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

On 9/21/2024 4:11 PM, WM wrote:

On 21.09.2024 19:26, Jim Burns wrote:

On 9/21/2024 9:47 AM, WM wrote:

On 20.09.2024 22:02, Jim Burns wrote:

On 9/20/2024 2:27 PM, WM wrote:

And there is no gap before ω.

>
ω-1 requires impossibilities:
a gap between ω and (hypothetical) ω-1

>
No.
A gap is where something could be
but is not.

>
Gaps are where you locate your darkᵂᴹ numbers.

>
Yes.
>

However,
'things always well.ordered' is
what we mean by 'ordinals'.

>
That is a potentially infinite collection.

If α is a finite ordinal, then α ∈ WWMB ω
If α isn't a finite ordinal, then α ∉ WWMB ω
WWMB "what we mean by"
By extensionality,
no other facts about WWMB ω exist.

There are no non.well.ordered WWMB ordinals.

>
That may be.

That is what we mean.

There is no finite WWMB ordinal α not.before
the first transfinite ordinal, WWMB ω
>
There is no finite WWMB ordinal α without
its WWMB successor α+1 before ω
>
There is no transfinite WWMB ordinal ξ before
the first transfinite ordinal, WWMB ω

>
That may be. But then
there is no complete set of natural numbers
and no ω.

WWMB ω is the first transfinite ordinal.
Existing or not.existing, WWMB ω isn't
anything other than the first transfinite ordinal.
As long as it's WWMB ω which you discuss,
WWMB ω-1 doesn't exist.

there is no complete set of natural numbers

If we discuss Boolos's theory ST
⎛ ∃{}
⎜ ∃z=x∪{y}
⎝ extensionality
then
each of WWMB the natural numbers exist.
Showing that involves _saying_
in the language of ST, what that _means_
which we can do and I have done before, here.
If we accept plural.quantification
⎛ ∃∃{y:P(y)}
⎜ ∀x: x∈{y:P(y)} ⇔ P(x)
⎝ plural.extensionality
then
a predicate ℕ(y) "y is WWMB a natural number" exists
in the language of ST
and thus
the plurality {y:ℕ(y)} exists, of
all and only the natural numbers.
If we accept that, ω={y:ℕ(y)} and exists
OTOH, You could deny {} or x∪{y} or {y:P(y)}
and your denial would not be _illogical_
However, if you did, it would be a bad fit with
your argument that
darkᵂᴹ.number.deniers are matheological fanatics,
blind to "common sense".
{} not.exists?
x∪{y} not.exists?
{y:P(y)} not.exists?
But _we're_ the fanatics.