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

On 1/8/25 4:04 AM, WM wrote:

On 08.01.2025 00:30, Richard Damon wrote:

On 1/7/25 7:51 AM, WM wrote:

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No, ZF doesn't have as an axiom that the set of Natural Numbers exist.

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AoI: There exists an infinite set S.

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Which isn't that the NATURAL NUMBERS are an infinite set.

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The infinite set has been designed by Zermelo according to Dedekind's definition of the natural numbers, as Zermelo noted. https:// gdz.sub.uni-goettingen.de/id/PPN235181684_0065? tify=%7B%22pages%22%3A%5B276%5D%2C%22pan%22%3A%7B%22x%22%3A0.461%2C%22y%22%3A1.103%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.884%7D

 So? it doesn't mean that ZF has made it an axiom that the set of Natural Numbers exist, he has made his Axiom of Infinity to be designed so that the existance of the Natural Numbers can be derived from it. You confuse cause from effect.
 IT is good to know where you are trying to go, or it can be hard to get there.
 IF you claim that the axiom of infinity is NOT valid, then why do you keep on using the results of it in your logic? One of your problems is you have ADMITTED that you "logic" isn't axiomized (since you admit you can't provide a set of actual axioms to define it) and thus you admit that your "logic" isn't actually LOGIC. Your "Theorem" can't be actually a Theorem, as you don't have any axioms on which to prove it.
 

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Every union of FISONs {1, 2, 3, ..., n} which stay below this threshold stays below this threshold too.

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But not the union of *EVERY* FISON, the FULL INFINITE set of them.

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All are below 1 %.

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No,

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Show one FISON that is larger than 1 %.

 There isn't one, but that doesn't matter,
 Show me a Natural Number that is bigger than Aleph_0 / 100?
 It doesn't exist, because Aleph_0 is infinite, and an infinite number divided by ANY finite value is still that infinite value, and thus there is no finite value greater than that.
 This doesn't give you your "dark numbers" as "non-defined finite numbers", but shows that your logic is just broken.