Re: The set of necessary FISONs

On 3/5/25 4:05 PM, WM wrote:

Am 05.03.2025 um 13:25 schrieb Richard Damon:

On 3/4/25 4:07 AM, WM wrote:

 

Wrong, Cantor shows that the number of Natural Numbers generated by the iterative method of, we have 0, and for every number we have its successor, is not one of those finite numbers, but is another number Aleph0

 Below you contradict yourself.

Where?

 

We call that phenomenon potential infinity.

>
WHich is just infinity,

>
Cantor denies your claim.
"Nevertheless the transfinite cannot be considered a subsection of what is usually called 'potentially infinite'. Because the latter is not (like every individual transfinite and in general everything due to an 'idea divina') determined in itself, fixed, and unchangeable, but a finite in the process of change, having in each of its current states a finite size; like, for instance, the temporal duration since the beginning of the world, which, when measured in some time-unit, for instance a year, is finite in every moment, but always growing beyond all finite limits, without ever becoming really infinitely large." [G. Cantor, letter to I. Jeiler (13 Oct 1895)]
Here he is right.

>
Which doesn't mean what you think it means.

 He means what he says.

Right, the "Transfinite" (the numbers beyond the finite) are not members of the finite, the numbers used to count up towards infinity.

>
He is pointing out that these "transfinite" concepts aren't part of the infinite set built by iteration (the "potential infinity") but is beyond it.

 Therefore iterartion fails to produce actual infinity.

Where do you get that? The memeber of the set we iterate in are all finite, but the resultant set is infinite itself.

>
We SEE the "potentially infinite" via a process, where each step is finite, but the final result of it *IS* an infinite thing.

 There is no final result. You are unable to understand infinity.

Sure there is, it is the set of the Natural Numbers.
You just can't limit your logic to logic that requires creating things by finite work to work on a infinite set.

 Because the latter is a finite in the process of change, having in each of its current without ever becoming really infinitely large.

No, it is an infinitely long process of change, that none of the number created are ever not finite, but the full set is.

>
None of the members of N are themselves infinite, but the set itself is.

 Not by recursion or induction! Therefore UF is a proper subset of ℕ.
UF = ℕ  ==> Ø = ℕ
 

Nope, because you defined your UF to be an INFINTE union of FISONS that made up the Natural Numbers.

Regards, WM