Re: Replacement of Cardinality

Le 02/08/2024 à 14:52, joes a écrit :

Am Fri, 02 Aug 2024 11:38:33 +0000 schrieb WM:

Don't claim the contrary. Define (separate by an eps from 0) all unit
fractions. Fail.

Well, there's no epsilon that separates all positive numbers from zero.
But every fraction has an epsilon that is smaller.

Note that epsilons must be chosen. You cannot choose an eps that separates more than few unit fractions.
If you object, then simply apply all of them. Then every fraction could be separated and none would remain. Then you had contradicted the existence of accumulation points.

Hence there is only the sequence of natnumbers.

And no sequence of omega - k.

Those are natnumbers.

 

We just condense the whole of N into one concept and call that omega,

That is nonsense. ω is the first number following upon all natural
numbers.

Following the WHOLE of the natural numbers. The successor of a natural
is also one.

Not for all.

 

If k did not have a successor, what would k+1 be?

ω

Ah, a natural number.

No, ω is the first transfinite number. But we cannot count from 1 to ω.
Regards, WM