Re: The set of necessary FISONs

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

On 04.03.2025 02:05, Richard Damon wrote:

On 3/3/25 4:10 AM, WM wrote:

 

They are just an artifact that your logic blew up when N_def became infinite, blinding you to the truth.

 By induction the number of numbers can never become an actually infinite quantity, larger than all finite numbers.

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
You just can't understand the concept, as you logic can't handle infinityies.

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What is the highest expressable number?

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That does not exist because with n also n+1 is expressable.

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So, why isn't N_Def the same as N?

 Every FISON contains only a finite set of numbers. ℕ is more.

Right, and no one says that there is A FISON that is the set of Natural Number, just that when you take the union of an infinite number of them, you get the Natural Numbers.

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We call that phenomenon potential infinity.

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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.
 Regards, WM

Which doesn't mean what you think it means.
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.
We SEE the "potentially infinite" via a process, where each step is finite, but the final result of it *IS* an infinite thing.
None of the members of N are themselves infinite, but the set itself is.