Re: how

On 4/25/2024 3:01 PM, Chris M. Thomasson wrote:

On 4/25/2024 1:20 PM, WM wrote:

Le 25/04/2024 à 01:13, Richard Damon a écrit :

On 4/24/24 1:16 PM, WM wrote:

Le 24/04/2024 à 04:01, Richard Damon a écrit :

On 4/23/24 3:34 PM, WM wrote:

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].
If [0, ω) --> [0, ω) and ω*2 --> ω*2, then ω*2 is the only image point in (ω, ω*2]. Infinitely many points remain empty. Crippled mathematics. Ugly. Inacceptable.

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Why?

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Continuity.

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Who says we have that sort of "Continuity"?

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I.
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If the preimage has intervals of 1, then the image has intervals of 2.

 A =       { 1, 2, 3, 4, ... }
 A*2 =     { 2, 4, 6, 8, ... }
 (A*2)/2 = { 1, 2, 3, 4, ... }
 There is a one-to-one direct mapping back and fourth and both sets are infinite. They have the same size...

Also, any element of any of those infinite sets is already in the set of natural numbers... :^)

    

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ω+n is existing. How is it approached if not from the finite domain?
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Regards, WM
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