Re: The set of necessary FISONs

On 2/20/25 5:32 AM, WM wrote:

On 20.02.2025 03:23, Richard Damon wrote:

On 2/19/25 11:57 AM, WM wrote:

 

Note, your subject line uses the word you mean, "necessary", but you ignore the fact that a set of necessary elements doesn't need to exist.

 Assume a set of sufficient FISONs. |ℕ \ {1, 2, 3, ..., n}| = ℵo is true for all FISONs. That contradicts the assumption.
 Regards, WM

But that doesn't define a set by modern set theory.
For ALL Finite values of n, that statement is true.
Thus showing that ANY individual FISON isn't needed.
You never showed that you could remove ALL FISONs at once, as you aways had a highest n in the set to remove.
All induction says, is that this means you can remove ANY element of the full set of FISONs.
You can't step from ANY FISON to the full set of FISON by your arguement, as that isn't what induction does, it just proves that the set of things with that properties includes the set of Natural Numbers.
Not that the Set of Natural Numbers, as a set, has that property.
Properties of Sets and properties of their elements are distinct.
Of course, you are too stupid to understand that,