Re: Replacement of Cardinality

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.

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?

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

What is immediately before ω? Nothing? The empty set?
Regards, WM