Re: Replacement of Cardinality (infinite middle)

On 08/19/2024 04:18 PM, Jim Burns wrote:

On 8/19/2024 6:21 PM, Ross Finlayson wrote:

On 08/19/2024 02:43 PM, Jim Burns wrote:

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[...]

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[...] eg where your limit ordinal comes from
as a matter of definition or axiomn if necessary,

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Traced back, the limit ordinal comes from an axiom,
at least, it has if we haven't screwed up.
"Where things come from" is the job of axioms.
"What things mean" is the job of definitions.
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It's an important distinction.
You are the best authority possible on what you mean.
That gives you a certain freedom of action,
when it comes to _definitions_ which
you don't have when it comes to _axioms_
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Axioms have a tendency to being boring.
"Of course THAT exists. Duh!"
That is entirely the point.
You CAN disagree with an axiom,
where you CAN'T with a definition, not sensibly.
I don't WANT you to disagree.
Whence "Duh!"
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Given Boolos's axioms for ST, (AKA 'Duh!')
⎛ ∃{}
⎜ ∀x∀y∃z=x∪{y}
⎝ and extensionality
we get that
the natural numbers exist,
finite sequences granting addition and multiplication
exist.
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But we can't prove 'Duh!', except from other axioms,
or, if we can, we've proved it's worthless.
(I don't _believe_ it's worthless, but...)
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[...] now that you've definitely declared that
your hedgerow has an inside and an outside,

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I have declared that my hedgerow has an inside.
I am agnostic with regard to its outside.
That suffices because
I am only talking about its inside.
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Note: All is not lost.
There are other hedgerows,
possibly outside of this one.
However, when I talk about another hedgerow,
I'll talk about only its inside.
That's all part of The Plan.
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>

I must imagine you've heard of the inversions
of the strip of Moebius or the bottle of Klein,
where mathematics provides examples that there
is a track over both sides at once.
Then, about that the class of ordinal is an ordinal
and needn't be given by axiom or relation to an axiom,
yet instead as a matter of comprehension over the class,
has that Burali-Forti is the guy for whom is named
the idea that the order type or ordinals would be an
ordinal thus extra-ordinary, just like Mirimanoff arrives
at that comprehension over ordinary sets results an extra-ordinary,
that it's some sort of ubiquitous ordinals in a theory of ordinals,
and that sets modeling ordinals must thusly have that these
are some of the land outside Goedel's Frege's completeness
or "the incompleteness", theorems _true_ about the _objects_
of _the theory_, yet, not consequent its axioms and comprehension
their derivations.
Of course, "Goedelian incompleteness" is, inside the usual theory,
because it just employs anti-diagonalization just like Russell,
Cantor, Church-Rosser, and about that much all make their inner tent.
That though it's allowed as basically saying something at all
about either consistency or completeness, of the theory at all,
has that it's about objects of arithmetic or a _number_ theory,
and so not necessarily as otherwise about sets, at all, except
of course as so modeled by them as what's a model theory the
descriptive set theory the meta-theory of all what we have here
as ZF and ZFC and ZFC with classes or NBG, all that as ZF plainly
then that if you introduce Choice a.k.a. Well-Ordering, then
you've also added an ordering theory and one where all the
elementary objects are ordinals, contra sets. (And there's
another one about parts, and another one about classes, and so on.)
We should be pretty grateful for Goedel except insofar as
just says something negative about what would otherwise
be the extra-ordinary, that it's so that your hedgerow
could be a great Moebius topiary, it is so though that
we can sort of line up Burali-Forti against Russell,
Knight and Jeffreys against Bayes, Combinatorics against Borel,
that some how in some constructive account following mutual
deconstructive accounts of otherwise their conflicting accounts,
there's a meeting in the middle, middle of nowhere,
right in the middle of one theory of it all, at all.
Now, this, "Plan", sounds interesting I suppose,
here it's "Foundations" which can only be one theory
at all, what have you got in mind over there? This
Foundations includes all of "standard mathematics"
then has some more besides, is yours the same
"Russell's big-top with Goedel on the high-wire"?
Here that eventually fills up with the Elephant output.
Or, you know, keeping the Relephant, ....