Re: Replacement of Cardinality (ubiquitous ordinals)

On 07/31/2024 01:21 PM, Jim Burns wrote:

On 7/30/2024 4:56 PM, Ross Finlayson wrote:

On 07/30/2024 11:18 AM, Jim Burns wrote:

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[...]

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The idea that there's one theory for all this theory,
has that otherwise there isn't
and you're not talking about any of them.

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If I remember correctly,  your (RF's) name for
not.talking about
what's outside the domain of discussion
is hypocrisyᴿꟳ.
>
That sounds like you're delivering a value.judgment:
that we _should not_ not.talk about
what's outside the domain of discussion,
that we _should not_ for example, not.talk about
_all_ triangles when we discuss whether
the square of its longest side equals
the sum of the squares of the two remaining sides.
>
However,
it is because we are hypocriticalᴿꟳ (in your sense?)
that such discussions produce results.
"Conclusions", if you like.
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We make finite.length.statements which
we know are true in infinitely.many senses.
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We can know they are so because
we have narrowed our attention to
those for which they are true without exception.
Stated once, finitely, for infinitely.many.
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Non.hypocrisyᴿꟳ (sincerityᴿꟳ?) throws that away.
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You're talking about a field, I'm talking about foundations.
... Of which there is one and a universe of it.
Then, when these examples of just carrying forward inductive
inference get out, they do. That there's a class of expressions
that are outside of tertium non datur is always a thing.
Let's talk about completions and the infinite limit and
the continuum limit, there's always an inductive counterargument
that it's not so, not complete, not the sum, not continuous.
About triangles and right triangles, and classes and sets in
an ordinary theory like ZFC with classes, now your theory has
classes that aren't sets.
That analysis is sometimes catalysis, for anaphora and cataphora,
is a thing, and two things.
In foundations, there's a universe to account for,
there's nothing outside, or vice-versa, ..., and vice-versa.
Then ubiquitous ordinals you can also find in where, for example,
"ordering theory" is fundamental instead of "set theory", it's
a theory altogether with entirely different elements, then with
the decriptive approach of model theory, making models and giving
them names, the "equi-interpretability" here has that simply
the ordering theory's a bit simpler than set theory, and in
a fundamental theory the elements are simple.
So, hypocrisy, like Russell's retro-thesis, a restriction of
comprehension that goes along with other axioms that would
build for themselves a confounding confusing conflating
consterning counter-example, doesn't just go without saying.
Yeah, my mathematical conscience demands that hypocrisy is bad.