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

Am Sat, 11 Jan 2025 13:07:31 +0100 schrieb WM:

On 11.01.2025 11:25, joes wrote:

Am Sat, 11 Jan 2025 11:04:56 +0100 schrieb WM:

On 11.01.2025 01:28, Richard Damon wrote:

On 1/10/25 4:48 PM, WM wrote:

On 10.01.2025 21:08, Jim Burns wrote:
>

Where OUR infinityⁿᵒᵗᐧᵂᴹ "doesn't work",
it's you who's saying it doesn't work,

You are inconsistent. You claim that all natural numbers are an
invariable set. But when all elements are doubled then your set
grows,
showing it is not invariable. That is nonsense.

But the set doesn't grow.
Which element is in the doubled set that wasn't there in the first
place?

The number of elements remains constant. All odd numbers of ℕ are
deleted. That implies that new even numbers are added.

Nope. This is all just your conception of Aleph_0 as finite.

Not finite but complete.

Infinite things don't have an end.

It does not behave like that. All countable sets are bijective to each
other.

It appears so. But it is wrong. Construct a bijection between natural
numbers and even natural numbers. f(n) = 2n. Then double the even
numbers by multiplication. Many are not in the bijection.

"Doubling" IS multiplication by 2, turning the naturals into the evens.

If Cantor has constructed a sequence containing all even numbers of
the original set ℕ, then the doubled even numbers are missing.

What? Doubled even numbers are also even numbers.

But not natural numbers, since *all* natural numbers have been doubled.

All even numbers are naturals. Multiplication by any natural number
always results in another natural.

When we mark all natural numbers with an astrisk, then none remains.
When wie double them, then none remains. The odd numbers leave. All even
numbers remain. But the same number of of even numbers is larger than ω.

It is not. The sequence 2*n does not go beyond infinity.

1, 2, 3, 4, 5, ..., ω becomes 2, 4, 6, 8, 10, ..., ω, ω+2, ω+4, ..., ω2.

No. There is no x e N such that 2*x >= omega. You have listed two
consecutive infinities on the right.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.