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

On 5/10/2024 11:00 PM, Moebius wrote:

Am 11.05.2024 um 01:44 schrieb Jim Burns:

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".

That makes sense to me.
Without transfinite set theory,
what good are transfinite ordinals?
Transfinite flying rainbow sparkle ponies?

But I get your idea (I think).
>
We may imagine (!)
infinitely many natural numbers
independent of set theory:

This is perhaps a small point:
I would prefer "We may describe..."
Our imaginations are too limited for
the objects of even quite mundane mathematics.
But we can describe them and reason about them,
without imagining them.

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?

Agreed.
I would say "describe", though.
I think that the imagine/describe distinction
gets at the source of WM's troubles.
WM _imagines_
a natural number after all natural numbers.
But imagination is -- _should_ be -- flexible.
So flexible, we can imagine flying rainbow
sparkle ponies and largest natural numbers
_without their existing_
Descriptions written in ink or pixels
aren't as flexible as that.
I think descriptions aren't so user.friendly
as imaginings. That's my explanation for how
descriptions, with their great advantage,
"lose" in the struggle for our attention.

(Actually, I sometimes didn't like the term
"infinite ordinal" in this connection.
My thought was:
"It's just an additional number." :-P)

But what is a number?
Sometimes this, and sometimes that.
The devil is in the details.