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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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.
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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.
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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.
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... Of which there is one and a universe of it.
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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.
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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.
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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.
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That analysis is sometimes catalysis, for anaphora and cataphora,
is a thing, and two things.
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In foundations, there's a universe to account for,
there's nothing outside, or vice-versa, ..., and vice-versa.
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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.
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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.
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Yeah, my mathematical conscience demands that hypocrisy is bad.
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