Re: Simple enough for every reader?

On 30.05.2025 11:36, Mikko wrote:

On 2025-05-29 14:47:49 +0000, WM said:
 

It is in certain mathematical structures but not in all.

>
Anyhow a reader in sci.logic should understand it.

 Everybody should understand at least arithmetic induction and its
limitations. But everybody doesn't.

"Not everybody does" would be correct.

 Cantor did not use ℵo for infinity in general but only for a particular
kind of infinity.

For all infinite sets of natural numbers he used it. That's what I discuss here.

P[n]: {1, 2, 3, ..., n} has infinitely many (ℵo) successors.
P[n+1]: {1, 2, 3, ..., n, n+1} has infinitely many (ℵo) successors.

 But P[n] -> P[n+1] is not there.
 

Do you doubt ℵo - 1 = ℵo?

 That can't be said in Peano arithmetic.

Here I use induction in Cantor's set. That is allowed. Cantor did it too.

 

It is basic mathematics as you learn it in the first semester.

 No, it is not that basic. There are no infinities there, and no
induction, either.

Where did you study? In general courses of mathematics Cantor is taught during the first semester.

 

As I said the theory must be specified.
>
In Peano arithmetic the induction axiom is applicable to everything.
If you want something else you must specify some other theory, perhaps
some set theory.

>
Induction is applied to every natural number of the Peano set. The proof shows that it cannot be applied to every natural number of the Cantor set.

 You have shown no proof that shows that.

Here it is again: All natural numbers can be manipulated collectively, for instance subtracted: ℕ \ {1, 2, 3, ...} = { }. Here all have disappeared.
Could all natural numbers be distinguished, then this subtraction could also happen but, caused by the well-order, a last one would disappear.

It is not a contradiction that at least one disappears when all disappear.

It is a contradiction however that a last one disappears.
Regards, WM