Re: how

Richard Damon wrote :

On 4/23/24 3:51 PM, WM wrote:

Le 23/04/2024 à 21:45, Tom Bola a écrit :

WM schrieb:
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Le 23/04/2024 à 01:01, Richard Damon a écrit :

On 4/22/24 10:15 AM, WM wrote:

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The results cannot be compressed to the interval (0, ω) of the set { 1, 2, 3, ...}. This shows that new numbers are generated by multiplication.
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Of course they can be compressed into the interval (0, ω), as every finite number n < ω, when doubled results in a finite number 2n which is also < ω.

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Try to map the closed interval [0, ω]*2 = [0, ω*2].

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f = (0, 0),
(1, 2),
(2, 4),
(3, 6),
(..., ...),
(w, w*2)

 Indeed. When n reaches ω, 2n reaches ω2.
 Regards, WM

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But n, the finite number, never reaches ω.
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ω isn't just some unimaginably big number, it is a value beyond finiteness, a transfinite value, and infinity.
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Of course, if you logic tries to make that transfinite value finite because that is all it can handle, your system just blows up in your face.

IMO, it might be best not to think of it as a value at all. Little omega represents (or can be represented as) the ordered set of natural numbers. The 'plus one' simply means the next in rank to them as an element, not really an addition operator as in the finites. Each new element in the ranking represents or can be represented by all previous elements in their respective positions.