Prof. Hugh Woodin | Rothschild Lecture: Beyond the infinite

Speaker(s): Professor Hugh Woodin (Harvard)
Date: Monday 5th October 2015 - 16:00 to 17:00
Venue: Isaac Newton Institute for Mathematical Sciences
Title: Rothschild Lecture: Beyond the infinite
Programme: Mathematical, Foundational and Computational Aspects of the Higher Infinite
The modern mathematical story of infinity began in the period 1879-84 with a series of papers by Cantor that defined the fundamental framework of the subject. Within 40 years the key ZFC axioms for Set Theory were in place and the stage was set for the detailed development of transfinite mathematics, or so it seemed. However, in a completely unexpected development, Cohen showed in 1963 that even the most basic problem of Set Theory, that of Cantor's Continuum Hypothesis, was not solvable on the basis of the ZFC axioms.
The 50 years since Cohen's work has seen a vast development of Cohen's method and the realization that the occurrence of unsolvable problems is ubiquitous in Set Theory. This arguably challenges the very conception of Cantor on which Set Theory is based.
Thus a fundamental dilemma has emerged. On the one hand, the discovery, also over the last 50 years, of a rich hierarchy axioms of infinity seems to argue that Cantor's conception is fundamentally sound. But on the other hand, the developments of Cohen's method over this same period seem to strongly suggest there can be no preferred extension of the ZFC axioms to a system of axioms that can escape the ramifications of Cohen's method.
But this dilemma was itself based on a misconception and recent discoveries suggest there is a resolution.

[@nikitasmarkantes5046]

Σας ευχαριστώ. Τώρα κατάλαβα τι εννοούσε ο Πρόκλος όταν ανέλυε τον Παρμενίδη. Όποιος δεν σας έχει ακούσει, είναι πολύ δύσκολο ως ακατόρθωτο να συλλάβει τη διδασκαλία του Πλάτωνα στον Παρμενίδη. Σας ευχαριστώ πολύ.

[@tokajileo5928]

he is an extremely smart man

[@Achrononmaster]

@59:45 this was a beautiful vision. (Yet I am glad for my sleep Hugh said he did not want to get technical.) The Ultimate-L route to absence of justification for the large cardinal hierarchy beyond supercompact is not what I see as a problem, I think it's a feature (if U-L is true). The idea we can pin down the infinite from where we stand on the small cardinals is what would be troubling. If the "gift" of ultimate-L is to avoid a multiverse of set theory I think sacrificing justification for large cardinals is worth it.
It means harder work to justify larger cardinals, and "morally" perhaps that's what a good honest platonist aught to expect. (Excuse my language.)

[@tappetmanifolds7024]

If Ultimate L is true.
Godel's incompleteness theorems may disprove a subjective probability.

[@nunoalexandre6408]

Love it!!!!!!!!!!

[@nodders1]

"The mathematical story of infinity begins with the empty set" is such a triggering statement... truth to be told we (humanity) started with 1, the number 1, the count of 1, the smallest non-empty set. Then we asked ourselves, where did it come from? And thus worked out the origin, the empty set, from which all is generated.
Seriously, no small child starts counting from zero... they all start from 1 and so on.
Zero is very counter-intuitive!
It's when you start to think that 1 is the power set of the empty set... then you lost all sense of reason imo...

[@travisfitzwater8093]

Holomorphic projections?

[@j.503]

Not to be a wet blanket, but at a certain point this all seems to sound like metaphysics.

[@Achrononmaster]

That's a very funny comment. It certainly is not physics, but nor is plain old Peano Arithmetic.

[@synaestheziac]

Infinity raises metaphysical questions whether you like it or not

[@tombouie]

Thks buts infinite tend not to conform to fundamental algebraic properties.

[@Achrononmaster]

Not so. Transfinite arithmetic is not the same as finite arithmetic. If you claim finite number theory is "fundamental" then your statement is a little vacuous. Like saying Banana properties do not conform to Apple properties.

[@tombouie]

@@Achrononmaster
I'm a retired military physicist, not a member of the dreaded academic math cult, & clueless to your point, sorry.
However pls just follow me for a few steps:
Consider this number with left/rightward repeating digits: ...123123.123...
1st ...123123.123... is not finite numberr & then it's infinite ; & therefore all other such numbers like-it are also infinite (sooooooo infinite is a set).
2nd (...123123.123...) x 10^3 = itself (...123123.123...) ; then at least one infinite number contradicts basic algebraic properties.
In a nutshell when doing algebra, avoid those god-damned infinite numbers like the plague ;)

[@tappetmanifolds7024]

​@@Achrononmaster
If the apple was yellow and the banana was red and both are fruits then the argument for infinite and transfinite sets is eventually reduced to a philosophical argument.
On a clear day you can see forever.

[@tappetmanifolds7024]

​@@tombouie
An Army officer in the electrical and mechanical engineers would admit that Noether's IST (Invariant Set Theory) describes symmetries of infinite sets of polynomial equations describing infinite sets of polynomial equations.
There exists infinite sets of mirrors near black holes.
Do the research.

[@tombouie]

@@tappetmanifolds7024
Hmmmmm .... complete above my head;
I'm just an old-fashioned retired military physicist ; theories especially fancy-pants ones must conform to the credible empirical observation/measurement data it describes (& not vice-versa damn-it ;). Sooooooooo I search-for & dos fancy-pants math tos-dos-justs that (likes geometric algebra)
You might enjoy my experience with the cult-of-mathematicians over the last few decades (enjoy ;)
Physics Major vs Math Class th-cam.com/video/_qdP0pd3idQ/w-d-xo.html
Mathematicians vs. Physics Classes be like... th-cam.com/video/xPzR_D9qKeo/w-d-xo.html
Math Professors Be Like th-cam.com/video/vC1GTBm4ehQ/w-d-xo.html