Re: The set of necessary FISONs

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:

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Note, as I understand it, that initial Zermelo Set Theory didn't even HAVE "induction",

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You don't understand it.

 He understands it better than you do apparently. Zermelo's theory didn't have transfinite induction.

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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.

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That is induction.

 That is the successor function, it doesn't extend to the transfinite.

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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;

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It does.

Not explicitly, it is a theorem.

Key points:

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AI is not suitable for mathematics.

It is okay when it is correct though.