Re: Replacement of Cardinality

On 8/28/2024 1:12 PM, WM wrote:

Le 28/08/2024 à 18:55, Jim Burns a écrit :

On 8/28/2024 9:25 AM, WM wrote:

Le 28/08/2024 à 08:13, Jim Burns a écrit :

On 8/27/2024 3:11 PM, WM wrote:

The function exists if
actual infinity exists.
The function does not exist if
only potential infinity exists.
>

¬∃ᴿx>0: NUF(x) = 1

>
Then NUF(x) does not exist
and infinity is not actual
and sets are not complete.

>
A potentially.infiniteᵂᴹ set is
an infiniteⁿᵒᵗᐧᵂᴹ set.

>
A collection.

>
A flying.rainbow.sparkle.pony.
>

An actually.infiniteᵂᴹ set is
a not.potentially.infiniteᵂᴹ set with
a potentially.infiniteᵂᴹ subset.

>
Subcollection.

>
Sub.flying.rainbow.sparkle.pony.
>
Merely changing a term doesn't change
what is referred to.

>
Potentially infinite sets are called collections
in set theory.

And I call them flying.rainbow.sparkle.ponies.
It doesn't matter what they're called.
Except that it matters that you are claiming that
something is potentially.infiniteᵂᴹ.
What do you mean by potentially.infiniteᵂᴹ, exactly?
I think you (WM) mean
finiteⁿᵒᵗᐧᵂᴹ:
  each subset {} or 2.ended
potentially.infiniteᵂᴹ:
  total order not finite
  one subset not( {} or 2.ended )
actually.infiniteᵂᴹ:
  not potentially.infiniteᵂᴹ and also
  one subset potentially.infiniteᵂᴹ
I suspect that finiteⁿᵒᵗᐧᵂᴹ isn't
what you want finiteⁿᵒᵗᐧᵂᴹ to be.
Definition.
For each B ⊆ A,finiteⁿᵒᵗᐧᵂᴹ: B is {} or 2.ended.
For each B ⊆ A,finiteⁿᵒᵗᐧᵂᴹ:
for each C ⊆ B: C ⊆ A,finiteⁿᵒᵗᐧᵂᴹ
for each C ⊆ B: C is {} or 2.ended
B,finiteⁿᵒᵗᐧᵂᴹ
For each B ⊆ A,finiteⁿᵒᵗᐧᵂᴹ: B,finiteⁿᵒᵗᐧᵂᴹ
Each subset of a finiteⁿᵒᵗᐧᵂᴹ is finiteⁿᵒᵗᐧᵂᴹ.
Likewise,
each superset of an infiniteⁿᵒᵗᐧᵂᴹ is infiniteⁿᵒᵗᐧᵂᴹ.
Potential.infinityᵂᴹ can't completeᵂᴹ.

I propose a very conservative answer:
that we accept at least
the empty set existsᴲ,

>
Does it?

>

Georg Cantor [...]
"Further it is useful to have
a symbol expressing the absence of points. [...]

>
Exactly. "It is useful".
>
However,
ordaining a symbol as "means this thing"
does not assert that this thing exists.
>
{} means "the absence of points".
Is there an absence of points?

>
Is this absence a set?

Is there an object in our domain of discourse
which "this absence" refers to?
Maybe. It depends on the discourse.
Is there a broader consideration preventing it?
If so, what is the broader consideration?

𝔊 means "the last natural number".
Is there a last natural number?

>
What is immediately before ω?
Nothing?
The empty set?

{} = 0 (von Neumann)
The set {ω-1} = {} of ordinals immediately before ω
is not an ordinal immediately before ω
I think that a good.sized chunk of mathematics
becomes unintelligible if we darkenᵂᴹ the distinction
between set and element.
I can't tell if that darkeningᵂᴹ is your goal or
merely a side.effect of the WM.cannot.be.wrong axiom.