Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 1/16/2025 8:32 AM, WM wrote:

On 16.01.2025 02:23, Jim Burns wrote:

On 1/15/2025 1:17 PM, WM wrote:

On 15.01.2025 16:16, Jim Burns wrote:

On 1/14/2025 4:07 AM, WM wrote:

On 13.01.2025 20:31, Jim Burns wrote:

On 1/13/2025 12:17 PM, WM wrote:

therefore creates even numbers.
They do not fit below ω.

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No.
They fit below ω

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In completed infinity
all available places are occupied.

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In each of our sets,
each of its elements is in the set,
each available place is occupied.

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Therefore new numbers are not accepted.

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And all even numbers fit below ω
None are created.

A potentiallyᵂᴹ infiniteˢᵉᵗ set
has all available places occupied,

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But it can grow.

No, it can't.

Multiplying all its elements by 2
creates new elements.

No. it doesn't.

the same as any other set,
which is to say,
it is (has been, will be) completeᵂᴹ.

>
Then new numbers must be outside.

There are no new numbers.
A step is never from finite to infinite.
Therefore, a step never crosses ω
Therefore, a sum never crosses ω
Therefore, a product never crosses ω
Therefore, a power never crosses ω