Re: Replacement of Cardinality

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

Le 27/08/2024 à 02:06, Jim Burns a écrit :

On 8/25/2024 3:45 PM, WM wrote:

This function exists because
nothing contradicts its existence.

>
Except for the contradicting
consequences of its existence.

>
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.

In a finiteⁿᵒᵗᐧᵂᴹ order ⟨B,<⟩
each non.empty S ⊆ B is 2.ended.
In a finiteᵂᴹ order ⟨B,<⟩
no one can say
what a setᵂᴹ is,
what an orderᵂᴹ is,
what finiteᵂᴹ is.
Perhaps, in 30 more years,
these question will have answers.
An infiniteⁿᵒᵗᐧᵂᴹ order ⟨B,◁⟩ is
trichotomous and not finiteⁿᵒᵗᐧᵂᴹ.
No set B has both
finiteⁿᵒᵗᐧᵂᴹ ⟨B,<⟩ and infiniteⁿᵒᵗᐧᵂᴹ ⟨B,◁⟩
A potentially.infiniteᵂᴹ set is
an infiniteⁿᵒᵗᐧᵂᴹ set.
Perhaps. When it's convenient.
An actually.infiniteᵂᴹ set is
a not.potentially.infiniteᵂᴹ set with
a potentially.infiniteᵂᴹ subset.
Perhaps. When it's convenient.
However,
no actually.infiniteᵂᴹ set is
a not.infiniteⁿᵒᵗᐧᵂᴹ set with
an infiniteⁿᵒᵗᐧᵂᴹ subset.
Definitely. Always.
Because
no setⁿᵒᵗᐧᵂᴹ of any kind is
a not.infiniteⁿᵒᵗᐧᵂᴹ set with
an infiniteⁿᵒᵗᐧᵂᴹ subset.
Definitely. Always.
The quest continues for
any kind of meaning that would fit your words.

¬∃ᴿx>0: NUF(x) = 1

>
Then NUF(x) does not exist

What exists?
I propose a very conservative answer:
that we accept at least
the empty set existsᴲ,
adjuncts of existingᴲ sets existᴲ,
definable pluralities of existingᴲ sets existᴲᴲ,
definable meta.pluralities of
existingᴲᴲ pluralities existᴲᴲᴲ.
⎛ ∃{}
⎜ ∀x∀y∃z=x∪{y}
⎜ ∃∃{z:P(z)}: ∀x:(x∈{z:P(z)} ⇔ P(x))
⎜ ∃∃∃{zz:P(zz)}: ∀∀xx:(xx∈{zz:P(zz)} ⇔ P(xx))
⎜ set.extensionalityᴲ
⎜ plurality.extensionalityᴲᴲ
⎝ meta.plurality.extensionalityᴲᴲᴲ
 From that very conservative proposal,
making the usual definitions,
there follows definitely always
-- because not.first.falsely --
our usual claims, claims which contradict
your usual claims, those allegedly.uncontradicted claims.
----
https://en.wikipedia.org/wiki/General_set_theory
| Axioms
| ST is GST with
| the axiom schema of specification replaced by
| the axiom of empty set.
https://en.wikipedia.org/wiki/Plural_quantification