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

On 11/29/24 10:37 AM, WM wrote:

On 29.11.2024 14:57, Richard Damon wrote:

On 11/29/24 8:44 AM, WM wrote:

On 29.11.2024 01:06, Richard Damon wrote:

On 11/28/24 12:50 PM, WM wrote:

If for all intervals 1, 2, 3, ..., n the covering is 1/10, then there are no natnumbers outside of all intervals and there are no hats
outside of all intervals.

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You are making the error of assuming that the infinite set is just like a finite set that has part of it.

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No. Analysis concerns infinite sequences and sets.

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You are looking at FINITE sets, and then trying to extrapolate to an infinte set, which doesn't work.

 Analysis is basic.

Analysis needs to be done on a correct basis.
Thinking of an infinte set as finite is just wrong logic.

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The problem is that the actual problem is defined on the INFINITE set, and in that case, there ARE enough hats to cover.

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No. The limit of the sequence f(n) of relative coverings in (0, n] is 1/10, not 1. Therefore the relative covering 1 would contradict analysis.

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And 0^x is 0, and x^0 is 1, which shows that just because you have a constant sequence, it limit is not necessarily the final value.

 The limit of the sequence 1/9, 1/9, 1/9, ... is 1/9.

But f(x) is not necessarily the limit of F(y) as y approaches x.
Thus, your logic of looking at the limit of the "fraction used" of the finite sets does not need to be the "fraction used" of the infinte set.
Just like some infinite series can have differing limits based on the order you add the terms, you can't just assume that if a limit can be computed, that is the value "at" that limit.
You are just showing your ignorance of how infinity actually works, because you logic blew itself to smithereens when you applied it to something it could not actually handle.
You are just showing why simple logic can not deal with "actual infinity", but you need logic that actually understands its properties.
Sorry, you are just proving your ignorance and stupidity.

 Regards, WM