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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Sat, 22 Feb 2025 07:13:36 -0500
Organization: Peripheral Visions
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WM formulated on Saturday :
> On 22.02.2025 12:14, FromTheRafters wrote:
>> WM explained on 2/22/2025 :
>>> On 22.02.2025 02:05, Richard Damon wrote:
>
>>>> Note, as I understand it, that initial Zermelo Set Theory didn't even 
>>>> HAVE "induction",
>>>
>>> You don't understand it.
>> 
>> He understands it better than you do apparently. Zermelo's theory didn't 
>> have transfinite induction.
>
> Who claimed that?

Certainly not you. Induction is a theorem there, and not extended to 
the infinite sets.

>>>> but its last axiom was the Axiom of Infinity that say that there exists 
>>>> in this domain the set Z that contains the null set as an element and is 
>>>> so constituted that to each of its element a, there corresponds a further 
>>>> element of the form {a}, in other words with each of its elements a, it 
>>>> also contains the corresponding set {a} as an element.
>>>
>>> That is induction.
>> 
>> That is the successor function, it doesn't extend to the transfinite.
>
> Induction covers the infinite set of elements. Transfinity is not included. 
> First element and the successor function is induction.

First element and successor function is 'inductive hypothesis' not 
induction.

https://www.cs.cornell.edu/courses/cs2800/2011fa/Lectures/induction.pdf

>>>> Note, this is NOT the statement of induction.
>> 
>> AI Overview
>> 
>> No, Zermelo's set theory by itself does not explicitly include the 
>> principle of finite induction;
>
> It does.

Not explicitly, it is a theorem.

>> Key points:
>
> AI is not suitable for mathematics.

It is okay when it is correct though.