Re: The set of necessary FISONs

On 3/11/2025 5:28 PM, WM wrote:

On 11.03.2025 20:25, Jim Burns wrote:

On 3/11/2025 2:01 PM, WM wrote:

Zermelo defines Z to be an inductive set.

>
in order to ensure the existence of
an infinite or inductive set.

No.
There is a definition and there is an axiom.
They are different kinds of things.
The definition of Z states that
Z is inductive.
The Axiom of Infinity (better: of Inductivity)
states that something exists which satisfies
the definition of Z.
We (some of us, anyway) are careful to
keep definitions and axioms apart.
As long as only axioms have logical consequences,
we can continue our discussion
_about the same things_
while giving free rein to our imaginations
in our definitions.

in order to ensure the existence of
an infinite or inductive set.

A definition doesn't ensure existence.
That's what an axiom does.