On 22.01.2025 19:01, Python wrote:

If you have three coins of 2 euros not a single one is "necessary" to

pay a 3 euros drink

This failing analogy has been repeated again an again, first by
Rennenkampff, because their authors do not understand the principle:
Cantor's theorem concerns the set of indices or ordinal numbers, not a
set of sets.

Therefore first you have to enumerate the euros or the sets {1, 2}, {2,
3}, {1, 3}. Then Cantor's theorem can be applied. In the given order the
first necessary set is the second one.

Also in the case of coins the first one is not necessary.

Let me know whether you will be able to understand this simple explanation.

Regards, WM

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On 25.01.2025 18:03, FromTheRafters wrote:

WM pretended :

On 22.01.2025 19:01, Python wrote:

If you have three coins of 2 euros not a single one is "necessary" to

pay a 3 euros drink

This failing analogy has been repeated again an again, first by
Rennenkampff, because their authors do not understand the principle:
Cantor's theorem concerns the set of indices or ordinal numbers, not a
set of sets.

Then how are these 'Cantor's Theorem' ordinals contructed?

That can be done in an arbitrary way. For the above sets of euros or for
{a, b}, {b, c}, {c, a} the first necessary is the second. For FISONs
F(n) I use the index n. But we can attach the ordinals in an arbitrary
way. There is no first necessary and no first sufficient either. All
FISONs fail to complete the set ℕ if it is fixed and greater than all
FISONs. The potentially infinite collection UF(n) however is obviously produced.

Regards, WM

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On 26.01.2025 10:08, FromTheRafters wrote:

WM wrote :

On 25.01.2025 18:03, FromTheRafters wrote:

WM pretended :

On 22.01.2025 19:01, Python wrote:

If you have three coins of 2 euros not a single one is

"necessary" to
pay a 3 euros drink

This failing analogy has been repeated again an again, first by
Rennenkampff, because their authors do not understand the principle:
Cantor's theorem concerns the set of indices or ordinal numbers, not
a set of sets.

Then how are these 'Cantor's Theorem' ordinals contructed?

That can be done in an arbitrary way.

Arbitrarily constructed ordinals? Tell me more!

Which of {a, b}, {b, c}, {c, a} are required for the union {a, b, c}?

Indexing: 1. {a, b}, 2. {b, c}, 3. {c, a}.

The first set is not required because

{a, b, c} = {a, b} U {b, c} U {c, a} = {b, c} U {c, a}.

The second set is the first required one because

{a, b, c} = {b, c} U {c, a} =/= {c, a}.

Therefore Cantors's theorem supplies the set of ordinals {2, 3}.

By the way, every other choice of indices would yield the same
Cantor-set {2, 3}.

Regards, WM

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Am Sun, 26 Jan 2025 09:31:45 +0100 schrieb WM:

On 25.01.2025 18:03, FromTheRafters wrote:

WM pretended :

On 22.01.2025 19:01, Python wrote:

If you have three coins of 2 euros not a single one is "necessary"
to pay a 3 euros drink

This failing analogy has been repeated again an again, first by
Rennenkampff, because their authors do not understand the principle:
Cantor's theorem concerns the set of indices or ordinal numbers, not a
set of sets.

Then how are these 'Cantor's Theorem' ordinals contructed?

That can be done in an arbitrary way. For the above sets of euros or for
{a, b}, {b, c}, {c, a} the first necessary is the second.

Only in that order. You only mean that one can drop only one of the sets,
but not two.

For FISONs
F(n) I use the index n. But we can attach the ordinals in an arbitrary
way. There is no first necessary and no first sufficient either. All
FISONs fail to complete the set ℕ if it is fixed and greater than all FISONs.

Every single FISON is not equal to N. But not if you take all of them.

The potentially infinite collection UF(n) however is obviously
produced.

And it is equal to N.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.

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On 26.01.2025 11:47, joes wrote:

Am Sun, 26 Jan 2025 09:31:45 +0100 schrieb WM:

The potentially infinite collection UF(n) however is obviously
produced.

And it is equal to N.

According to set theory it is invariable.

Regards, WM

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On 26.01.2025 14:04, FromTheRafters wrote:

After serious thinking WM wrote :

On 26.01.2025 10:08, FromTheRafters wrote:

WM wrote :

On 25.01.2025 18:03, FromTheRafters wrote:

WM pretended :

On 22.01.2025 19:01, Python wrote:

If you have three coins of 2 euros not a single one is

"necessary" to
pay a 3 euros drink

This failing analogy has been repeated again an again, first by
Rennenkampff, because their authors do not understand the
principle: Cantor's theorem concerns the set of indices or ordinal >>>>>> numbers, not a set of sets.

Then how are these 'Cantor's Theorem' ordinals contructed?

That can be done in an arbitrary way.

Arbitrarily constructed ordinals? Tell me more!

Which of {a, b}, {b, c}, {c, a} are required  for the union {a, b, c}?

Indexing: 1. {a, b}, 2. {b, c}, 3. {c, a}.

The first set is not required because

{a, b, c} = {a, b} U {b, c} U {c, a} = {b, c} U {c, a}.

The second set is the first required one because

{a, b, c} = {b, c} U {c, a} =/= {c, a}.

Therefore Cantors's theorem supplies the set of ordinals {2, 3}.

By the way, every other choice of indices would yield the same
Cantor-set {2, 3}.

I see set manipulations but no ordinals at all, arbitrary or not.

The ordinals of necessary sets are 2 and 3. The first necessary ordinal
is 2.

Regards, WM

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Am Sun, 26 Jan 2025 13:35:40 +0100 schrieb WM:

On 26.01.2025 11:47, joes wrote:

Am Sun, 26 Jan 2025 09:31:45 +0100 schrieb WM:

The potentially infinite collection UF(n) however is obviously
produced.

And it is equal to N.

According to set theory it is invariable.

Yes, it is.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.

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