Re: Replacement of Cardinality

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.
Also,
'set', 'collection' and 'flying.rainbow.sparkle.pony'
don't even enter the formal language.
Being absent, their change is no.change.
∀S ⊆ ℕ: S ≠ {} ⇒ ∃k ∈ S: k = min.S
What are those?
Sets?
Collections?
Flying.rain.bow.sparkle.ponies?
Something else?
All of the above, at once?
1. There is no way to answer.
2. There is no need to answer.

¬∃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,

>
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?
𝔊 means "the last natural number".
Is there a last natural number?
Questions about the language, answered.
Questions about what the language is about,
not answered, not yet.

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

>
Does it?

Is there an absence of points?
Set aside useful '{}':
Do you (WM) reject the axiom  ∃x∀y:y∉x  ?
It isn't logic which encourages you to accept the Axiom of Empty.
Not all consequences are logical consequences.
You have it within your power to make
what you are talking about useless, pointless,
completely uninteresting.