From Newsgroup: sci.logic

Am 10.06.2026 um 06:44 schrieb Moebius:

Am 03.06.2026 um 23:00 schrieb WM:

All ordinals are said to exist. Therefore somewhat must be below -e.

Yeah, M|+ckenheim: All finite ordinal numbers (i.e. the natural numbers)
are "below -e".

Cantor's enumerations are based on potential infinity.

No, they are not. (Mikko)

Right. No, they are not.

Then naturals and rationals could not seem to be in bijection
Possible bijections of actually infinite equinumerous sets like n <--> n
and n <--> 1/n exclude n <--> q.

Regards, WM

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From Newsgroup: sci.logic

Am 10.06.2026 um 06:41 schrieb Moebius:

Am 03.06.2026 um 22:56 schrieb WM:

Am 03.06.2026 um 11:11 schrieb Mikko:

-e exists as an infinite ordinal number. But there is no infinite ordinal >>> number before it, just like there is no natural number before zero.

In other words:

There are inifinitely many finite ordinal numbers before -e but none of
them is "-e - 1".

What is immediately before -e[?]

Nothing, M|+ckenheim, nothing. At least no ORDINAL NUMBER.

There is no eason for a gsp. THere are dark ordinals - iff omega exists.
Note that also before zero there is no gap.

Regards, WM

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From Newsgroup: sci.logic

On 18/06/2026 23:10, wm wrote:

Am 10.06.2026 um 06:44 schrieb Moebius:

Am 03.06.2026 um 23:00 schrieb WM:

All ordinals are said to exist. Therefore somewhat must be below -e.

Yeah, M|+ckenheim: All finite ordinal numbers (i.e. the natural
numbers) are "below -e".

Cantor's enumerations are based on potential infinity.

No, they are not. (Mikko)

Right. No, they are not.

Then naturals and rationals could not seem to be in bijection

Yet a bijection between them can be shown.
--
Mikko
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From Newsgroup: sci.logic

Am 19.06.2026 um 09:15 schrieb Mikko:

On 18/06/2026 23:10, wm wrote:

Then naturals and rationals could not seem to be in bijection

Yet a bijection between them can be shown.

It seems so because it can be shown for the first few pairs. Infinitely
many naturals and far more rationals remain.

Regards, WM
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From Newsgroup: sci.logic

On 19/06/2026 15:31, WM wrote:

Am 19.06.2026 um 09:15 schrieb Mikko:

On 18/06/2026 23:10, wm wrote:

Then naturals and rationals could not seem to be in bijection

Yet a bijection between them can be shown.

It seems so because it can be shown for the first few pairs.

Every proof that a mapping is a bijection proves about all
members of both sets.
--
Mikko
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From Newsgroup: sci.logic

Am 20.06.2026 um 10:33 schrieb Mikko:

On 19/06/2026 15:31, WM wrote:

Am 19.06.2026 um 09:15 schrieb Mikko:

On 18/06/2026 23:10, wm wrote:

Then naturals and rationals could not seem to be in bijection

Yet a bijection between them can be shown.

It seems so because it can be shown for the first few pairs.

Every proof that a mapping is a bijection proves about all
members of both sets.

But there has never been a proof. Dark numbers cannot satisfy Cantor's
claims like the folowing:

"If we think the numbers p/q in such an order [...] then every number
p/q comes at an absolutely fixed position of a simple infinite sequence"
[E. Zermelo: "Georg Cantor rCo Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 126]

"The infinite sequence thus defined has the peculiar property to contain
the positive rational numbers completely, and each of them only once at
a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

"thus we get the epitome (-e) of all real algebraic numbers [...] and
with respect to this order we can talk about the nth algebraic number
where not a single one of this epitome (w) has been forgotten." [E.
Zermelo: "Georg Cantor rCo Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 116]

"such that every element of the set stands at a definite position of
this sequence" [E. Zermelo: "Georg Cantor rCo Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 152]

The clarity of these expressions is noteworthy: all and every,
completely, at an absolutely fixed position, nth number, where not a
single one has been forgotten.

And that is wrong!

Regards, WM

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From Newsgroup: sci.logic

On 20/06/2026 16:05, wm wrote:

Am 20.06.2026 um 10:33 schrieb Mikko:

On 19/06/2026 15:31, WM wrote:

Am 19.06.2026 um 09:15 schrieb Mikko:

On 18/06/2026 23:10, wm wrote:

Then naturals and rationals could not seem to be in bijection

Yet a bijection between them can be shown.

It seems so because it can be shown for the first few pairs.

Every proof that a mapping is a bijection proves about all
members of both sets.

But there has never been a proof. Dark numbers cannot satisfy Cantor's claims like the folowing:

"If we think the numbers p/q in such an order [...] then every number p/
q comes at an absolutely fixed position of a simple infinite sequence"
[E. Zermelo: "Georg Cantor rCo Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 126]

Cantor is not talking about "dark" numbers.

A bijection between natural numbers and positive fractions is sufficinet
to show that a bijection between them esixts. Nothing else matters.
--
Mikko
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From Newsgroup: sci.logic

Am 21.06.2026 um 12:47 schrieb Mikko:

Cantor is not talking about "dark" numbers.

No, he assumes that all can be uswd. Therefore he is wrong.>

A bijection between natural numbers and positive fractions is sufficinet
to show that a bijection between them esixts. Nothing else matters.

Since almost all natural numbers cannot be used, there is no bijection applying all natural numbers. Proof: Betwen every applied unit fraction
and 0 there are almost all unit fractions not applied and not applyable.

Regards, WM

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From Newsgroup: sci.logic

On 22/06/2026 17:41, WM wrote:

Am 21.06.2026 um 12:47 schrieb Mikko:

Cantor is not talking about "dark" numbers.

No, he assumes that all can be uswd. Therefore he is wrong.

No, he does not say anything about using them. He is simply talking
about numbers and pairs of numbers, much like Galileo talked about
integers and squares.

A bijection between natural numbers and positive fractions is sufficinet
to show that a bijection between them esixts. Nothing else matters.

Since almost all natural numbers cannot be used, there is no bijection applying all natural numbers.

Your "since" is false. There is a bijection because various bijections
have been written. Whether almost all natural numbers can or cannot be
used is irrelevant. There is no need to use almost all of them.

Proof: Betwen every applied unit fraction
and 0 there are almost all unit fractions not applied and not applyable.

That is not a proof. That is an irrlevant claim about applicability.
--
Mikko
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From Newsgroup: sci.logic

Am 23.06.2026 um 08:42 schrieb Mikko:

On 22/06/2026 17:41, WM wrote:

Am 21.06.2026 um 12:47 schrieb Mikko:

Cantor is not talking about "dark" numbers.

No, he assumes that all can be uswd. Therefore he is wrong.

No, he does not say anything about using them.

You are a liar. I have quoted his statements. "such that every element
of the set stands at a definite position of this sequence" [E. Zermelo:
"Georg Cantor rCo Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts", Springer, Berlin (1932) p. 152]

A definite position!

Proof: Betwen every applied unit fraction and 0 there are almost all
unit fractions not applied and not applyable.

That is not a proof. That is an irrlevant claim about applicability.

Your misunderstandings are unlimited. No surprise that there remain
dregs of mathematics pursuing set theory.

Regards, WM
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From Newsgroup: sci.logic

On 23/06/2026 15:48, WM wrote:

Am 23.06.2026 um 08:42 schrieb Mikko:

On 22/06/2026 17:41, WM wrote:

Am 21.06.2026 um 12:47 schrieb Mikko:

Cantor is not talking about "dark" numbers.

No, he assumes that all can be uswd. Therefore he is wrong.

No, he does not say anything about using them.

You are a liar. I have quoted his statements. "such that every element
of the set stands at a definite position of this sequence" [E. Zermelo: "Georg Cantor rCo Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts", Springer, Berlin (1932) p. 152]

A definite position!

You are a liar. Nothing about using there. And the source is not his
own text but Zermelo-?s.

Proof: Betwen every applied unit fraction and 0 there are almost all
unit fractions not applied and not applyable.

That is not a proof. That is an irrlevant claim about applicability.

Your misunderstandings are unlimited. No surprise that there remain
dregs of mathematics pursuing set theory.

You can say "misunderstanding" baut you can't show one.
--
Mikko
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From Newsgroup: sci.logic

Am 24.06.2026 um 12:25 schrieb Mikko:

On 23/06/2026 15:48, WM wrote:

Am 23.06.2026 um 08:42 schrieb Mikko:

On 22/06/2026 17:41, WM wrote:

Am 21.06.2026 um 12:47 schrieb Mikko:

Cantor is not talking about "dark" numbers.

No, he assumes that all can be uswd. Therefore he is wrong.

No, he does not say anything about using them.

You are a liar. I have quoted his statements. "such that every element
of the set stands at a definite position of this sequence" [E.
Zermelo: "Georg Cantor rCo Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts", Springer, Berlin (1932) p. 152]

A definite position!

You are a liar. Nothing about using there. And the source is not his
own text but Zermelo-?s.

Why stands every element at a definite position? It has been put there.
But I proved that most elements cannot be put there. You are also very illiterate. The source is Cantor's collected works edited by Zermelo. >

Proof: Betwen every applied unit fraction and 0 there are almost all
unit fractions not applied and not applyable.

That is not a proof. That is an irrlevant claim about applicability.

Your misunderstandings are unlimited. No surprise that there remain
dregs of mathematics pursuing set theory.

You can say "misunderstanding" baut you can't show one.

See above: Cantor's text.

Regards, WM>

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From Newsgroup: sci.logic

On 24/06/2026 18:19, WM wrote:

Am 24.06.2026 um 12:25 schrieb Mikko:

On 23/06/2026 15:48, WM wrote:

Am 23.06.2026 um 08:42 schrieb Mikko:

On 22/06/2026 17:41, WM wrote:

Am 21.06.2026 um 12:47 schrieb Mikko:

Cantor is not talking about "dark" numbers.

No, he assumes that all can be uswd. Therefore he is wrong.

No, he does not say anything about using them.

You are a liar. I have quoted his statements. "such that every
element of the set stands at a definite position of this sequence"
[E. Zermelo: "Georg Cantor rCo Gesammelte Abhandlungen mathematischen
und philosophischen Inhalts", Springer, Berlin (1932) p. 152]

A definite position!

You are a liar. Nothing about using there. And the source is not his
own text but Zermelo-?s.

Why stands every element at a definite position?

Cantor defined a function that assigns a position to each element.
Therefore each element has a definite position.
--
Mikko
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From Newsgroup: sci.logic

Am 24.06.2026 um 12:25 schrieb Mikko:

On 23/06/2026 15:48, WM wrote:

Am 23.06.2026 um 08:42 schrieb Mikko:

On 22/06/2026 17:41, WM wrote:

Am 21.06.2026 um 12:47 schrieb Mikko:

Cantor is not talking about "dark" numbers.

No, he assumes that all can be uswd. Therefore he is wrong.

No, he does not say anything about using them.

You are a liar. I have quoted his statements. "such that every element
of the set stands at a definite position of this sequence" [E.
Zermelo: "Georg Cantor rCo Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts", Springer, Berlin (1932) p. 152]

A definite position!

Nothing about using there. And the source is not his
own text but Zermelo-?s.

The source is Cantor's, collected by Zermelo from G.Cantor: Ein Beitrag
zur Mannigfaltigkeitslehre.[Crelles Journal f. Mathematik Bd. 84, S. 242
- 258 (1878)].

Proof: Betwen every applied unit fraction and 0 there are almost all
unit fractions not applied and not applyable.

That is not a proof. That is an irrlevant claim about applicability.

Your misunderstandings are unlimited. No surprise that there remain
dregs of mathematics pursuing set theory.

You can say "misunderstanding" baut you can't show one.

Above you said that is not Cantor's text. But it is.

Regards, WM
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