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?
>
>