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

On 10/25/2024 12:57 PM, WM wrote:

On 25.10.2024 16:53, Jim Burns wrote:

On 10/25/2024 7:42 AM, WM wrote:

The whole interval (0, ω) is not finite,
let alone the doubled interval.

>
⟦0,ω⦆ is the set of finite ordinals.
That is the definition of finite ordinal.
That is the definition of ω,
the first ordinal after all finite ordinals.

>
Correct so far.
>

γ before ω: γ is finite.
γ ∈ ⟦0,ω⦆ ⇒
∀β ∈ ⦅0,γ⟧: ∃α: α+1=β
>
ω before ξ: ξ is not finite.
ω ∈ ⦅0,ξ⟧ ⇒
¬∀β ∈ ⦅0,ξ⟧: ∃α: α+1=β
>
(Keep in mind that ¬∃α: α+1=ω )

>
That is wrong in complete infinity.

For ordinals,
γ ∈ ⟦0,ω⦆ ⇒
∀β ∈ ⦅0,γ⟧: ∃α: α+1=β
∀β ∈ ⦅0,γ⟧: ∃α: α+1=β  ⇒
γ ∈ ⟦0,ω⦆
If you want to discuss a "different" ω
give it a different name.

A better question is:
why do you (WM) support it?

>
I support it in order to show that
your infinity is inconsistent.

My infinity? You mean
γ ∈ ⟦0,ω⦆  ⇔  ∀β ∈ ⦅0,γ⟧: ∃α: α+1=β
?
How did you manage to show that
_what I'm saying_ is inconsistent
without even accepting that
that is _what I'm saying_ ?

Example:
Almost all unit fractions
cannot be discerned by definable real numbers.

All unit fractions are reciprocals of
positive countable.to.from.0 numbers.

If they are existing,
they are indiscernible, i.e. dark.

If it is existing,
it is a reciprocal of
a positive countable.to.from.0 number.
I haven't found a reason to discern.
I am talking about each thing I am talking about.
I am not.talking about each thing I am not.talking about.