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

On 5/3/2024 9:57 AM, WM wrote:

Le 03/05/2024 à 11:07, Jim Burns a écrit :

WM call various things
"actually infinite" and "potentially.infinite".
What does he mean by those terms?
NOT "What does Cantor mean? Euclid mean?"

>
By actual infinity I mean what Cantor [...]

It is the idea that infinite sets are fixed
such that no element can be added or removed.
It is the idea that
the natural numbers reach immediately from 0 until ω

If every natural nunmber is increased by 1,
then the set is shifted from {1, 2, 3, ...} to {2, 3, 4, ..., ω}.

Your version of actual.infinityᵂᴹ skips
ℵ₀.many numbers which CAN be counted.to from 0
Only you (WM) call skipping.ℵ₀.many "completing".
Maybe your mis.taught students call it that as well.
Each number which CAN be counted.to from 0
is before
ℵ₀.many numbers which CAN be counted.to from 0
Your description of ω is a description of
a number which CAN be counted.to from 0
and thus
ω as you've described it is before
ℵ₀.many numbers which CAN be counted.to from 0
which skips ℵ₀.many.

The increase of every number mentioned above can only be applied collectively.

Each number which CAN be counted.to from 0
is a number which CAN be counted.to from 0