On 3/4/2024 6:16 PM, Ross Finlayson wrote:
On 03/04/2024 01:20 PM, Jim Burns wrote:
On 3/4/2024 1:52 PM, Ross Finlayson wrote:
On 03/04/2024 10:31 AM, Jim Burns wrote:
On 3/3/2024 6:19 PM, Ross Finlayson wrote:
And Leibniz is like, "thanks, I got this".
Leibniz has this.
We have this.
Nobody enters Cantor's Paradise.
n = 1, 2, 3, ...
0^1/n = 0, 0, 0, ..., 1
What, not first first, not ultimate untrue?
Perhaps you'd like some sort of response to that?
Perhaps you'd be interested to know that
I don't know what you mean by
not.ultimately.untrue.
Here this was
0 ^ 1/1 = 0, not.first.false
0 ^ 1/2 = 0, not.first.false
0 ^ 1/3 = 0, not.first.false
When I started to use "not.first.false"
I intended it as a short, a very.very.short,
explanation why we all should trust
the conclusions of a correct logical argument
no less than we trust its premises.
Our goal is to distinguish true ⊤ claims
from false ⊥ claims about points.in.a.line or
widgets or flying.rainbow.sparkle.ponies or Bob.
Consider a finite sequence of truth.values, ⊤ and ⊥
any finite sequence of truth.values.
⟨ ⊤ … ⊤ ⟩
It's a finite sequence, therefore,
if the value 'false' exists in ⟨ ⊤ … ⊤ ⟩
then the value 'false' exists a first time.
∃⊥ ⇒ ∃₁⊥ in ⟨ ⊤ … ⊤ ⟩
That's what I wanted to say:
∃⊥ ⇒ ∃₁⊥ in ⟨ ⊤ … ⊤ ⟩
We know that's true because
it's true in general of anything
in a finite sequence.
In a finite sequence of playing cards,
If one card is a club
then one of them is the first club.
∃♣ ⇒ ∃₁♣ in ⟨ ♠ … ♥ ⟩
And so on.
∃⊥ ⇒ ∃₁⊥ in ⟨ ⊤ … ⊤ ⟩
¬∃₁⊥ ⇒ ¬∃⊥ in ⟨ ⊤ … ⊤ ⟩
∀¬₁⊥ ⇒ ∀¬⊥ in ⟨ ⊤ … ⊤ ⟩
∀¬₁⊥ ⇒ ∀⊤ in ⟨ ⊤ … ⊤ ⟩
∀⊤ in ⟨ ⊤ … ⊤ ⟩ is our Holy Grail,
wherein all the truth values are ⊤
The lemma ∀¬₁⊥ ⇒ ∀⊤
reduces the problem to finding finite ⟨ ⊤ … ⊤ ⟩
such that
each claim is not.first.false in ⟨ ⊤ … ⊤ ⟩
That is why "not.first.false"
First.false is false, thus
true is not.first.false.
₁⊥ ⇒ ⊥
⊤ ⇒ ¬₁⊥
Some claims in some sequences must be
not.first.false in their sequences.
In ⟨ P P⇒Q Q ⟩ Q is ¬₁⊥
⊥ ⊤ ⊥
⊤ ⊥ ⊥
⊥ ⊤ ⊤
⊤ ⊤ ⊤
In ⟨ P Q P⇒Q ⟩ Q might not be ¬₁⊥
⊥ ⊥ ⊤
⊤ ⊥ ⊥ !
⊥ ⊤ ⊤
⊤ ⊤ ⊤
Our goal, the Holy Grail, is to find/construct
a sequence of claims such that
each claim in that sequence is not.first.false,
and we can see it is, like Q in ⟨ P P⇒Q Q ⟩
or we already know that claim is true, ⊤ ⇒ ¬₁⊥
0 ^ 1/1 = 0, not.first.false
0 ^ 1/2 = 0, not.first.false
0 ^ 1/3 = 0, not.first.false
0 ^ 0 = 1, not.ultimately.untrue
I don't see how 'not.first.false' and
'not.ultimately.untrue' have
anything to do with each other.
The most important use of 'not.first.false'
is in finite sequences of not.first.false claims,
some of which we wouldn't know are true
except for being located in that sequence.
The claim.sequence
⟨ 0¹ᐟ¹=0 0¹ᐟ²=0 0¹ᐟ³=0 ... 0⁰=1 ⟩
isn't a finite sequence,
and
we know those claims for reasons other than
being located in that sequence.
You may be familiar with this as a definition
in fractional powers with respect to zero
the radix and zero the power, just showing that
as a sort of example that = 0 is not.first.false,
but, that not.ultimately.untrue, is different.
Setting aside
'not.first.false' and 'not.ultimately.untrue',
limit(0¹ᐟⁿ) ≠ 0ˡⁱᵐⁱᵗ⁽¹ᐟⁿ⁾
Which is to say 0ˣ is discontinuous at 0
That could be overlooked, because
arithmetic is continuous _almost_ everywhere.
But that is an exception.
In these cases, it's not _jumping_ cases
so much as, _spanning_ cases.
The Intermediate Value Theorem works quite well
when that each of:
extent <- you allow this [0,1], or where LUB = 1
density <- you allow this
completeness <- you don't allow this
The intermediate Value Theorem implies
Dedekind completeness.
Dedekind completeness implies
the Intermediate Value Theorem.
That's why I have been so free in
switching between the two when describing ℝ
measure <- it would so follow
holds up,
That in this case it also exactly is that
dom(EF) is discrete and ran(EF) is continuous,
a continuous domain,
s/continuous/connected
Then, about completeness as above,
"iota-completeness" if you will,
the LUB of a subset of ran(EF) is in ran(EF)
quite trivially, so, that's the usual definition.
That's the usual definition, Jim.
It is inconsistent for positive.iota to exist
which equal.spaces infinitely.many points
from 0 to 1
0 is the greatest.lower.bound of
finite.ordinal.reciprocals
If iota is positive,
a finite.ordinal.reciprocal exists
between 0 and iota,
and
some finite ordinal is larger than
the set of iota.spaced points from 0 to 1
Yet, in "Pre-Calculus", then of course
there was the notion of limit, and
it was about mentioned in passing exactly that
the course-of-passage of numbers zero through one,
"constant monotone strictly increasing",
was just being put aside,
as the later work has to be all stood up,
and that it has its own way,
and it's a pretty good way,
and it's standard,
and it's a linear curriculum,
and it's the way everyone would know.
Limits are poorly characterized as
infinitieth elements of infinite series.
Here is an unnecessary "paradox":
What is limit sin(x)/x as x -> 0 ?
There is no sin(0)/0
There is no sin(0)/0 anywhere,
and it's also not at some putative infinitieth entry.
There is a point 1
1 is near almost all of
⟨ sin(⅟1)/⅟1 sin(⅟2)/⅟2 sin(⅟3)/⅟3 ... ⟩
1 is a synecdoche for
⟨ sin(⅟1)/⅟1 sin(⅟2)/⅟2 sin(⅟3)/⅟3 ... ⟩
1 can't be in
⟨ sin(⅟1)/⅟1 sin(⅟2)/⅟2 sin(⅟3)/⅟3 ... ⟩
but it doesn't need to be.
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