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

On 10/12/24 1:49 PM, WM wrote:

On 10.10.2024 21:54, joes wrote:

Am Thu, 10 Oct 2024 20:53:07 +0200 schrieb WM:

On 10.10.2024 20:45, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

>

If all natnumbers are there and if 2n is greater than n, then the
doubled numbers do not fit into ℕ.

For any finite n greater than zero, 2n is greater than n.  The same
does not hold for infinite n.

There are no infinite n = natural numbers.

Exactly! There are furthermore no infinite doubles of naturals (2n).

 But the doubles are larger. Hence after doubling the set has a smaller density and therefore a larger extension on the real line. Hence not all natural numbers have been doubled.

The doubles are larger than the element they replace, but that value was always in the set to begin with, so it never creates a "new" term.

>

Numbers multiplied by 2 do not remain unchanged.

Either doubling
creates new natural numbers. Then not all have been doubled. Or all
have been doubled, then some products fall outside of ℕ.

No.  Not even close.

Deplorable. But note that all natural numbers are finite and follow this
law: When doubled then 2n > n. If a set of natural numbers is doubled,
then the results cover a larger set than before..

Additionally: if n is finite, so is 2n. It cannot go beyond w.

 Then there is no complete set. The doubling can be repeated and repeated. Always new numbers are created. Potential infinity.

You got it! The complete set of infinity is not available to finite beings to directly observe and handle, because it is just too big for use to work on.
That was the conclusion that Aristotle came up millennia ago.

 Regards, WM

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