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

On 1/7/25 5:20 AM, WM wrote:

On 07.01.2025 02:36, Ross Finlayson wrote:

On 01/06/2025 02:43 PM, Jim Burns wrote:

 

It would be great if you (WM) did NOT
find lemma 1 weird,
but it is what it is.

 It is not weird. But your conclusions are weird.
 

The inductive set being covered by
initial segments is an _axiom_ of ZF.

 And the existence of the set ℕ is also an axiom of ZF. Therefore ZF is incompatible with mathematics.

No, ZF doesn't have as an axiom that the set of Natural Numbers exist. It may have axioms that we can see allow for such a set, but it is not an axiom of ZF.
If your "naive mathematic" is inconsistant witj ZF, then you can't use them in a system based on ZF. All you are doing is proving that your "mathematics" (which you can't actually provide a full definition of) is incapable of handling infinite sets, just like naive set theory broke when applied to systems with certain self-reflective properties.

 |ℕ| is a fixed quantity larger than every n, by any factor, for instance by the factor 100.

Right, but it is also an INFINITE quantity that doesn't obey some of the laws of FINITE mathematics.

 All FISONs stay below the threshold |ℕ|/100, or in other words, multiplication of any FISON by 100 is insufficient to cover |ℕ|.
 

So? The union of an infinite set of them can have properites different that any set that is a union of only a finite number of them. That is a nature of infinity.

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

But not the union of *EVERY* FISON, the FULL INFINITE set of them. SOething your "logic" can't handle, as it can't handle infinity.

 Regards, WM