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

On 10/28/24 6:40 AM, WM wrote:

On 27.10.2024 19:00, Richard Damon wrote:

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?

 I investigate this basis of Cantor's theory showing that, if it was true, it would nevertheless fail to agree with his theory.

No, all you are showing is that Cantor's theory about infinity shows that those infinite values have properties different from the finite numbers.

>

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.

 

So, all you have done is shown that YOUR CONCEPT of Acutual Infinity can't be correct.

 I am not yet convinced, but it may be.

It doesn't matter if you are convinced, it just is fact.
YOU don't control what is truth, and when you disagree with truth, you are just showing yourself to be either ignorant or a liar (or both).

 Regards, WM