Harvey Friedman bringing incompleteness and infinity out of quarantine
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Re: Harvey Friedman bringing incompleteness and infinity out of quarantine
#2His recent posts on equivalence theory:
http://www.cs.nyu.edu/pipermail/fom/2017-February/020299.htm...
http://www.cs.nyu.edu/pipermail/fom/2017-February/020300.htm...
http://www.cs.nyu.edu/pipermail/fom/2017-February/020301.htm...
They still don't strike me as particularly natural or simple, particularly when compared to my favorite example of incompleteness, the surreal numbers. The surreal numbers are defined via transfinite induction and can grow as large as whatever cardinal you choose, and there are good reasons to pick large cardinals (closure under various operations), with the associated complications.
> “The idea that there’s absolute solidity, a right and wrong, in mathematics—that mathematics has no real conceptual philosophical issues that have to be dealt with … I’m interested in completely blowing that up.”
First-order logic is sound and complete; it's only second-order and higher logic that supports Godel's incompleteness theorem. Since most math is done in first-order logic (plain ZF, not even C; I think this is what the 85% figure in the article is referring to) I'm pretty sure this is not going to happen.
Re: Harvey Friedman bringing incompleteness and infinity out of quarantine
#3> Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes...
If anyone else's train of thought got wrecked by the proposition of mutually-exclusive classes sharing terms, here you go: http://www.encyclopedia.com/people/science-and-technology/ma....
Re: Harvey Friedman bringing incompleteness and infinity out of quarantine
#4It's especially interesting given that P vs NP is suspected to be unprovable in ZFC. Waiting for further development!
Re: Harvey Friedman bringing incompleteness and infinity out of quarantine
#5It's especially interesting given that P vs NP is suspected to be unprovable in ZFC. Waiting for further development!
I hugely enjoyed a survey article by Scott Aaronson on P ≟ NP. Section 3.1 contains a discussion on the possibility of the question being independent from ZFC. (He doesn't believe this possibility to be likely.)
Re: Harvey Friedman bringing incompleteness and infinity out of quarantine
#6If you want to hear of Friedman in real debate there is great discussion in the foundations of mathematics mailing list archives that are public. There there is real lively yet high-standards scholar figth of first rate experts from all viewpoints. I liked for instance the 'myth of second order logic' theme initiated by S. Simpson.
It seems to me that the article is wrong when it says that the spheres recompounded bigger are a problem of large cardinal axioms. I think it derives from Choice, the C in ZFC instead.
Re: Harvey Friedman bringing incompleteness and infinity out of quarantine
#7It's especially interesting given that P vs NP is suspected to be unprovable in ZFC. Waiting for further development!
I'm curious why you say "P vs NP is suspected to be unprovable in ZFC".
What would make mathematicians suspect that (if you know)? I mean ZFC is not a toy system, it is obviously used in complex and deep ways to prove or resolve long-standing questions in ways that build on tons of deep results that themselves took tons of research. (By the way for anyone else reading, just so you don't get the wrong idea: ZFC is actually the standard set of axioms mathematicians work with every day. It's the normal way to do math.)
What would make someone suspect this one is impossible to prove/decide under ZFC? (Or what would make someone think it is independent under ZFC?)
"A lot of people have tried to prove it" doesn't seem a convincing argument. (Consider advances toward the twin prime conjecture, or the now-proven Fermat's last theorem, etc.) There must be something more that makes you say that.
Thank you for any answer.
Re: Harvey Friedman bringing incompleteness and infinity out of quarantine
#8There is a lot of ordinary mathematics that is outside of ZFC, which Friedman is well aware of but the writer of this article may not be. Grothendieck was not interested in abstract set theory when he introduced what are now called "Grothendieck Universes". He merely wanted to do algebraic geometry at a high level of abstraction. Similarly, Conway was apparently a bit dismayed when the formalization of his simple idea of "surreal numbers" required deeply unnatural encodings in ZFC...
My opinion is still that ZFC itself is unnatural as a foundation for mathematics, precisely because we have to do so much encoding to get anything useful out of it. And whenever you iterate "large" encodings you leave the universe of "ordinary ZFC". Conway suggested - and this is realized in modern type theory - that we should instead allow arbitrary "free" constructions, such as his surreal numbers, to extend the basic universe of mathematics. To some extend this can be encoded in set theory, but only with ridiculously large (Mahlo) cardinals.
This is not a direction that Friedman considers worthwhile, because he thinks that first-order logic and ZFC are inevitable. It's a shame that so many people on the FOM mailing list share the same view. There are good reasons why you don't find a lot of category theorists on that list anymore...
Re: Harvey Friedman bringing incompleteness and infinity out of quarantine
#9> Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes... If anyone else's train of thought got wrecked by the proposition of mutually-exclusive classes sharing terms, here you go: http://www.encyclopedia.com/people/science-and-technology/ma... .
Yeah. Typically you don't even use Choice though, instead it's Zorn's Lemma: https://en.wikipedia.org/wiki/Zorn's_lemma. Zorn's Lemma has interesting weakenings, like https://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem (they all amount to statements about particular types of ultrafilters: https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonst...).
Re: Harvey Friedman bringing incompleteness and infinity out of quarantine
#10> Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes... If anyone else's train of thought got wrecked by the proposition of mutually-exclusive classes sharing terms, here you go: http://www.encyclopedia.com/people/science-and-technology/ma... .
The "there is at least one class which.." doesn't mean "there is at least one class in that class of mutually exclusive classes which..."; it means "there exists at least one class at all which...". This is one way of phrasing the axiom of choice (although Russell uses "class" instead of "set"; in modern-day usage, we distinguish classes from sets, and AC applies to sets only).