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

On 1/19/25 5:47 AM, WM wrote:

On 18.01.2025 14:46, Richard Damon wrote:

On 1/17/25 4:56 PM, WM wrote:

 

That "definition" violates to definition that set don't change.

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So it is. But if infinity is potential, then we cannot change this in order to keep set theory, but then set theory is wrong.

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So, you are just agreeing that your logic is based on contradictory premsises and thus is itself contradictory and worthless.

 No, set theory claims actual infinity but in fact useses potential infinity with its "bijections". They contain only natnumbers which have ℵ₀ successors. If all natural numbers were applied, there would not be successors:
ℕ \ {1, 2, 3, ...} = { }.
 Regards, WM

Set theory doesn't distinguish the two, because it doesn't matter to set theory, The set has either the final result of the "potential" infinity description, or just is the actual infinite set.
Note, even in actual infinity, every Natural Number has Aleph_0 successors, as there is no "last" Natural Number(s), even though all exist.
Your concept of having a last (or even a finite tail) is just a property of finite sets, and doesn't apply to infinite sets.
All you have shown is that "Actual Infinity" can't be looked at by a logic that only deals with finite sets.
Such a logic can potentially look at "Potential Infinity", as each set allong the way is a finite set, and getting to infinity is only at the eventual completion, so the logic only breaks "in the limit".