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

Am 11.05.2024 um 01:44 schrieb Jim Burns:

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 [...] exists
independently of set theories.

Quite an interresting view.
Historical fact: Infinite ordinals were introduced by Cantor in the context of his "transfinite set theory".
But I get your idea (I think).
We may imagine (!) infinitely many natural numbers independent of set theory:
1 < 2 < 3 < ...
Of course, no set IN which "contains" these numbers in this case.
In addition to all these numbers we may imagine an additional "number" which is larger than all these (natural) numbers:
1 < 2 < 3 < ... < ω.
Right? (Actually, I sometimes didn't like the term "infinite ordinal" in this connection. My thought was: "It's just an additional number." :-P)