Panel Discussion - The Role of Foundational Studies in Mathematics

well good afternoon everyone welcome to our special panel session on the topic the role of foundational studies in mathematics so the format this afternoon is as follows I will invite each of the panelists to speak for about 10 minutes and then after that there will be a discussion among the panelists and then following that we will invite the audience to ask questions to any one of the three panelists before we start let me just briefly and quickly introduce our three distinguished guests here I think many of you will know them already especially those who are attending this logic summer school so from right to left professor Hugh Wooden of Harvard University's professor philosophy and mathematics in Harvard University he wouldn't graduated from with PhD from Berkeley and he was professor at California Institute of Technology and then you see black Berkeley and then currently at Harvard University I should also say that he was already professor for professor at Caltech before he got his PhD so that is a bit in the wrong order professor Angus McIntyre emeritus professor at Queen Mary College University of London he was formerly professor at Yale University and also at the University Oxford and I got to know him very briefly when I was a graduate student there but then I left and then he was a year and after Oxford he retired from Oxford in 1999 and then was a professor at Edinburgh and then later at Queen Mary University as she also said that professor wooden is a specialist worth world authority in set theory per set professor McIntyre is one of the foremost experts in the field of model theory with applications to to algebraic geometry and in number theory and lastly professor that's Lehman University of California oh sorry Angus McIntyre got his PhD and Paul Cohen at Stanford University Tesla Danis colorful what we Paul Cohen okay I'm stand corrected professors Lehman receive his PhD under Jerris X at Harvard University and he was professor at the University of Chicago before he joined UC Berkeley in 1995 six professors layman is again one of the leading experts of maybe the leading experts in recursion theory and recently with additional interest in applications or recursion theory to their frontier approximations and analytic number theory so to start may I invite professor and guitar Angus McIntyre to give you his opening remarks please welcome professor in McIntyre [Applause] thank you very much I've taken part in such things before I don't have particularly happy memories of them but let's try to do better I think I it's probably best for me to begin on them so autobiographical load and touch from seven things which I've liked and some things I haven't liked and then we do discuss a relevant particularly happy to be at this panel of these two very old friends and very esteemed so I grew up in Scotland in a rural area over knowing the mathematicians or philosophers in my family but we had an extremely good educational system excellent libraries basically facilities were getting critical things from libraries my name is somewhat precocious I went very early to primary school away from classes and so so with the time I was a young teenager it was reading very widely and they I suppose via things like from Russell s is a logic in Norwich I got in and learned about mathematical logic and I found it very fascinating somewhat easy to get started trying to do things and of course I was very fascinated by girls work and set the in general but it and I was already doing some slightly original things as a fairly young teenager if I didn't really intend to be a philosopher I was fascinated by these things I didn't intend to be a mathematician I intended to be a writer and I applied the University Joanna I said I was going to be with each other the system is covered allowed you to change the document at the last minute and I changed it to do mathematics and astronomy so I did two years of that I did some radio smaller and so on and in the meantime I was buying all of the chief and I started reading serious things logic and I became fascinated by set theory then when I'd like to plan to go after two years because the logic they are seems to me to be straight from the Middle Ages they don't we just got young the syllogism you along but I was taught by very very good mathematicians ranking a famous mathematician I went to see an analyst but they had a month of philosophy again again I had the good fortune to meet good mathematicians and good philosophers but inspiring people I also had the advantage that I access all hours of the day and night to the university library because the librarian had been in school with my father was remote parallel on the west coast of Scotland and there I began to meet seriousness that was tops he spooks so so I is much more interested in watching but there was also I was very much interested in philosophy I thought a great deal about it I didn't have philosophy courses and I was half my philosophy I did well in them and I was persuaded to go to Cambridge to do philosophy but this I do regret now I could have gone I guess I could have gone to the state's very young but the two years at Trinity College Cambridge which is the elite UK educational base I did very well in in philosophy and looking back on it now it seems to me as if I was living in some sort of comedy film or things that were being discussed see Antony unrealistic to me but I went to mathematics courses as well just for the parents age then I went to Stanford largely because I had heard about kocoum and also because I read things at leas for in crisis I could have gone to Berkeley and some other places but I chose to go to Stanford this fact has been held against me even in recent years by John Addison Burt he know that I didn't come to query and went to Stanford I said and they don't seem to understand why at the time it was cold it was new and so on and fascinated me that frightened we did on my own and so I went to the philosophy department at Stanford there was no real distinction from nothing philosophy there I worked within a scholar for this apply fantastic pseudo class yeah but after a couple of weeks I decided that couldn't stay with Russell I already didn't like what I was hearing but I like the math courses I was doing and in those days at least it was easy to move by honking traffic suface and said it needed to leave and he just said well just a little math department will send the money after I went over there they didn't experiment buried rapidly yeah I was offered a job in the philosophy department I'll akatosh my bed at Stanford but I once I started seriously on the mathematics again I I followed up they my many years of reading philosophy haven't had no influence on me at all most of it meant nothing you know most of the people I read who were writing with Austrian mathematics might as well have been talking about them the philosophy of monkey it was nothing specific to mathematics in my innocence a few remain had some marvelous courses at Stanford and separation after a lady and I had lost contact with my woman Dana Scott was a barber supervisor that saw me at this time but gradually things had learned in the remoter faster but algebraic number theory began to intrude and I wanted to try and use but I knew and while Jake Owen twins right without algebraic geometry and that is not particularly congenial to any of the people that logic stands at that but Dana was a very helpful supervise that he introduced little Abraham Robinson I found him very very congenial it was a very primitive timeframe for lodging for Roger applied number theory but I felt it was gonna blossom on the teaching at Stanford in so many areas was so fantastic I learned from many many things which others even to be able to to Jerusalem the modern theory but I have to say that overall my feeling is oh the one great event for me at Stanford was meeting Chrysler who was dogged in various interests in my life my intellectual life some of you may have met him I don't know but it was a very difficult personality who generally fought with every friend he ever had but with him I I spoke on the phone twice a week and some weeks received or some days and see five letters and they were wrong about everything but frequently they were about very specific lines in some recent paper by somebody with philosophical pretensions and these potentials didn't survive very long price of of going which he had been trained he was of course one of the biggest HEA have been infinity and then he determined my my philosophical wife and suppose it exists but it's nicely with no taste whatsoever for any organized activity the little ones eating and ie I mean I suppose we may be expected to talk here about foundational crises and things of this kind for me there's no foundational crisis that hasn't been for a very long time and under the influence of Fryzel I tend to regard their their prices that are in Russell sort of overblown people make too much drama and then they gradually just faded away that's probably enough for me to say just now I would like to say that I love say 50 many people think that having an enemy have said and I love recursion if you have a dabbled in both these are the real thing for me but even take a more to persuade me that [Music] there's very much philosophy mathematics but rather than that says he has got very much to do with the way mathematics evolves and housing oh thanks thanks CT oh it's fine Gus Angus all started with a personal some memories oh I had one one oh there's a lot of serendipity in mathematical life and when I was in the undergraduate I was an undergraduate at Penn State and my degree was in physics but I found me in the moment my last year or two is it I saw at university I my interest gradually I realized what I really liked about physics was math and so I decided I was gonna I was gonna leave the sciences and move into them into just pure math and I applied to graduate school and I got into various places and it came down to either going to Harvard which is where I went or going to Yale and it's because of Steve Simpson that I ended up in recursion Theory instead of model theory so it's it small small effects have made small forces can have major effects in your intellectual life so about about foundations and the role foundations in mathematics I I confessed to Hugh yesterday I said I don't know what I'm gonna talk about in this in this panel discussion because I don't this was yesterday afternoon I don't really have an explicit well formulated and defensible position on the role on foundations of math so he looked at me and he gave me you just look like just like that basically he gave me this kind of sardonic smile and he said to me well you have 12 hours to think of one so I thought that was a little harsh yeah but seeing that it was only it was four o'clock in the afternoon when he said it and he gave me the I said I kid I had 12 hours to think of it I realized it wasn't as bad as I thought because he was giving me time to eat and to sleep and to do other things I saw it was a being generous it didn't demand that I do it get a perfect one spending all my time on it so about about and I did I did take it to heart and I spent the rest of the day and to end some some time this morning before the lecture started thinking about it you know what is it that I really think about foundations of math and the way that that foundations gate gives insight into mathematical practice so it's I truly did have have not tried to distill an explicit philosophy of math but I do operationally have an implicit an implicit way of behaving and so that's what I think all that's about all I can report on so I see foundations is there's two kinds of foundations in that what one is I think common mathematical practice oh you have some emerging mathematical subjects as some some piece of a mathematical investigation and it initially it starts out intuitive and and you can be working on mathematical things without actually you're exploring a mathematical phenomenon and then there's a foundational effort that goes into saying what is it that we're talking about what are the primitives of the area that work that we are investigating what what are the what are the basic constituents of this of this area and we know a lot of examples but what are the essential ingredients right and then giving those essential ingredients all pulling them out isolating the properties which distinguish them from other mathematical situations and and arranging the subjects so that you have economy you've made everything very efficient because you're working with the with the heart at the heart of the matter right that's foundational it doesn't necessarily have to do with mathematical logic it depends on the area in which you're working get getting the math together so that that's all you know like measure theory like what the size mean right so what's what is extent mean that those kinds of things were them great triumphs in math and I think that they qualify as being foundational so so there's that there's the aspect that it's most familiar to me which is the other aspect of foundational is the part that that's one experiences as a mathematical logician so it's I think when meta mathematical considerations come into play there is a foundational aspect to that to whenever that happens it's it's all now my experience is my training and my my perspective as a mathematical logician is recursion theoretic and in in in recursion theory it's the the ingredients of defined ability or the ingredients of the sorts of constructions that you can use on a complexity of the objects that you employ now what what it what are the ingredients that you bring to bear on some problem and the recursion theoretic analysis is typically those ingredients will allow you to analyze problems of this complexity and no further that's the boundary on what did what the mathematical means you're employing can achieve like such and such theory is decidable it can be computed by algorithm if there's a good if there is a good example so first there's there's the logical aspect of saying such and such mathematical situation can be algorithmically or determined like the decidability of a theory that's one foundational investigation for from the point of view of involving metamathematics because you're analyzing you're analyzing what computations mean and seeing the limit of what they can accomplish the other kind of foundational activity would be to specify what is an algorithm not what is a program not what is a computable function but what is an algorithm and how do you identify that two programs are using exactly the same algorithm and so the the sort of precise definition of algorithm is yet I think to be satisfactorily accounted for okay so so there's there's the know if if you're thinking about recursion theory so I gave an example of where you recursion theory lets you tell distinguish between things that are decidable or undecidable it also lets you distinguish between when certain statements can be proven fanatically or require something beyond the usual usual fine artistic methodology Thanks all right so that's that's so the the best example I know the canonical examples that Paris Harrington theorem and as a side Harrington who stopped work on bottles of arithmetic and recursion theory and set theory model theory he once told me he never understood set theory and he never understood model three everything he ever did with recursion theory and it's because in also he's a genius it's he could have been working with I don't know he could have been working with two water bottles and in a peanut and you could have put proving those theorems it's but okay so it's it's on recursion theory gives you this kind of meta mathematical analysis I'll give you another example on where something that's it's foundational gives you mathematical insight Oh you've analyzed some you've analyzed some situation you prove that theorem here's a complicated situation like under what circumstances is a theory Omega categorical okay so there's a beautiful theorem at representing how many types it happens for each number of variables so then you could ask well is there a better characterization is there yet a simpler criteria to tell whether a theory is Omega categorical but there's a beautiful one you know all the accountable models are isomorphic that sounds like it's two quantifiers over the reals but then you get something about the feeling which is arithmetic could you have an even better characterization are you going to prove that there's no simpler characterization of that property of a theory well you have to show that it's complete at the level of the definition of the at the criteria that you already had so you can you can prove that so you can show that certain analyses mathematical analyses are optimal oh you can show that certain certain set of tools are inadequate you can you can make precise the notion of well here is some some proof that brings in all kinds of machinery is there an elementary proof that question is is made precise that's a foundational question in my mind but you have to define what it means to be Elementary in order to say yes there is one and more interestingly no there is no element we prove at this attack all right so formulating those kinds of questions involves understanding of having a mathematical grip on a meta mathematical investigation and I think that that getting at getting those those precise answers in mathematical terms to meta mathematical questions sheds a lot of light on the nature of math don't well I was an undergraduate at Caltech and I started out in analysis I loved analysis and I was given as a summer research project Collatz keys conjecture which turns out to be independent so that's why I ended up in set theory and so I mean it was a granted student at Berkeley so on the roll of philosophy in mathematics or foundational issues I agree with Angus I think the original role of set theory in foundations is obsolete we understand the basic notions that set theory was designed to clarify but there's a serious foundational issue for me now in set theory and and therefore also mathematics and that has to do with all of the independence phenomena so we have the conception of the universe of sets and of and in fact inception is meaningful it would seem to demand answers to questions like to continue my policies and that's clearly a foundational issue because what does it answer me you can just assume it so what criteria what's that how do you analyze that investigation and how do you make the case one way or the other I mean Colin at least the last time them felt that the independence of see the formal independence of Ch from the CSE axioms was the last word on the matter didn't well you know to me that's maybe not the last word there are other questions which are independent which in some sense have answers determinacy provides a good example of that so the basic questions about the projective SATs not in the algebraic geometric sense but simple sets of real numbers just look at close the open sets under continuous images and complements those are the projective sets and what can you say about them by the lebesgue measurable do you have selection principles or projective subsets of the plane those are all independent but deterministic is the answer and the entanglement with determinists with large Cardinals is a powerful case those are the correct answers so in some sense what's happened is we now have the correct axiom to second-order number theory is it what's the axioms don't give full information to second order number three second organ amber theory is the theory of the integers with all sets of integers and projective determinacy is arguably the correct theory for that structure much as the piano axioms of the correct theory for arithmetic so now the question becomes can one expand that understanding to be itself or can one make a case that there can be no such expansion well that to me is a foundational issue and the answer whatever that answer is is going to involve some philosophical issues there could be a dynamic there so that's the those are the issues them very much interested in now you could just say there are no answers there may be columns right ignore the determinacy success then you have another issue to deal with the conception of the universe of sax with large Cardinals makes a prediction that those large Cardinal notions are consistent now whatever your view on mathematical truth is the consistency of a theory is a meaningful statement it's an arithmetic statement hi zero one sentence now you could take the view that it's not interesting but that doesn't mean it's not meaningful and so it's an interesting fact that today at least as far as I'm aware no one has been able to make a credible argument for the consistency of large Cardinal axioms except for the consistence conception of the universe of sets in which those large Cardinals or something like them exist so that's interesting why can't there be another way of verifying consistency so if it is in the conception of the universe of sets and that is how you verify consistency existence gives you consistency then you can't leave ch as indeterminate it makes no sense to have a conception of the universe of sets with large Cardinals and give up on questions about small sets so there's a challenge there so what should the correct view of truth and set theory be and that's to me a genuine foundational issue whether it's relevant to other areas of mathematics who's to say you know we've only been doing mathematics for a few thousand years in a million year set theory will be the same ages number theory hopefully connections will up and down so anyway the independence and set theory the ubiquity of unsolvable problems in set theory is a serious philosophical issue to me in the very conception of the universe of sets and how does one deal with that I don't think about this the whole time Bethenny I mean you've obviously spent a lifetime in amazingly 5/3 possible solutions which you've seen in this garden I mean what did Cohen say so describe my first time I met I gave a talk at Stanford I was actually I went to Berkeley in 89 but I can give a talk at Stanford until around 2000 and at that time so the nice thing about an unsolvable problem is you can change your mind so at that time I had decided maybe CH was false because there was a candidate axiom that was settled the theory of P Omega 1 and as robust away as projective determinacy settles the theory of second-order number theory and so I'm just going to Stanford to give a talk on that that's I presume why was I asked so I looked at Cohen's book no I better read out bomb what he said as I read at the back of the book set theory and the Continuum Hypothesis and there's a phrase something like it's it's speculate that eventually it will become to be accepted that CHS poems so I gave my talk and all that stands up and says well I just want everyone to know H is independent has no answer and that's the last word on it and so then I said well wait a minute that's not what she said in your book or wrote in your book he said that you thought CH was false and I should know better and Cohen said no that's not what I said I said I think it will eventually become accepted that CH is false I did not say that I thought CHS false so at that time and him based on my later conversations with caller ID are very long conversation with him in Vienna at the girdled meeting he was a formalist he thought everything was in he speculated that you know Goldbach's conjecture was independent because you're mixing the additive and multiplicative notions of integers he literally said that so that was his view and I don't know what his view was in the 60s when he discovered for saying but as so forcing axioms was an appealing route and for not CH in the end I felt compelled that this move away from that and the trouble is it wasn't fitting into a framework for a theory of V having a solution to CH to me it can't just be CA or CH you have to have a global very V and so they're not many candidates for a global theory of E and you know it's kind of you I mean you could just play the logic game the axiom V equals L is a robust taxi very hard to get independence from that axiom gelasa nice phrase it's a semi completed taxi but it's false because it's a nice large cartons so you can ask the logical question is there a semi complete axiom for B which is compatible with all our cardinals and how would you verify that well that's something you could investigate on the consistency question there is another twice I've been on panels like this more than that but there was one in Munich the Russell meeting and the pfefferman was in the audience better men was it was a set theoretic skeptic and so in the course of the discussion I asked Fefferman is PD consistent and he said yes and then I said how can you believe it's consistent if you don't think it's true and he said well if it was inconsistent people are all very smart you would have found the inconsistency okay now it's funny because I had exactly the same conversation with Jensen I asked Jensen is PD consistent this is that oboro hawk and he said yes and I said how can you believe it's consistent if you don't think it's true same answer so those raised an interesting question you look at determinacy axioms you look at large Cardinal axioms cuz there's a certain almost syntax that is used to formulate those axioms some have been formulated that are do me to a contradiction but that contradictions never been difficult why there are difficult theorems in set theory every theorem is proved that something is inconsistent namely the negation of the theorem but why is it that the large Cardinal axioms that have been probed proposed and fallen determinacy actions that have been proposed and fallen or not many of those why isn't there a deep structural I think that's what was lying behind [Music] maybe even intuition so that's an interesting question and it comes back to the consistency claims how is it is there any way of justifying the consistent claims why is he up see consistent if you're a set theoretic skeptic no basis to believe in the CFC fancy no universe presumably the belief and the consistency of arithmetic it traces back to the strongly held view that the conception of the integers is a coherent conception but you can't really make that case as powerful as powerfully for separated because of all the independence is always so to a set theoretic Scott tip why is it to be consistent why are measurable Cardinals consistently why are super compact Cardinals consistent are they consistent so I'm how could you is there a way of authenticating the consistency of those notions that doesn't trace back to a conception of the universe obsess and this is a real difference I get asked a lot about category theory why McClain and I used to debate I would visit Ted in Chicago and how we have these discussions and you like to talk what he like they're talking so you know McLain strongly felt the categorical framework was appropriate framework for a lot of mathematics vinos interesting category theory can interpret set theory so it can interpret large Cardinal maybe don't use any attorney none of mine using Orlando there's there's not in the same games yeah so so why isn't if category theory is correct framework for mathematics then mathematical knowledge discovered in set theory should be rediscovered shouldn't it okay and this correct some reason it doesn't do the handle oh and I was just making the point that the the discovery of the hierarchy of large Cardinals which is a hierarchy of consistency strength seems to be unique to set theory that's all and and so one can speculate why doesn't that higher why hasn't it emerged in other areas that's all I thought I'm not making a judgment I'm just pointing out this is my response to the claim it is curious and we a category theory hasn't discovered its own strong principles it borrows them from set theory right but it hasn't discovered its own that's not enough it's not saying anything negative about category theory it's just saying that why'd it if the large cardinal hierarchy is so intrinsic one could I wonder what I have verged in other error other areas of mathematics I'd like to say something about the hierarchy which is one thing that really uh that I find very compelling about a a let's say a distinguished conception of seven is the idea that there's a hierarchy of complex a hierarchy of define ability hierarchy of strength and it's not only is it one-dimensional but it's well-worn and it's reflected in the wage hierarchy in Steel's theorem about uniformly degree invariant functions and the consistency strength hierarchy that fact that that there's this one one way to go and things seem to fitting everything seem to be measured by it in one metric needs explanation but to me that's a that's an astounding fact if there's so much independence why is it that so many things are comfortable in and things that did not come from the same sources of mathematical investigation yeah that's like saying what would I do if a large cartons were inconsistent for example that makes it kind of a crock because almost passes and what's my view set theory is at a critical juncture faces two very different futures depending on certain technical conjectures and whether there's some kind ultimate version and up to about two years ago three years ago it looked like they were going to be two different versions kind of flavor a and flavor B which in that view would have led exactly to the dilemma that you described where you'd have to maybe semi-complete axioms to borrow that phrase again but which are incompatible and that really looked like it was going to be the case and it turned out not to be only one flavor and survived if he could reach all of us so so it's hard to answer I would say if it happens then that challenges the very conception of he because it shouldn't happen it would be like an inconsistent I mean you know the other thing that could happen is just another way to explain a lot to cardinal hierarchy maybe nonsense hasn't happened so I'm one could speculate what if you know someone came up with a theory of witches and here the axioms who widgets and lo and behold or large widgets the existence of large with its cap shirts a large Cardinal are yeah I mean you know so no I don't know I mean you could ask whether the large Cardinals are relevant to the so-called core problems in mathematics that's less of an issue to me [Music] [Music] [Music] it would be very charitable Center I think cetera has a serious issue out of the independence the trouble and set there is you write down a statement about the uncountable and very likely it's independent how do you build a mathematical structure theory in that situation all you can do is generate a theory of possibility but that's not a structure theory that's the model theory is that theory and to me the model theory of set theory is a different subject in set theory and it's the model theory of set theory that has been the primary focus of set theory since Cohen because the you know the technique for independence is so powerful and there are many many you know exploring the landscape so I don't think set theory or transport mathematics has really had a chance yet because there's no setting in which to do it because of the independence so maybe once if there is such a setting and it really has a chance to develop a rich structure theory who's to say what will happen so that's part of this the issue that's facing set theory now and you know so I'm probably the only person who holds this here you know there's certainly a view among set theorists that want to think that the model theory of set theory is set theory but I don't hold that view that's like there are questions they're very difficult questions in the model theory of set theory but they're not questions about fee you know like I'll give you an example suppose you have a sentence in the language of set theory and suppose there are well founded models of ZFC plus fee suppose for every alpha there's at most one model of height alpha suppose that there are mega one many models different heights because that sentence implied V equals L and you have to be a little bit careful you can come up a trivial counter example so P you can ask the first question suppose you had a sentence fee in the language of set theory and it has exactly one wall founded model does that model does that model think B is equal to out and the answer's no as a counterexample student enough of Harrington did that so that's part of this model theory of set theory and these are hard questions but they're not be questions if that's and you know the spectrum problem for a well founded models of set theory is a really difficult question what's the spectrum well as a counting model place okay like I mean you can get kind of silly insight like suppose you have a complete theory in the language of set theory and it has countably many well-founded models but not but no more okay what can you conclude that theory uh it's a complete theory B is equal to hottest in that period that theory you can limit the metal arch Cardinals that can be present so it's it's constraining a model you know so but that's those are not be questions those are questions in the model theory of some theory and there many many questions like that and some of them are incredibly difficult or seem to be but they're not being questioned so I think a challenge for set theory is to figure out what B is and then if you can do that try to develop true transfer mathematics if that can be done what do you think is evidence in this investigation how are you going to acquire evidence then B as a certain okay so this spring I don't want to dominate its discussion large Cardinals seem to be the clue large Cardinals are fog and also forcing in the sense that more large Cardinals you have envy the more me he is divorcing and there been some recent theorems over the last few years which is quite surprising and and these are theorem the picture that's emerging is almost that large Cardinals are trying to tell you that some version of B is equal to L by proving the best Conte so this begins a solid a stick from super compact if V is something like L 2 TCH but no large Cardinal why don't you see an image but wait a minute if you have a super compact Cardinal the singular Cardinals hypothesis hold above the super compact the best possible result you can hopeful and just I'll give another example another feat I'll get two more examples another feature of Al it's not a generic extension to large Cardinals give any insight into that question but first of all is that even first-order well it turns out that if B is a generic extension of an inner model that inner model is sigma2 definable from parameters uniformly so we can talk about the inner models and be that it's a generic extension of so you can do the theory of that and make it a long story short those the tsuba theorem of about two years ago if you have an extendable card know which is stronger than super composite and you look at all the inner models that v is a generic extension of there's at least one mm-hmm that's exactly the kind of thing you would expect you can't prove that from super compact is false you have to ramp up the large Cardinals and then you get more evidence and then the last bit of evidence how does it end with it another feature of al is V is equal to ha now the hottest hereditarily ordinal definable sets introduced by colonel so when I was a graduate student and in Lausanne or afterwards in Los Angeles and I was at Cal Tech it was the cabal's not indeterminacy the idea that B is equal to HOD was kind of universally rejected these should not be hard now I've always found that and in a kind of weird situation but it's like the argument against the accident of choice oh you say the reals can be well ordered show me a well ordering of the real all right if every real isn't HOD I can show you well order the reals so you show me a real it's not at hard mmm you know is that it's just anyway so what's the the so where does hard enter the picture again with an extendable cardinal so if you have an extendable cardinal there is something called a hot dichotomy theorem which should have been proved in the 70s but wasn't and it says that either V is extremely close to hot in the sense that all sufficiently large singular Cardinals are identified as singular in hot and their successors are correctly computed and much much more or V is extremely far from home in a way that we don't know how to force so this raises the speculation that it's not a dichotomy theorem but V must be close to HOD if you have large Cardinals but wait a minute that's another confirmation of it saying he is close to define ability exactly what V is an l-like model would predict so this seems to be this emerging picture that you bring in stronger and stronger large cardinal notions they are new to the making forcing powerless because forcing above a super compact or an extendable Cardinal and preserving the Cardinal is extremely difficult most of the techniques that we use to generate pathology you can't do that for us and these theorems like the dichotomy theorem and the others are symptoms of that so yeah so there's evidence emerging perhaps that's something like a V is an ultimate L is is true now you need the axiom and then you need to know that there's no sconce here and well we have the axiom of the conjectured axiom and we have the conjecture the ultimate all conjecture which would confirm there's no Scots theorem okay but we don't know which way it's going to go in other words it's it's clear in a certain perspective there is an ultimate L but that could be the ultimate L could simply be that's as far as its analysis of large Cardinals goes and that there's a level of the large cardinal hierarchy that it cannot reach so it's ultimate but only for a certain kind of analysis that would be the dark future now the other way it could be ultimate because it is genuinely an ultimate L and it captures all our Cardinals well these Ultima neljä formal axiom implies V is not a generic extension so if you think of V is having a width and height it binds the width to the height forcing is not powerless as forcing changes with preserving height and so in a context of an axiom like V is ultimate now there's no independence by forcing the only ambiguity and V is the height large Cardinals and so that opens the possibility for developing a rich structure theory which we haven't really had the axiomatic setting to do to do that well again it could go the other way so yes there could be evidence I mean the idea that ultimate ala seemed ridiculous 10 years ago but that's because there were fundamental misconceptions about the nature of these inner models and [Music] they were a node so you know there was a misconception that seemed to convincingly convincingly predict there could be no ultimate owl that you could build hence there is no ultimate now it's just an endless thing endless road but that was based on an assumption that turned out not to be correct that's where we are well perhaps I think this is a good time to invite the audience to come up with questions maybe I can start with the first one so so assuming that everything goes according to plan we do have ultimate L and and it's accepted as the the right model to have the right structure to help does it mean the end of set theory okay I'll make two comments if the ultimen all conjecture is true then the real debate begins is the you know let's suppose that there are some kind of mass gnosis but except terrorists and they all embrace ultimately will not is at the end of set right now I think it's the beginning of set theory because you'll have an opportunity to develop you'll be free of Independence you we have you know you can start developing through the lands of ultimate owl you will see patterns that you do not otherwise see I'll give you one example now I will shut up determinacy is often studied and if you so that if you look at large Cardinals and you look at this you can form the smallest inner model of B containing the reals that's called a Lavar and if you have infinitely many wooden Cardinals or with a measurable above so say a proper class then the axiom of determinacy holds in this minimal model containing all the organelles and they're real and you get a very rich structure but is a dl of are that's the assertion that this determinacy axiom holds true or do you need to resort to the large Cardinals okay well let's take another proposition if you're gonna ask about the axiom of choice you want to know how simple of a well ordering of the reals can there be is there a well ordering of the reals and Ella bar for example well know of the axiom the terminus e hole so we have two propositions the axiom of determinacy holds in ll are maybe just the axiom of choice fails now there is no connection between these except for the obvious one in ZF see in Collins model for not CH the axiom of choice fails in Ella var right so there's no strain and the axiom of choice failing in ela bar but now look at that through the lens of I'm and L like model so throw out all the universes except in this hierarchy of approximate generalizations of L and now in that very confined setting through that very narrow lens ask the L of our question and steal showed that the axiom of choice fails in l a-- VAR in that setting if and only if determinacy holds so now you have a completely different perspective on why the axiom of determinacy holds in LLR it's just simply LFR is unable to compute the well ordering of the reals so in the old Elle setting this might be the pattern that there are routes to new hierarchy new strong principles that aren't formulated in the language of that we use to describe large Cardinals now they're going to be formulated in terms of complexity and without the confined setting of something like B is ultimate now you won't see the patterns that are merged so that would be an issue will that happen and if it does and new principles of infinity come out of that analysis then I think that starts to make the case for vias ultimen l if the route to discovery Oh strong principles past the principles we know now is in the setting of ultimate L and as a set theorist that becomes compelling evidence for ultimate now so but that's a challenge for that picture even if you have the ultimen all conjecture and you're gonna make the case or v is ultimate out you have serious challenges because when you build in this picture of ultimen l when you build this model you're only using a super compact cardinal so why are two super compact Cardinals consistent you're not using the second one so there's a foundational challenge there and how that plays out will affect how the axiom is viewed by Seth Paris and it's too early to say we don't even know if the axiom ultimen L is compatible with these large Cardinals that's the point of this of the ultimate objective so yeah so it's hard to work too hard to say at this point the kale I would I mean if I can only ask one question I will start with because I have more I will start with one - professor Sleiman you gave examples of meta mathematical analysis of various pieces of mathematics it seemed that these were examples of meta mathematical analysis of in a sense known piece of mathematics so my question is do you imagine that can it be that foundational or logical analysis can lead to a genuine mathematical progress in the following sense of genuine that we can gather some mathematical understanding but not meta madam I mean mathematical understanding of a mathematical but not met a mathematical phenomenon that is not understood otherwise I don't know if the question is sharp enough but can it be that logic can lead to a better theory of some non Mathematica thematic mathematical set of concepts and that wouldn't be understood otherwise are there known examples or can there be for example so I think that so they all interpret that question so that I can answer it and so the in the way I want to so I think I'm gonna take the question is asking on for an example of a a meta mathematically inspired understanding of some traditional mathematical phenomena which led to insight about that phenomena which then evolved our understanding and and in particular produced results that were not meta mathematically motivated so I think that there's there's all well we've been hearing about the lecturers this week of one such example I'll tell you one that I know I know for myself because I was involved in it there was a this had to do with CT mentioned I've recently been interested in questions of dial Fantine apart nation so they they are sort of questions had that caught my interest were questions about normality and an approximation of transcendental numbers by rational numbers so the question that we started out with was to understand the complexity of those phenomena alright so it was an index set calculation this is joint work with Veronica Boettcher and Yanbu job ultimately so that there we wanted to understand how is it that normality means that the digits are equally distributed asymptotically relative to a particular base or relative to that all the blocks of digits are equally distributed asymptotically relative representation and a certain base so we were interested in then what's what's the index set how complicated is it is there a characterization of Avadh how complicated is it to tell about a real number that it is absolutely normal normal with regard to every base or how complicated is it to tell of a real number that it's normal relative to some base and so so that involved in a a recursion theoretic analysis of how how the satisfaction of those properties evolves how do the school functions work right now is there independence between realizing normality in this space and suspending normality in a different base and there were results from the 50s and early 60s by castles and Schmidt saying that you actually won another done on a number theory property of four multiplicative dependents so two numbers are independent if they don't have any common powers all right so so we investigated that we figured out how how well what was the recursion theoretic analysis what was the index set and in the course of that investigation you know one has to understand how the skolem functions work for the phenomena and you have to have a complete understanding of it because you have to show that if you're proving an index set theorem you have to analyze to what extent those skolem functions can behave generically like the skolem function if you show something in Sigma zero for complete you have to show that that's the way that the Sigma zero for your choice complete works can mimic any Sigma 0 for described situation okay so then well then in order to do that sort of recursion theoretic investigation we had to see how the satisfaction of the number theoretic property worked completely and it exposed various various dark corners in the way that that's all things can happen and that led to to a characterization of exactly under what conditions for which sets of in which sets of integers do there exist real numbers which are simply normal to exactly the elements of this given set and to no others so it allowed to an extension of the the Castle Schmidt theorem in full generality base by base so it's it's it was it was a traumatic 'el investigation which revealed a sort of complete control over the way that things can be can be satisfied you know so all the possible an exhaustive account of all the possibilities which is what a index set theorem has to do right so and exposed you know possibilities that had not been articulated before and you could combine them to give a complete account a finer account of what the transcendental number theorists said done earlier so I think the answer is yes I think that there's a kind of in my experience so what would I know most about is a recursion theoretic perspective which can be applied to two topics other than purely recursion theoretic topics and it's a certain point of view in a certain collection of ideas and but mostly it the point of view is the relevant then but what kind of understanding you have to have to understand it recursion theoretically and then that has consequences that understanding can lead to corollaries and applications I also let me just follow up on that last that last comment I think that all the attempts to go said I think it's more interesting to take a certain mathematical perspective like you could say well there's the meta mathematical perspective there's the algebraic perspective the geometric perspective a certain mathematical to the things that are intrinsic to it now but you know from the point of view that you've evolved in the way that you understand things and then it it will reveal it those things that it shows in high relief which depending on different perspectives make different things stand out so it's more interesting to take the perspective you have and apply it to some mathematical thing about which you're curious all right and then understand that thing as best you can rather than I am going to go look for situations where where I can so I'm this is an implicit I guess criticism of some of our colleagues in logic who go to look for independence results on in a certain mathematical content and warp next to suits the independence phenomenon I think that's less interesting than to use the meta mathematical point of view and try to understand what's happening there whether you find independents or large Cardinals or whatever you find sand of the Fields Medal is working home little groups and like other one or two other eminent French mathematicians he's not entirely comfortable with the excellent choice of aesthetic grounds rather than anything deeper but he may be central use in in his work which is competent with essential use of various dualities vector spaces and he was not happy the trials that the issue and study at the same time is answer and Sara asked can I do about this how do I get rid of the excess I said because if we just plunge in and look around and try to learn that you would succeed it's not easily and then that cries all this way said balloon it's quite easy much easier more successful quite easier to see the statement you improve other groups are arithmetic it's also quite easy to see that the proof for which in the Fields Medal is a set FC and then you pass him to air went back out with absolute listening to be eliminated so this was new mercenary was happy with well it's certified home you know you have to interact with other mathematicians yeah there's something here Mike you might be able to be logical with the idea that now I'm going to change this field and use acknowledging you always mimic anything you my question is actually about the role of the foundations of particularly the logical foundations of mathematics in the role of Education in mathematics for example would something like a logic last should that be held at the same level as the kind of Quinns essential algebra analysis topology sequence that typically grad students have I think I think both at least both you and Hugh should respond to this question because both of you had been chairs at Berkeley math department I think every graduates to understand independence at some level I mean yeah so I would gain patient knows how logic it I mean you'll find as you become mathematicians and give talks if you give a department colloquium you can't assume any knowledge of logic is know and so kind of hard to give a talk under those conditions whereas you know it's an it's an asymmetric situation a number theorist comes in and gets their talk and you know they assume you know what modular forms are and all of that and so you know I think independence the fact that host questions have been shown to be independent and that there is a technology for doing that I think there should be part of the basic knowledge graduate students gain in mathematics you know [Music] so I would be not an obvious I'm biased but can you like I'm not saying that I mean satisfaction and structure and basic things like that should be part of the standard toolkit whether you use it or not and whatever one view on logic the incompleteness of their on tell them the great theorems of the modern mathematics I think a professional mathematician should have some knowledge of that I think if you don't know what I would if you don't know if you can't distinguish between something being computable and not being computable then you're missing out on a huge huge piece of the understanding of modern now I told people and they appear to join some of the use of things later and it also with occasional really nice sighting things I mean a one occasion a guy came and I was teaching a course from the stability so it was quite advanced the guy whom I didn't know before it turned out he wanted to attend this course he was an engineer but this time another government professor emeritus by far the best you there's another pre-made guy who just fastest time medicine wasn't these guys what is it is only well it did everything from the other hand movement was a compulsory course the logic undergraduate level Kakutani had insisted upon it with general topology impermanence concerns line approach was a very volatile changing for individual here that one point things are good in at the breaker only photo posted by day so we had a big ordeal in the math department of whether or not every math student shouldn't do this material very general topology Serge was defeated in the fold department but he never came over an answer he simply went down to the department office asked this department secretary for the curriculum for the course tour about my through in the whispered the passing another were strong views of the distance but later he he was very helpful to boundaries we interacted along with him problem it's some of the things I'm lecturing on this week we learn from [Music] any other questions yeah first one comment so I think that sets you he has really many say influence of the model see we said three of other fields because I came across been reading or hearing talk but also in topology or other fields they are certainly open questions which I couldn't solve for 5000 years or so and then they found out they depend answer depends on the model of set theory one is using so this is what I observed so that there is indeed some impact in getting answers to questions where one didn't know what to do and the other thing is a one question I have do you see any parallels in the situation of set theory and computational complexity because in computational complexity there's also a big lieu of assumptions and then people study how does this assumption implies those assumption there's only one big difference in the model theory of set theory if they show that two is some sums don't imply each other it is absolute in computational complexity is it mostly say if these things are not equal we can show that those things are not equal but actually we don't know yeah so but do you see nevertheless a bit of parallel in this situation [Music] well parallel maybe in this following sense and I guess some complexity theory and you can't solve the main problem you study the kind of relationships if you can't figure out the P is not in P then you try to understand sick landscape so in set theory I don't know how strong the analogy is but until you figured out what B is you study the landscape of possibilities but the difference is there's not a driving question B presumably the question that P versus NP is not independent and we don't have an analogous question in set theory had it well maybe the ultimen I'll but that were headed for so it's a slight yeah it's not like in set theory the community has a problem they can't solve and so they're exploring the possibilities around what might happen in the attempt to solve that problem that's how I and I evilly interpret complexity periods you have the main problem P versus NP and these are attempts at understanding that problem but that's at a very nice level and then we don't have a homologous either the question is that the area no you know it's not like the community is striving to answer a question that's well articulated and so you explore both you know the possibilities to try to get insight into the solution to that question you know solving C ages have a question because however you solver it involves a new accent you're not going to prove it from you think that's a flaw is it a flaw in set theory they did if there isn't an articulated Holy Grail well I think there are articulated Holy Grails now or it was a flaw yeah I have insider information that's true you know it's like a big area of set theory model and all the generalizations about with large Cardinals well what are you doing we're trying to build the inner model for whatever what do you mean so it's using and you're quite sure the only way to solve that problem is to develop in a model or whatever that means that's a Holy Grail power and they're very good versions of Ashley and her model theory and it's just it has a practical effect a good Holy Grail problem can be understood by the wider community so they have some feeling of oh you guys are studying that problem and then hopefully when you solve it they'll remember we'd you guys been talking about that most of the years so it focuses attention good Holy Grail problem focuses attention but if the problem gets solved by means other than what you intend you have to take that solution it's not fair to go straight oh that's not what I meant I'll give you an example when I was a graduate student the problem of building interesting models of set theory where the continuum was large was well very was well studied problem and now we know why that problem was so difficult and that's because forcing axioms imply the continuous album team so that's a explanation why it was so hard to build interesting models where the continuum was large but there was a a test question and that was the Borel conjunction so the brow conjecture is a set of strong measure 0 it is countable and a set of strong measures 0 of given any epsilon sequence you can cover the set by intervals would shrink according to that sequence so CH and are you easily get a counter example so labor was the first to show the consistency of Borel conjecture but his model had the continuum without too so people asked it is the Braille conjecture consistent with the continuum large so I got interested in the problem I was a graduate student and it turned out that it wasn't a very good test question because we take Labor's model with a brown conjecture and add lots of random rifles we preserve them so you could easily get models with large continuum and the brown conjecture without addressing the fundamental issue which was how do you do count with support iterations of forcing and get the continuum large yeah so that was not a great test question or a Holy Grail question even though it was much discussed but for inner model theory we have questions which I'm absolutely sure cannot be solved easily articulated questions which cannot be solved by any means other than solving our model problem quite comfortable so Tessa you know these Holy Grail questions are useful and you know and it's just it's good for people outside the community to understand like proof theory now well you know do the proof theory for ZFC well what's the test question yeah and I've never heard one articulated so it's hard to you know it's like what do you mean by doing the proof or no analysis or whatever well good to have a yeah I'm sure there is I mean you know ordinal analysis can prove equivalences that are otherwise not know like for example Paris Harrington is equivalent to Goodstein sterile that's the equivalence of two arithmetic statements over piano but they it's proof theory that's used to show the equivalence so presumably and to do the you want to see generalizations of that over stronger theories so earlier there was some discussion about set theory and category theory and it seems that for especially maybe a lot of current developers of category theory a lot of the purpose is maybe not to analyze foundational issues but to provide a language which makes it easy to do investigations in certain part of matter do you think that's it's one of the duties of logic to find foundations that are good for developing other parts of mathematics even if they're not so good for analyzing foundations itself I'm not doing that I think what people have done in Thomas TV providing something extremely coherent which fitted the analysis of sheaves and reach of this I know but very general situations not possible spaces anything like that and this length the language product introduction of diversity was beginning to pervade the geometry and it was already beginning to show us officer truly powerful method for obtaining the formulation most of the people were dominated with something that time range but in India just thought it was the most natural thing in the water they related matters to the kinds of things they were doing I don't I don't know what the thought of our safety efforts at writing in the several universes so I'm not very attractive and I thought suppose Lavinia is really reformed me for example he has petition all star witness in the paper and the main conjectures to which is a big big thing again Fields Medal paper word one of them that tried to calculate you need to count it for the million generously the size of the absolute values of Romania's fact eliminate alcohol knowledge it's very very sophisticated material and the at some point you have to embed it in numbers deleted through the line in doing that he said that the accident Rose is abhorrent to and of all costs he must avoid this doing this suppose it's not that difficult for the purposes he was doing he used have a finite approximation just a little bits of the mechanics nothing for what we're trying to prove of that woman get them into it but of course if I continuously try it was has something and so on this model it you suddenly do Norman bad so I think it might community buying large actual choice was no other thing they wanted they wanted much more explicit calculations I think though they develop something entirely natural about sheep situations of continuously marrying sponges it's huge number people came in and what I mean morticians team and small informal institutions in volunteer or higher order models I think that was a perfect thing to do one I don't think it is implicitly that this was the foundation for all of my something that should be devised thing which fitted perfectly the kinds of general abstract arguments they were doing by that anything else but I think it is the general hewed idea for people interested in for what hit my mines which I do take to be really understanding how mathematics has involved and I what changes are happened and so on are certainly needed and only fashion I was a command Ellison the achieved 240 civilians people do things like this in the common domains of another constant mathematics something we haven't discussed here I mean there's a great deal of talk now about this hold on tight previous foundation mathematics we have with the alleged abandoned over other foundational schemes that it carries within a means of giving we formalized proofs and verified something never claimed or sent me I don't think anybody would ever imagine finding today but at some point the mathematical logic community should discuss this more I mean the idea of the dominant ideas came at least one very great mathematician - I would like to see that kind of thing I don't know what other areas imbalance will be finding from new parts of mathematics cool reveal no social independent reps and connections from computer other kinds of computability for me in the world to be able to look out but not the idea that giving a mobile phone there she is changing is there any in the audience who works on moment of privacy love to have our super temperament mmm maybe in a thousand years at least super tech write your paper yeah and then instead of you know tacking it it'll proof check and fill in all the gaps correct but they do better tend to emphasize this aspect a little sensor emphasize this aspect to the fact that these big groups may have errors of them by even those errors okay I think that's exaggerated but that yell fire so there's two things you might warm up such a system that it may be for the first time I was some kind of general confirmation correctly audience a fool but also that great new ideas and events and prevalence it's not so clear that this is anything Americans well that's evanescent one of them is very Oh what's up we've we've overdone the business about it's been too much of a burden on set feeding to talk of it as the foundation for mathematics it was general language was appropriate at the time raiga muscle insomnia was achieved by was a very it was probably indispensable for the proper development of firstly measured feelings about very confident believing in the state I don't think Percy cities know so it's obviously don't I don't I don't see set theory as that kind of foundations for math I think we've grown up we don't need that kind of foundations for math that's well I mean it's the issues that set theory clarified in the early days are well understood now and those aren't issues anymore the foundational issues at set theory seems to be addressing which happened each other's and mathematics that are out of being dependent resolves that's a different issue you know if independence makes its way into classical mathematics like number theory you know if a PI zero to sentence is independent then you have a serious issue and trying to figure out whether it's true or not but we don't really have techniques I mean there's this one example it's the biggest gap I know of and in terms of what you need to prove something versus what you use to prove it and it's a really good example where a large Cardinals basically discovered something about pi night algebra and that begins and Labor's work on the left distributive algebra and so left distributive algebra you have one operation is left distributed and labor use large Cardinals to solve the word problem for the free left distributive algebra on one generator and then the horny came up with a classical group using brave groups but under the large Cardinal perspective something interesting was discovered and if there improved so turns out that for every n there's a unique stricted algebra with two to be n elements of dice and wharfing and and there's a natural homomorphism he just do a mod to the end from me lot to lot to the end from n plus 1 to a n so he just need to go on so then you take the inverse limit and you take that generator so these are all you know singly generated algebra so you take the inverse limit so some big left distributive algebra but you look at that generator and you ask is the southerner borough the inverse limit generated by that generator the free algebra hmm okay and it comes down to a very concrete question and these the sequence of finite out of us so you soon on the order of a rank to rank a little bit less that's near the top of the large Cardinal higher could the answer is yes ladies they're not known why is here the relevant statements a PI 0 to sentence and not know and it's known that it can't be done easily because really what there is is associated to this sequence of algebra so there's a function and it no it's no it's not primitive recruits so you have to analyze something that's not primitive recur and so it's a whole area it's called labor tables and everything so it's an example where large Cardinals because give a natural model for the left distributive algebra discovered a phenomenon down in the finite algebras prove the theorem there and it's still not known it art cardinal approval pi zero two sentence and that PI zero two sentence could be equivalent to the one consistency of the theory of large Cardinals it's a theory of large Cardinals and huge for each N and if that there is one consistent you get a yes well those are two pi zero two sentence that go on that would be like a because that's never happened but it's a good example where the large Cardinal perspective and as a route to discovery and you know maybe in the end a very complicated calculation gives the proof without using Oh number theory whatever we can it was the large Cardinals and it's uh you know I mean it's the concept of the demand question he gave an answer the question that comes to my mind in this context is why should we think that we had what why should we have a conception of university my finger yeah so what I was saying is if only just for the consistency of large Cardinals is a conception of the universe of sets in which those large Cardinals something like them is it then that seems to demand a resolution to the CH how you argue that magical Arden alarcon C exists in the universe of sets so you're saying there it's clear what's going on but by the way we have no idea what's going on down here at the bottom that somehow feels suspicious to me that's all so it's good I mean it's the consistency claims of large Cardinals falsifiable uh it's in how does one account for those you know you could just say it's uninteresting but you could say you could take no position I mean there's something really interesting about consistency claims so you know most famous problem in mathematics I'll write the remodel offices PI zero one sentence hey my favorite large card wooden card consistency of a wooden Cardinal or PI zero one I will declare wooden Cardinals are consistent okay now what about the Riemann hypothesis the Millennial Prize Committee had so little confidence in the truth Riemann hypothesis that in the initial rules they would not pay out for counter example what's going on I mean you know so imagine that someone proved the Riemann hypothesis was independent of PA or applied compiègne without resolving it are they suddenly gonna say it's true so what is it about large Cardinals that we're willing to make the claim that the consistency is true having now having verified we can't prove it so there's something there's something going on there and I have it clear I mean isn't there a man not a number theorist I'll have to ask this but seems to me there's overwhelming evidence for the hypothesis these things are usually very long personally so I've been working about some of the other papers which are very difficult and he was very much interested in being on hypothesis and I think I think he was probably the first person to ring analyze the universe's history I mean they were doing the same thing that we constructed analysis of the price of one father a few methods taken up by numbers in this way the abstract and I studied these carefully by him with the tooling and it seems pretty clear the touring at the end of his life when he was an expert so if you put a set of a bedding marker on my hypothesis now how would the shares are trading at maybe I mean I talked to are a lot of people who prove theorems based on the remodeling process so I think that the prevailing view is that it's true well there's this good here it's again although things are what else uses on predictions I saw a picture of enormously personally that's a few but in fact you've got a squint picture and you spread out over time and you get deeper and deeper into patience of course you get these two marvelous things and the list goes on they do they can locate the Zener is up to all of that's how far but there's many questions they cannot answer even little download the nervous thing that tries and was obsessed with but little ensure that the number of primes is connected to this one anymore what I'm going to go and they say other further as far as I could see in the beginning and then as a switch inside between the living the crew first of all using I use indicators and - all McGarry this isn't going to have to look the best in assistant where the social what happens the years to people that don't lease almost but twice on these libraries only two three years left to live I mean that almost suddenly weekly I was getting this is our night or someone remember a number of years and they spoke to them in detail a little reminiscent very much something that worked on things like that just you know where they are now in the tempo that there's some gigantic bomb and such that would change what happens before life but they kind of get it down even the people who guy he's stuffing live eternity gold buyers or I asked him they can't get it down to do really knowing nothing that'll blow that one blow down I mean so I think it's like the illusion even one us when those are also in terms of Compton areas you theory that doesn't mean that knowledge and I think quite explicit thanks the sectors we're willing to say many many pies are one sentences like a common large Cardinals are true but they mean I think if they think that me soundings and fantastical things of a movie distribution with Brian's in the bias in the numbers would be complete without seeing or take the distribution of things but in find a detail there are big differences which can be but is to make the distribution but you'll get more deeply on another scale is moment there's a bias to favor one over the other they can hold the least thing so they have tremendous and sophisticated math not suppose it doesn't anywhere near the natural question is about the when you have all this stuff and run the matrices which is too bad it's a picture fantastically suggest but it doesn't advance the computational so I still think they should pay off for Cameron yeah I mean of course I agree in the yeah I mean this is a please you know I may have all I haven't looked at the rules my many days no I did it my son I gave a talk on me when I processor 1 of the ICMS maybe he this sort of Express body maybe he was on the committee if they don't nobody he made it clear that it was goodbye newsrooms all he said it in a sort of disparaging waivers they didn't say windy mathematics but it was a there was a clear sense there quite a bit befit the large Cardinals are finding through three couple equals hell his goals because it is incompatible with large white destiny compatibility Cardinals according to the sign of well in compatibility is a song a falsehood it would be my dear it seems to me set theory the conception the universes so it's all about infinity and having large Cardinals and if you have a framework for V which is incompatible at large Cardinals seems inconsistent with the founding principles of the conception of the universe etcetera and so you're compelled to reject I mean for example if you soon V is equal to L all you I mean you gain nothing except a lack of information you no longer see the projective sets in their true form you I mean it's it's it seems to me absolutely compelling them he is not al you just wipe out or obscure a rich structure theory has been discovered so that's why would say compatibility with large Cardinals in some form know is essential and I proposed axiom for something like generalization if E is equal to Al the litmus test that is it better be compatible with large Cardinals or it's just it's after rejecting so that's that's all I mean it's a yeah that's set there is about except a set theory without big sets not set there and it's an interesting question whether it's kind of like we seem to have a robust conception of the integers right most people agree that arithmetic statements are true or false determinacy extends that the second order number theory fact it's harder to get independence in second-order number theory in the context of determinacy because then to get independence and number theory so there are examples of arithmetic statements independent from the piano accidents like Paris Harrington those analogs don't exist for PD it's much much harder so now raises a question well we have a conception a second-order number theory that's robust why can't we have a conception of V that's that robust and ISO immune to forcing compatible you know so is there such a conception of e how would you make the case that there is what criteria would have to be satisfied how could you make the case that there is no such consumption those are interesting questions I mean it's it can't be a slog up you can't be you understand the power of the integers and the power of the reals and the power of the power of the real that's a long way it's if there is a robust conception of he it's just got to be one axiom is there such an axiom and how could you verify that it does provide that conception or maybe you can I make a case that there can be no such accident and how would you make that case now in that latter case okay I'm so close with this I said that consistency traces back to the existence it seems to be the only mechanism we have well if you look at the large cardinal hierarchy in the context of the accident choice it goes up to one level and just stops the Coonan contradiction and it's a very ad hoc just runs into a wall if the axioms go further now the usual view is those axioms are inconsistent but that's not entirely accurate as far as we know they just imply the accident of choices Falls so we have in the natural language of large Cardinals these axioms which we now know imply the consistency of all the ZFC compatible larger cardinal axioms that we have so we have no basis for the consistency of these super large Cardinal axioms which deny the axiom of choice there's no ZFC candidate that's going to match among those that we know so if consistency traces back to existence and these axioms are consistent does that mean we have to give up the oxime choice so there's a serious issue here or maybe these axioms are inconsistent outright and they'll be the first examples of inconsistency in the natural language of large Cardinal axioms where the inconsistencies result of a deep structural analysis the point is without the axiom of choice the universe is a very difficult thing to analyze so so it could be that these actions are inconsistent but they involve really profound insight into the structure of me now it has to work in a choiceless context and there's that hodkin dichotomy theorem I said that if you have an extendable Cardinal this is in ZFC hot is either very close to be are very far from be and far from being away we cannot do so the conjecture is HOD must be close to B now that's a ZFC connection if that is provable in other words EFC plus and extend of a cardinal implies hot as close to B that wipes out the choices hierarchy so a deep structural theorem HOD must be close to V in ZFC with large Cardinals would translate into Z Z in consistency result and if it goes that way that's a good candidate for a deep inconsistency result in the natural language of large Cardinals pushing back on the idea that if you formulate a large Cardinal axiom and form terms of elementary embeddings or whatever in the natural language you just stare at it you'll either see that it's inconsistent right away or it's probably consistent and it would be a very strong counter example to that intuition so that's part of the dichotomy I think that set theory well it's always a great experience to be at a firm like this to listen to leading mathematicians to talk about the subject of mathematics and in this case from the foundational foundational point of view and from their own perspective in my entire career so far I've only had the chance of attending such forums on very few occasions I've counted not more than three or four and and today's forum I think was intellectually very stimulating and certainly I think personally I benefited greatly from from this the discussions that that took place and I believe that that the audience here especially the students the young people in many years from now in you reflect you remember the session today and it might even have some impact on your intellectual development so please join me in thanking professor wooden professor MacIntyre and professors Lehmann [Applause] so that answer session don't forget it Protestant class to attend

Channel: Institute for Mathematical Sciences

Views: 1,806

Rating: 5 out of 5

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Id: uPyb7LeCUks

Length: 127min 6sec (7626 seconds)

Published: Mon Aug 05 2019