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

On 9/30/2024 11:12 AM, WM wrote:

On 29.09.2024 21:56, Jim Burns wrote:

On 9/27/2024 2:54 PM, WM wrote:

On 25.09.2024 20:40, Jim Burns wrote:

There are numbers (cardinalities) which increase by 1
and other numbers (cardinalities), which
don't increase by 1.

>
No.

 You invoke _axiom.1_

Every countable set is countable,
i.e., it increases one by one.

 Axiom.1 _describes_
what you are currently discussing.
 Axiom.1 means
⎛ If
⎜ the set of unit fraction can't increase by 1
⎜ then
⎝ we aren't discussing that set.
 Axiom.1 does not mean
⎛ If
⎜ we are discussing the set of unit fractions
⎜ then
⎝ that set can increase by 1

What you want is
to tell us we've been wrong about sets.

>
You have been wrong about sets.

What you are talking about aren't _our_ sets.
Compare to:
You (hypothetically) decide that
triangles should have _four_ corners.
And we carry on discussing three.cornered triangles.
Therefore,
we (hypothetically) would be wrong about
_your_ four.cornered triangles.
Which we would be.
I freely admit it.
But we wouldn't have any reason to _care_
about _your_ "triangles".
We have no more reason to care about _your_ "sets".

You assume that sets are invariable

...for the best of reasons:
_what we mean by set_ is invariable.
In other news,
we assume that triangles have three corners.
Which is what we mean.

but you don't assume that all elements,
here unit fractions, can be detected.

We typically assume nothing about detectability
and non.detectability.
Also,
we typically _make no claims_ about detectability
and non.detectability, so
that non.assumption isn't a problem _of ours_.

For every x NUF increases by not more than 1.

>
For every x>0 and x′>0
NUF increases by not more and not less than 0.
>
⅟⌈1+⅟x⌉ → ⅟⌈1+⅟x′⌉
⅟⌈2+⅟x⌉ → ⅟⌈2+⅟x′⌉
⅟⌈3+⅟x⌉ → ⅟⌈3+⅟x′⌉
⅟⌈4+⅟x⌉ → ⅟⌈4+⅟x′⌉
...

>
Wrong.

⅟⌈1+⅟x⌉ → ⅟⌈1+⅟x′⌉
⅟⌈2+⅟x⌉ → ⅟⌈2+⅟x′⌉
⅟⌈3+⅟x⌉ → ⅟⌈3+⅟x′⌉
⅟⌈4+⅟x⌉ → ⅟⌈4+⅟x′⌉
...

🎜 Aleph.naught bottles of beer on the wall,

 ℵo unit fractions cannot come into being
without a first one.

Unit fractions do not come into being.