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

On 01/02/2025 01:55 PM, Jim Burns wrote:

On 1/2/2025 2:07 PM, Ross Finlayson wrote:

On 01/02/2025 09:35 AM, Jim Burns wrote:

>

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.

>

"The Logic", ?

>
First.order.logic, AKA predicate.logic
variables which range over some non.unique universeⁿᵒᵗᐧᴿꟳ
of what we are talking about.
>
Claims true.without.exception for
members of the universeⁿᵒᵗᐧᴿꟳ ('axioms')
>
Finite sequences of claims, each claim of which
is true.or.not.first.false (axiom.or.inference).
>
Given all that, each claim in such a sequence
is true.without.exception in that universeⁿᵒᵗᐧᴿꟳ
>
A wider, all.inclusive universeᴿꟳ plays no role.
>

Is it, "not.ultimately.untrue"?

>
⎛ Student:
⎜ Does a dog have Buddha.nature?
⎜ Zhaozhou:
⎝ Mu.
>
I don't see how to make sense of
"not.ultimately.untrue" in the context of
true.or.not.first.false claims.
Following a revered (though different) tradition,
I un.ask the question and answer:  Mu.
>
There is no ultimate.possible not.first.false claim.
So, to be perfectly literal,
no claim can be ultimately.untrue.
Or ultimately.true.
>
Each claim is not.ultimately.untrue and
each claim is not.ultimately.true.
>
I doubt that my answer actually answers
anything even a little bit like
what you intend, so I un.ask the question.
>
>
>

That's very logicist positivist of you,
yet here there' a strong mathematical platonism
and it also arrives at a stronger logicist positivism.
Imagine ordinals in ordering theory
so they are primary
as sets are primary in set theory
then they're ubiquitous
and ordinals model sets as sets model ordinals.
"Numbering" and "Counting" are quite different,
though it's lost in either the other.
"Mu" is agreeable, "not yes, not no, maybe both".
Yet, it applies to itself.
"Inverse", here, is the rule, not "Contradiction",
the Principle of Sufficient Reason
needn't any Principle of Contradiction.
Also some strong constructivists don't consider it so, ....
So, besides notions like "zero'eth order logic",
how about agreeing for a strongly modal relevance logic?