Re: because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

On 5/10/2024 3:34 PM, Moebius wrote:

Am 10.05.2024 um 20:56 schrieb Jim Burns:

ℕ_def  ℕⁿᵒᵗᐧᵂᴹ  ω  ⋃ₙ⟨⟨0…n⟩⟩  ♃  🐎  is
the set of all and only numbers which CAN be counted.to
and as such its elements are fixed,
because nothing exists which is
partly.countable.to and partly.not.countable.to.
Nothing is partly.in and partly.out.

>
At least in the context of set theory.

Talking about the set ℕⁿᵒᵗᐧᵂᴹ
in the context of set theory
is most of what I can hope to accomplish.
However,
I think a little more might be available.
I have the impression that
the ordinals which are _represented_ in set theory
are, in some sense, _prior to_ set theory or
orthogonal to set theory, at least,
with neither prior.
I'm thinking of
the von Neumann cumulative hierarchy of sets.
V₀ = ∅
Vᵦ₊₁ = 𝒫(Vᵦ)
Vᵧ = ⋃ᵝᑉᵞ Vᵦ
It seems advisable that we grant ordinals
their existence before we head down that road.
And natural numbers are finite ordinals.
I think it can be argued that
ω (sometimes called ℕⁿᵒᵗᐧᵂᴹ) exists
independently of set theories.
For ω
I suggest the definition:
γ < ω  ⟺
∀β: 0 < β ≤ γ  ⇒
∃α: 0 ≤ α < γ  ∧  α+1 = β
I'm pretty sure we can derive induction from that.