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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Sat, 31 May 2025 12:59:27 +0300
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On 2025-05-30 14:25:22 +0000, WM said:

> 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.

Only after proving that all infinite subsetssets of the set of natural
numbers are equinumerous.

>>> 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.

No, there is no artithmetic induction and no set induction there.

-- 
Mikko