Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 09.01.2025 11:06, FromTheRafters wrote:

on 1/8/2025, WM supposed :

On 08.01.2025 18:32, FromTheRafters wrote:

WM explained :

On 08.01.2025 12:04, FromTheRafters wrote:

WM formulated on Wednesday :

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If ω exists, then ω-1 exists.

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Wrong.

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A set like ℕ has a fixed number of elements. If ω-1 does not exist, what is the fixed border of existence?

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Who says that there has to be a  fixed border of existence?

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According to set theory every set has a fixed set of elements, not more and not less.

 This applies to only finite sets.

No. Ask a set theorist of your choice. Sets are absolutely invariable.

 

Omega is a limit ordinal not a successor.

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But the natural numbers are invariable. For every n, there is n+1 which is not created but simply exists.

 Great for every n which are elements of N but omega is not an element of N. In the naturals, n+1 is usually axiomatic and n-1 is a theorem. In the infinite ordinals omega+1 is axiomatic and omega-1 is undefined or not existent.

Axiomatic? It is simply mathematical reality, but only for visible numbers. The axioms have been designed to reproduce this mathematical reality.
Regards, WM