Re: The set of necessary FISONs

WM explained on 2/22/2025 :

On 22.02.2025 02:05, Richard Damon wrote:
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Since that is the earlier work, and not the later refined work, maybe you should update your studies. (Note this creates Z, not N)

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Zermelo's Z_0, the set of numbers is a subset of Z.

The set of natural numbers, but what is the powerset of Z?

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.

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.

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; however, the ability to perform finite induction can be derived from the axioms of Zermelo set theory, specifically the "Axiom of Infinity" which guarantees the existence of infinite sets, allowing for the construction of the natural numbers and subsequently the application of mathematical induction on them.
Key points:
    No direct axiom for induction: Zermelo's axioms do not explicitly state a principle of finite induction.
Derived from other axioms:
By constructing the natural numbers using the "Axiom of Infinity," you can then prove the principle of mathematical induction within the framework of Zermelo set theory.
Importance of the Axiom of Infinity:
This axiom is crucial for establishing the existence of infinite sets which are necessary to use induction on natural numbers.

You are wrong. Remember Peano's successors.

The successor function alone is not induction.
AI Overview
No, the successor function is not the same as induction, but it is a key component of the concept of mathematical induction; the successor function simply gives the "next" number in a sequence, while induction is a proof technique that relies on the successor function to prove statements about all natural numbers by showing that if a statement holds for a number, it must also hold for the next number in the sequence.
Key points:
Successor function:
This is a function that takes a natural number and returns the next natural number, essentially adding 1 to it.
Mathematical induction:
This is a proof method where you demonstrate that a statement is true for a base case (usually 0) and then show that if the statement is true for any number "n", it must also be true for the next number "n+1".
How they relate:
Basis for induction: The successor function is used in the "induction step" of a mathematical induction proof, where you assume a statement is true for a number "n" and then use the successor function to show it's also true for "n+1".
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