Re: how

Am 04.05.2024 um 00:18 schrieb Chris M. Thomasson:

On 5/3/2024 7:17 AM, Moebius wrote:

Am 03.05.2024 um 15:23 schrieb FromTheRafters:
>

omega is the first infinite ordinal. It is not larger than the natural numbers, it *is* the natural numbers.

>
Well, actually, it is larger than _each and every_ natural number.
>
Using symbols: An e IN: n < ω.
>
Using a common depiction: 0 < 1 < 2 < 3 < ... < ω.
>
See: https://en.wikipedia.org/wiki/Ordinal_number
>
On the other hand, you are right, in the context of axiomatic set theory (if the natural numbers and the ordinals are defined due to von Neumann) we do have IN = ω.

 0 < ω
1 < ω
2 < ω
3 < ω
...
 This holds true for infinity...

Sure. Using classic words (oldies but goldies):
 > 0 < ω
 > 1 < ω
 > 2 < ω
 > 3 < ω
 > ... ad infinitum.
Using modern math lingo we would rather state:
An e IN: n < ω.
:-P
(After all, there are infinitely many elements in IN. Namely ALL natural numbers.)