Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 24.10.2024 13:40, Richard Damon wrote:

On 10/24/24 7:07 AM, WM wrote:

On 24.10.2024 05:04, Richard Damon wrote:

On 10/23/24 10:46 AM, WM wrote:

On 23.10.2024 13:37, Richard Damon wrote:

On 10/22/24 12:12 PM, WM wrote:

On 22.10.2024 18:03, Jim Burns wrote:

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∀n ∈ ℕ:  2×n ∈ ℕ

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Not if all elements are existing before multiplication already.

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IF not, then your actual infinity wasn't actually infinite

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It is infinite like the fractions between 0 and 1. When doubling we get even-numerator fractions, some of which greater the 1.

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But from 0 to 1 isn't an infinite distance

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Measured in rational points it is infinite.

 But values are not measured in "rational points".

We can measure in rational points. Between two such points there are many dark ones, but that is the same between 0 and omega.

 

It is claimed that there are all numbers. "That we have for instance when we consider the entirety of the numbers 1, 2, 3, 4, ... itself as a completed unit, or the points of a line as an entirety of things which is completely available. That sort of infinity is named actual infinite." [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 167]

 So?
 If you HAVE all the numberes, 1, 2, 3, 4, ... that set goes on FOREVER and doesn't have a upper end.

But it has all. All can be doubled.

 If you have the COMPLETE unit, it doesn't have a highest number.

But it has all. All can be doubled.

 Your operation of doubling the values on the line from 0 to 1 isn't operating the property that that set is infinite on, so doesn't follow the law of the infinite.

Infinite sets can be mapped completely, according to set theory.

There is nothing about being complete that means it needs to have an "end"

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Whatever, it is complete and all its numbers can be doubled. Some are resulting in larger numbers than have been doubled.

 Nope, as every number (A Natural Number) doubles to another number in that set (The Natural Numbers) so you never left the set.

I take all of it.
Regards, WM