Re: The Circles

On 06/29/2025 09:40 PM, Ross Finlayson wrote:

On 06/28/2025 09:24 PM, Ross Finlayson wrote:

On 06/28/2025 08:38 PM, Ross Finlayson wrote:

Oh, been a while, figure I'll post.
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Watching a speech of Dr. Woodin the other day, sort of like,
"well Scott keeps generalizing his theorem and our closed
bounded universes if you don't mind me calling the cumulative
hierarchy that, sort of make that large cardinals sort of reiterate
and I'm not sure whether a large supercompact cardinal is going
to read right when the ordinary inductive set has neither compactness
nor is it extra-ordinarily super, while we're calling Cohen's method
blueprints now and really that's Skolem when in model relativization
we're pretty sure we can't add any axioms without them confounding
each other".
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And it's like, try less axioms, arrive at the extra-ordinary immediately
and resolve the paradoxes up front, since there are at least three
different rulial regularities the foundness, ordering, and dispersion,
otherwise you're not going to have a good time, and yes that's provable.
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Watching some Dr. Tao, "me and my mental collaborators are really
pretty happy about being able to divide-and-conquer proofs, though
one may aver that the implicits in the derivation aren't included in
the usual sort of dimensionless analysis, as that with regards to the
general awe of Breen-Deligne, we've sort of neglected quadratic
reciprocity".
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So, "foundations", on the one hand, and "number theorems", on the other,
both sort of seem needing some ways to look at them, a little different,
to help that otherwise there's "the circles" and "the going around in
the circles".
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Pretty good from Mark van Atten on Brouwer and models of continua.
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When it comes to foundations it's me who I trust.  It's gratifying
that sufficient mental reasoners find it quite thorough.
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Been enjoying some of this Yufei Zhou,
at least he knows there's a difference
between the ergodic and combinatoric,
sort of like "Borel vs. Combinatorics",
and when he mentions Ramsey theory
he knows there's a difference between 2 and 3,
and mentions Bergelson and Liebman or Oprocha,
about the quasi-invariant measure theory,
and differences between finitary and infinitary, and
when discussing Roth and Rusza, at least
entertains the various conjectures, and mentions
when things aren't done yet, then though as with
regards to various recently "decided" conjectures,
there's yet for various "independent" features of
number theory, in the infinitary, about things like
quadratic reciprocity and higher geometry.
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So, my foundations is pretty much great for all that,
descriptive set theory and all.
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Of course it's know since antiquity that merely-inductive accounts
aren't their own grounds, and the other day ten years ago at a
festival for Charles Parsons, Van Atta quotes even Dummett
as something like "inductive set defines the integers, integers
define the inductive set: it's circular", in a speech about the
intuitionism of LEJ Brouwer.
There Brauwer, Linebbo, and Shapiro, they got a thing getting
figured out, "free choice sequences" or ubiquitous ordinals,
"well our S4+ is sort of S4 though it's really S4 minus, ...".
Then I learned from a talk of Menachem Magidor that these
days it's Paul Erdos who described this feature of that what
the independence theorems of CH are avoiding is Erdos'
"the Ugly Monster of Independence", since it allows one to
draw all sorts of contradictions.
Then people are always looking for "the next axiom" when
they're going to have to do without some that already are,
and quite calling "the cumulative hierarchy" the "universes",
they're not, they're just scopes.
Anyways "Foundations" knows there's stuff to figure out,
so something like my "Borel vis-a-vis Combinatorics" of
2003 is pretty great, when I discovered the super-standard
"Factorial/Exponential Identity".