Re: The set of necessary FISONs

On 2/3/2025 8:21 PM, Ross Finlayson wrote:

On 02/03/2025 11:48 AM, Jim Burns wrote:

On 2/3/2025 1:36 PM, WM wrote:

On 03.02.2025 19:06, Jim Burns wrote:

On 2/3/2025 7:41 AM, WM wrote:

How can Peano create
the complete set by induction?

>
Peano describes a set with induction.

>
Without axioms
nothing must be used oin formal mathematics.

>
Axioms describe the domain.
Describe a finite ordinal:
⎛ Sets of them are minimummed or empty.
⎜ Each has an immediate predecessor or is zero,
⎜ and each of the priors of each
⎝ has an immediate predecessor of is zero.
>
That description is the axioms of the finite ordinals.
(There are other ways to describe them.)
>

Therefore Peano, Zermelo, or v. Neumann
create ℕ as well as the set of all FISONs
for use in set theory.

>
Axioms describe.
Magic spells create.

Axioms that are not _false_, define a domain.

Axioms describe a domain.
If they are self.contradictory,
then what's described doesn't exist.
If they aren't self.contradictory,
then what's described exists.
Maybe what exists isn't
what you wanted to exist. Nonetheless.
⎛ It's a consequence of Gödel's completeness theorem
⎜ (not to be confused with his incompleteness theorems)
⎜ that a theory has a model if and only if it is consistent,
⎜ i.e. no contradiction is proved by the theory.
⎜ Therefore, model theorists often use "consistent" as
⎜ a synonym for "satisfiable".
⎝
https://en.wikipedia.org/wiki/Model_theory

Others define a mere contingent contrivance.

If there is a mere contingent contrivance which
describes what I'd like described,
I will use it and be grateful for it.
What.I'd.like.described is very, very often
(I'd say, for nearly everyone, it's 'always')
NOT everything.
What.I'd.like.described differs
from time to time and from place to place.
It is, in a word, contingent.
And it is a contrivance, too! Yes!
What else did you think it might be, Ross?
Wafted from the sky by angels?