Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)

On 1/1/2025 6:50 PM, Ross Finlayson wrote:

On 01/01/2025 01:14 PM, Jim Burns wrote:

On 07/29/2024 12:46 PM, Ross Finlayson wrote:

Or, you know, "infinity plus one".

>
Consider the definition of a finite.cardinal as
the cardinal #A of a set A
smaller.by.one than sets fuller.by.one
#A ∈ ⟦0,ℵ₀⦆  :⇔  (#A < #(A∪{a}) ⇐ A ≠ A∪{a})
>
If,
as might be expected,
infinity.plus.one is different from simple.infinity,
then,
under that definition,
infinity is finite.

>
It's "well-ordering the universe".

Please complete this sentence:
⎛ In "It's 'well-ordering the universe'",
⎜ "it" refers to
⎝

Yeah, I know,
you don't have a universe in your theory,
as you say that
there's no meta-theory your theory,
yet, what's that then, all one theory?

I think that a universeᴿꟳ and a universeⁿᵒᵗᐧᴿꟳ
are different.
⎛ In the formal sciences, the domain of discourse,
⎜ also called the universe of discourse, universal set,
⎜ or simply universe,
⎜ is the set of entities over which
⎜ certain variables of interest in some formal treatment
⎝ may range.
https://en.wikipedia.org/wiki/Domain_of_discourse
I have universesⁿᵒᵗᐧᴿꟳ for my theories, as I must
wherever there are variables, and
there are lots and lots of variables in 'my' theories.
I take your universeᴿꟳ to be
a unique, all.inclusive universeⁿᵒᵗᐧᴿꟳ.and.domain.
The logic (FOL) of variables and universesⁿᵒᵗᐧᴿꟳ
does not require an all.inclusive universeᴿꟳ
We only need to be able to talk about
what we are talking about, the current universeⁿᵒᵗᐧᴿꟳ,
whichever that is.
There are pragmatic motivations for talking about
an all.inclusive universeᴿꟳ.
There are also pragmatic motivations for talking about
only what we are talking about, the current universesⁿᵒᵗᐧᴿꟳ.
For example, if someone denies the existence of infinities,
a good place to start might be the universeⁿᵒᵗᐧᴿꟳ of finites,
which is itself not finite, and
which can disobey rules designed for finites.

there's no meta-theory your theory,
yet, what's that then, all one theory?

In these discussions, my bottom.floor logic is typically FOL,
the logic of variables and their universesⁿᵒᵗᐧᴿꟳ.
My meta.theory of FOL is the theory of
finite sequences of claims, each claim of which is
true.or.not.first.false.
I think that I've mentioned that.