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

On 10/27/24 12:07 PM, WM wrote:

Am 27.10.2024 um 16:55 schrieb Alan Mackenzie:

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

On 26.10.2024 05:21, Jim Burns wrote:

>

The proof that all unit fractions can be defined
is to define them
as reciprocals of positive countable.to.from.0 numbers.

>

That describes all of them and only them.

>

No, you falsely assume that all natnumbers can be defined.

>
Wrong.  Natural numbers are defined as those defined by the Peano
axioms.  That is what we mean by "natural number".  Anything which
"can't be defined" this way isn't a natural number.

 Then there is no actual infinity.

Then why are all your arguements based on it?

Proof: If infinity is actual, then all elements of the set of unit fractions exist. The function NUF(x) = Number of Unit Fractions between 0 and x starts with 0 at 0. After NUF(x') = 1 it cannot change to NUF(x'') =  2 without pausing for an interval consisting of uncountably many real points. The reason is this: ∀n ∈ ℕ: 1/n - 1/(n+1) > 0. Of course x' and x'' are dark rational numbers, the smallest unit fractions.
 Regards, WM

So, all you have done is shown that YOUR CONCEPT of Acutual Infinity can't be correct.
Sorry, you just flushed your work down the toilet.