025 um 21:10 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
"so daß jedes Element der Menge an einer bestimmten Stelle dieser Reihe >>> steht" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts", Springer, Berlin (1932) S.
152]
"So that each element of the set stands at a definite position of this
sequence." That has no relevance to anything at issue here. [...]
WM is mixing up things Cantor expressed with his (WM's) own muddled "thoughts". In short: He's quite confused.
Am 03.04.2025 um 21:10 schrieb Alan Mackenzie:
WM <wolfgang.mueckenheim@tha.de> wrote:
"so daß jedes Element der Menge an einer bestimmten Stelle dieser Reihe >>> steht" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts", Springer, Berlin (1932) S.
152]
"So that each element of the set stands at a definite position of this
sequence." That has no relevance to anything at issue here. [...]
WM is mixing up things Cantor expressed
On 03.04.2025 21:10, Alan Mackenzie wrote:
It is not at all difficult to understand. Difficult to understand is
only why cardinality is used at all.
Those two sentences contradict eachother. Cardinality is used because it
is a sensible way of comparing the size of sets.
No.
It is worthless because it cannot describe changes of substance. If
there are |ℕ| natural numbers, then there are |ℕ|^2 positive fractions.
Yes, and aleph_0^2 = aleph_0. There are as many positive fractions as
natural numbers.
This is easily contradicted by observing that 1/2 is not a natural
number while all natural numbers are fractions.
This was proven by Cantor. That you don't understand the proof is
your problem, not ours.
I understand that you are duped. And I have explained why. Every pair of
the bijection has almost all elements as successors.
The cardinality is the same because it counts only the first elements.
That's a meaningless concatenation of words.
It is a pity that you can't understand.
Every natural number that you can use in a bijection has finitely many predecessors but infinitely many successors which will never be used.
An infinite process of lossless exchanges can cause loss, as we have
seen.
No. You have not seen it.
What makes you think you know what I have and have not seen?
I know your mistakes and their origin.
You are mistaken and try to maintain your mistakes by "limits" which
are not used in Cantor's theory:
I'm quite sure Cantor was enirely familiar with the theory of limits.
But he did not use them in his bijections.
"so daß jedes Element der Menge an einer bestimmten Stelle dieser Reihe >>> steht" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts", Springer, Berlin (1932) S.
152]
"So that each element of the set stands at a definite position of this
sequence." That has no relevance to anything at issue here. In
particular, it has no relevance to the loss of your favourite set element
caused by an infinite sequence of transpositions.
Just this is excluded. Only definite positions are admitted. No evasion
into the infinite.
Regards, WM
WM <wolfgang.mueckenheim@tha.de> wrote:
On 03.04.2025 21:10, Alan Mackenzie wrote:
It is not at all difficult to understand. Difficult to understand is
only why cardinality is used at all.
Those two sentences contradict eachother. Cardinality is used because it >>> is a sensible way of comparing the size of sets.
No.
You're wrong.
This is easily contradicted by observing that 1/2 is not a natural
number while all natural numbers are fractions.
It is not contradicted. There is a 1-1 correspondence between positive fractions and natural numbers.
I understand that you are duped. And I have explained why. Every pair of
the bijection has almost all elements as successors.
Eh?? What the heck are you going on about? Hint: a bijection is a set
of pairs. It is not ordered.
Every natural number that you can use in a bijection has finitely many
predecessors but infinitely many successors which will never be used.
Wrong.
Just this is excluded. Only definite positions are admitted. No evasion
into the infinite.
You mean, you're only allowing a finite number of transpositions?
that case, the distinguished element indeed does not disappear.
Regards, WM
WM <wolfgang.mueckenheim@tha.de> wrote:No, it's not. I mean, except for Mr. Mückenheim.
On 03.04.2025 21:10, Alan Mackenzie wrote:
Difficult to understand is only why cardinality is used at all. [WM]
Cardinality is used because it is a sensible way of comparing the size of sets.
No. [WM]
You're wrong.
Every pair of the bijection has almost all elements as successors. [WM]
Eh?? What the heck are you going on about? Hint: a bijection is a set
of pairs. It is not ordered.
Every natural number that you can use in a bijection has finitely many
predecessors but infinitely many successors which will never be used.
Wrong. Every natural number in a bijection of N with some set gets
"used", by definition of bijection.
WM <wolfgang.mueckenheim@tha.de> wrote:No, it's not. I mean, except for Mr. Mückenheim.
On 03.04.2025 21:10, Alan Mackenzie wrote:
Difficult to understand is only why cardinality is used at all. [WM]
Cardinality is used because it is a sensible way of comparing the size of sets.
No. [WM]
You're wrong.
Every pair of the bijection has almost all elements as successors. [WM]
Eh?? What the heck are you going on about? Hint: a bijection is a set
of pairs. It is not ordered.
Every natural number that you can use in a bijection has finitely many
predecessors but infinitely many successors which will never be used.
Wrong. Every natural number in a bijection of N with some set gets
"used", by definition of bijection.
WM wrote on 4/4/2025 :
You are caught in a world of stupidity. Set theorists have damaged the
honour of human intellect even more than Pope Pius XII.
When an element is added to a set, then this set is no longer the same
but different because the number of its members is different.
How do you know that they are not 'the same'?
No bijection perhaps?
WM was thinking very hard :
On 04.04.2025 23:12, FromTheRafters wrote:
WM wrote on 4/4/2025 :
You are caught in a world of stupidity. Set theorists have damaged
the honour of human intellect even more than Pope Pius XII.
When an element is added to a set, then this set is no longer the
same but different because the number of its members is different.
How do you know that they are not 'the same'?
A set containing element x is not the same as a set not containing
element x.
Then of course they are not "the same" but are they equivalent in size?
On 03.04.2025 21:10, Alan Mackenzie wrote:No, that is a subset relation („reality”). The identity function is obviously not a bijection from N to Q.
This is easily contradicted by observing that 1/2 is not a naturalIt is not at all difficult to understand. Difficult to understand isThose two sentences contradict eachother. Cardinality is used because
only why cardinality is used at all.
it is a sensible way of comparing the size of sets.
It is worthless because it cannot describe changes of substance. IfYes, and aleph_0^2 = aleph_0. There are as many positive fractions as
there are |ℕ| natural numbers, then there are |ℕ|^2 positive
fractions.
natural numbers.
number while all natural numbers are fractions.
Bijections aren’t ordered.This was proven by Cantor. That you don't understand the proof is yourI understand that you are duped. And I have explained why. Every pair of
problem, not ours.
the bijection has almost all elements as successors.
No, a bijection can be an infinite set of pairs - must be, if the setsIt is a pity that you can't understand. Every natural number that youThe cardinality is the same because it counts only the first elements.That's a meaningless concatenation of words.
can use in a bijection has finitely many predecessors but infinitely
many successors which will never be used.
Then you cannot talk about the limit."So that each element of the set stands at a definite position of thisJust this is excluded. Only definite positions are admitted. No evasion
sequence." That has no relevance to anything at issue here. In
particular, it has no relevance to the loss of your favourite set
element caused by an infinite sequence of transpositions.
into the infinite.
On 03.04.2025 16:40, Alan Mackenzie wrote:How else could it be.
WM <wolfgang.mueckenheim@tha.de> wrote:Fact. Ifff the natural numbers are an actually infinite set, then its elements are invariable and fixed.
On 28.03.2025 17:32, Alan Mackenzie wrote:Start out with a set of natural numbers. Multiply each member by four,
So tell us, O wise one, how many elements are there in {1, 3, 5, 7,|ℕ|/2 > |ℕ|/4.
9, ...}? And how many elements in {0, 4, 8, 12, 16, ...}? Which of
these two numbers is bigger, and why?
giving a new set. You'd have us believe that the new set contains
fewer elements than the original set.
By multiplication no larger numbersWhat do „fits” and „space” mean?
can be created. What you have in mind is a potentially infinite set.
Let me explain in detail:
Cantor created the sequence of the ordinal numbers by means of his first
and second generation principle:
0, 1, 2, 3, ..., ω, ω+1, ω+2, ω+3, ..., ω*2, ω*2+1, ω*2+2, ω*2+3, ..., ω*3, ... .
This sequence, except its very first terms, has no relevance for
classical mathematics. But it is important for set theory that in actual infinity nothing fits between ℕ and ω. Likewise before ω*2 and ω*3
there is no empty space.
According to Hilbert we can simply count beyond theIn which sense? There are infinitely many of either.
infinite by a quite natural and uniquely determined, consistent
continuation of the ordinary counting in the finite. But we would
proceed even faster, when instead of counting, we doubled the numbers.
This leads to the central issue: Multiply every element of the set ℕ by
2
{1, 2, 3, ...}*2 = {2, 4, 6, ...} .
The density of the natural numbers on the real axis is greater than the density of the even natural numbers.
Therefore the doubled natural numbers cover twice as many space thanAgain, how? How much?
before.
What is the result ofWhy? Every even number has a natural half its value.
doubling? Either all doubled numbers are natural numbers, then not all natural numbers have been doubled.
Natural numbers not available beforeNo, there is no natural number whose double is larger than ω.
have been created. This is possible only based on potential infinity. Or
all natural numbers have been doubled, then the result stretches
farther, namely beyond all natural numbers.
It is more suggestive to double the set ℕ U {ω} = {1, 2, 3, ..., ω}Where is ω?
with the result
{1, 2, 3, ..., ω}*2 = {2, 4, 6, ..., ω*2} .
What elements fall between ω and ω*2?
What size has the interval between 2ℕ and ω*2?What is the interval between two sets?
The natural answer is (0, ω]*2 = (0, ω*2] with ω or ω+1No, that would be two consecutive infinities. You can do that, but not
amidst.
The number of doubled natural numbers is precisely |ℕ|. But halfYes they do. The product of two naturals is also a natural.
of the doubled numbers are no longer natural numbers; they surpass ω. If
all natural numbers including all even numbers are doubled and if
doubling increases the value for all natural numbers because n < 2n,
then not all doubled even numbers fit below ω.
Natural numbers greaterThe „structure” (order type) of N is ω and not ω*2.
than all even natural numbers however are not possible.
Every other result would violate symmetry and beauty of mathematics, for instance the claim that the result would be ℕ U {ω, ω*2}. All numbers between ω and ω*2, which are precisely as many as in ℕ between 0 and ω, would not be in the result? Every structure must be doubled!
Like theHow do you define subtraction for ordinals?
interval [1, 5] of lengths 4 by doubling gets [1, 5]*2 = [2, 10] of
length 8, the interval (0, ω]*2 gets (0, ω*2] with ω*2 = ω + ω =/= ω where the whole interval between 0 and ω*2 is evenly filled with even numbers like the whole interval between 0 and ω is evenly filled with natural numbers before multiplication. On the ordinal axis the numbers
0, ω, ω*2, ω*3, ... have same distances because same number of ordinals lie between them.
This means that contrary to the collection of visibleNo, the union of the set called N and those numbers you have added is not closed. That’s your problem, not that of mathematics.
natural numbers ℕ_def which only are relevant in classical mathematics
the whole set ℕ is not closed under multiplication. Some natural numbers can become transfinite by multiplication.
This resembles the displacement of the interval (0, 1] by one point toWhat is „one point”? Can you give an explicit function?
the left-hand side such that the interval [0, 1) is covered.
Am Thu, 03 Apr 2025 22:28:44 +0200 schrieb WM:
On 03.04.2025 21:10, Alan Mackenzie wrote:
No,Yes, and aleph_0^2 = aleph_0. There are as many positive fractions asThis is easily contradicted by observing that 1/2 is not a natural
natural numbers.
number while all natural numbers are fractions.
Bijections aren’t ordered.This was proven by Cantor. That you don't understand the proof is yourI understand that you are duped. And I have explained why. Every pair of
problem, not ours.
the bijection has almost all elements as successors.
No, a bijection can be an infinite set of pairs - must be, if the setsIt is a pity that you can't understand. Every natural number that youThe cardinality is the same because it counts only the first elements.That's a meaningless concatenation of words.
can use in a bijection has finitely many predecessors but infinitely
many successors which will never be used.
are infinite.
Then you cannot talk about the limit."So that each element of the set stands at a definite position of thisJust this is excluded. Only definite positions are admitted. No evasion
sequence." That has no relevance to anything at issue here. In
particular, it has no relevance to the loss of your favourite set
element caused by an infinite sequence of transpositions.
into the infinite.
Am Thu, 03 Apr 2025 22:17:33 +0200 schrieb WM:
On 03.04.2025 16:40, Alan Mackenzie wrote:
Ifff the natural numbers are an actually infinite set, then itsHow else could it be.
elements are invariable and fixed.
The density of the natural numbers on the real axis is greater than theIn which sense? There are infinitely many of either.
density of the even natural numbers.
Therefore the doubled natural numbers cover twice as many space thanAgain, how? How much?
before.
Natural numbers not available beforeNo, there is no natural number whose double is larger than ω.
have been created. This is possible only based on potential infinity. Or
all natural numbers have been doubled, then the result stretches
farther, namely beyond all natural numbers.
It is more suggestive to double the set ℕ U {ω} = {1, 2, 3, ..., ω}Where is ω?
with the result
{1, 2, 3, ..., ω}*2 = {2, 4, 6, ..., ω*2} .
What elements fall between ω and ω*2?
Am Thu, 03 Apr 2025 22:28:44 +0200 schrieb WM:
On 03.04.2025 21:10, Alan Mackenzie wrote:
Bijections aren’t ordered.This was proven by Cantor. That you don't understand the proof is yourI understand that you are duped. And I have explained why. Every pair of
problem, not ours.
the bijection has almost all elements as successors.
Only definite positions are admitted. No evasionThen you cannot talk about the limit.
into the infinite.
on 4/13/2025, WM supposed :
Bijections with ℕ are ordered by the well-ordered set ℕ.
The set N is not ordered, you want omega for the ordered set of naturals.
On 13.04.2025 10:20, joes wrote:The set of naturals does not change; every number either is or is not
Am Thu, 03 Apr 2025 22:17:33 +0200 schrieb WM:
On 03.04.2025 16:40, Alan Mackenzie wrote:
The alternative is potential infinity.Ifff the natural numbers are an actually infinite set, then itsHow else could it be.
elements are invariable and fixed.
Aha! And for the one infinite length we get the undefined expressionFor all infinitely many finite lengths the densities are 2 to 1.The density of the natural numbers on the real axis is greater thanIn which sense? There are infinitely many of either.
the density of the even natural numbers.
So there are infinitely large even naturals?They cover the ordinal axis between 0 and 2ω evenly.Therefore the doubled natural numbers cover twice as many space thanAgain, how? How much?
before.
Just no. It would have to be larger than ω (=infinite) itself. Half an infinity is still infinite.Wrong if infinity is actaul.Natural numbers not available before have been created. This isNo, there is no natural number whose double is larger than ω.
possible only based on potential infinity. Or all natural numbers have
been doubled, then the result stretches farther, namely beyond all
natural numbers.
No, I mean in the second set? ω/2 is not even defined. Or do you thinkOn the ordinal axis immediately after all natural numbers.It is more suggestive to double the set ℕ U {ω} = {1, 2, 3, ..., ω}Where is ω?
with the result
{1, 2, 3, ..., ω}*2 = {2, 4, 6, ..., ω*2} .
What elements fall between ω and ω*2?
On 13.04.2025 10:25, joes wrote:You can do that, but what does it buy you? It would be infinite even
Am Thu, 03 Apr 2025 22:28:44 +0200 schrieb WM:
On 03.04.2025 21:10, Alan Mackenzie wrote:
Bijections with well-ordered sets are well ordered.Bijections aren’t ordered.This was proven by Cantor. That you don't understand the proof isI understand that you are duped. And I have explained why. Every pair
your problem, not ours.
of the bijection has almost all elements as successors.
Dude, you are talking about the result of an infinite process. AreThere is none.Only definite positions are admitted. No evasion into the infinite.Then you cannot talk about the limit.
On 14.04.2025 10:51, joes wrote:You could just say that the bijection is between infinite sets. Then your
Am Sun, 13 Apr 2025 17:37:57 +0200 schrieb WM:Order is necessary to convince intelligent readers that almost all terms
On 13.04.2025 10:25, joes wrote:You can do that, but what does it buy you? It would be infinite even
Am Thu, 03 Apr 2025 22:28:44 +0200 schrieb WM:Bijections with well-ordered sets are well ordered.
On 03.04.2025 21:10, Alan Mackenzie wrote:
Bijections aren’t ordered.This was proven by Cantor. That you don't understand the proof is >>>>>> your problem, not ours.I understand that you are duped. And I have explained why. Every
pair of the bijection has almost all elements as successors.
without an order. It’s not necessary for your argument.
are successors of any defined term.
So? What do you claim?It is claimed by Cantor that all pairs of the bijection exist at defined places without limit.Dude, you are talking about the result of an infinite process. Are youThere is none.Only definite positions are admitted. No evasion into the infinite.Then you cannot talk about the limit.
saying it diverges?
Am Sun, 13 Apr 2025 13:30:54 +0200 schrieb WM:
On 13.04.2025 10:20, joes wrote:The set of naturals does not change; every number either is or is not included in it.
Am Thu, 03 Apr 2025 22:17:33 +0200 schrieb WM:The alternative is potential infinity.
On 03.04.2025 16:40, Alan Mackenzie wrote:
Ifff the natural numbers are an actually infinite set, then itsHow else could it be.
elements are invariable and fixed.
Aha! And for the one infinite length we get the undefined expressionFor all infinitely many finite lengths the densities are 2 to 1.The density of the natural numbers on the real axis is greater thanIn which sense? There are infinitely many of either.
the density of the even natural numbers.
inf/inf.
So there are infinitely large even naturals?They cover the ordinal axis between 0 and 2ω evenly.Therefore the doubled natural numbers cover twice as many space thanAgain, how? How much?
before.
Just no. It would have to be larger than ω (=infinite) itself. Half an infinity is still infinite.Wrong if infinity is actual.Natural numbers not available before have been created. This isNo, there is no natural number whose double is larger than ω.
possible only based on potential infinity. Or all natural numbers have >>>> been doubled, then the result stretches farther, namely beyond all
natural numbers.
No, I mean in the second set? ω/2 is not even defined.On the ordinal axis immediately after all natural numbers.It is more suggestive to double the set ℕ U {ω} = {1, 2, 3, ..., ω} >>>> with the resultWhere is ω?
{1, 2, 3, ..., ω}*2 = {2, 4, 6, ..., ω*2} .
What elements fall between ω and ω*2?
Or do you think
the naturals form consecutive infinities like
1, 2, 3, …, ω/ω, ω/(ω-1), ω/(ω-2), …, ω/3, ω/2, ω ?
Am Sun, 13 Apr 2025 17:37:57 +0200 schrieb WM:
On 13.04.2025 10:25, joes wrote:You can do that, but what does it buy you? It would be infinite even
Am Thu, 03 Apr 2025 22:28:44 +0200 schrieb WM:Bijections with well-ordered sets are well ordered.
On 03.04.2025 21:10, Alan Mackenzie wrote:
Bijections aren’t ordered.This was proven by Cantor. That you don't understand the proof isI understand that you are duped. And I have explained why. Every pair
your problem, not ours.
of the bijection has almost all elements as successors.
without an order. It’s not necessary for your argument.
Dude, you are talking about the result of an infinite process. AreThere is none.Only definite positions are admitted. No evasion into the infinite.Then you cannot talk about the limit.
you saying it diverges?
Am Mon, 14 Apr 2025 13:56:01 +0200 schrieb WM:
On 14.04.2025 10:51, joes wrote:You could just say that the bijection is between infinite sets. Then your claim boils down to „those don’t exist”,
Am Sun, 13 Apr 2025 17:37:57 +0200 schrieb WM:Order is necessary to convince intelligent readers that almost all terms
On 13.04.2025 10:25, joes wrote:You can do that, but what does it buy you? It would be infinite even
Am Thu, 03 Apr 2025 22:28:44 +0200 schrieb WM:Bijections with well-ordered sets are well ordered.
On 03.04.2025 21:10, Alan Mackenzie wrote:
Bijections aren’t ordered.This was proven by Cantor. That you don't understand the proof is >>>>>>> your problem, not ours.I understand that you are duped. And I have explained why. Every
pair of the bijection has almost all elements as successors.
without an order. It’s not necessary for your argument.
are successors of any defined term.
It is claimed by Cantor that all pairs of the bijection exist at definedSo? What do you claim?
places without limit.
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