Re: Replacement of Cardinality

On 7/29/24 9:11 AM, WM wrote:

Le 28/07/2024 à 19:07, Richard Damon a écrit :

On 7/28/24 7:55 AM, WM wrote:

 

Mistake! ℕ U ω = {1, 2, 3, ..., ω}

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And who was using that set?

 I.

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yields the set
{2, 4, 6, ..., ω, ω+2, ω+4, ..., ω*2}.
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Why?

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See the correction.

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But what number became ω when doubled?

 ω*2

No, that is w double, what number in the first set became w?

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Every natural number when doubled is a Natural Number.

 No.

Why not? WHich ones don't?

 

Note, ω-1 doesn't exist in the base transfinite numbers, just as -1 doesn't exist in the Natural Numbers, you can't go below the first element.

 If all natural numbers exist, then ω-1 exists.

Why?

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But, just as we can expand the Natural Numbers to the Integers, and get negative numbers, we also might be able to define an extention to the transfinite numbers that can have a ω-1 element.

 ω-1 is not transfinite but cisfinite.

Nope, it is not finite, but beyound the finite.

 

Using the unit fractions itelligent readers understand that there must be a first one after zero. Others must believe in the magical appearance of infinitely many unit fractions.
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Nope, since that implies there is a highest Natural Number, which breaks their definition,

 That is unavoidable. You believe in the magical appearance of infinitely many unit fractions. That breaks logic and mathematics.

Nope, the fact that you admit that you logic breaks the definition of the Natural Numbers says you admit your logic can't be used with them.
It may be that you logic just can't handle a mathematics are big and powerful as the full definition of the Natural Numbers, but only arbirary but finite subsets of them.
You just need to understand your limitations.

 Regards, WM