 in progress seminar of today, and I will talk about the multiverse in mathematics. So, first of all, I will briefly describe what's the universe in mathematics, and then I will present three main view on the multiverse in mathematics. First of all, having some generalities on what it is and why we have this conception, and then three conceptions that I find interesting, that are also the main ones that are discussed in the philosophical literature. Woodin has both of you on the multiverse but also on the universe. So, I will present the arguments of the people, the structure of the conceptions, and then I will try to discuss and compare these views, and then I will give some conclusions and perspectives of the work. So, first of all, the universe in mathematics, from an informal point of view, the universe is a realm within which mathematics are taking place. It can be some conceived part of the field, but it can be all mathematics that take place in the universe. So, roughly speaking, it can be interpreted as the domain of discourse of mathematics. So, a universe is a collection that contains everything you need to deal with a given mathematical situation. There are different kinds of universes and different conceptions of it. First of all, it's a theory. There are two main universes of sets, the phenomena universe, which is the cumulative hierarchy of sets, and also the constructible universe of sets, I found the universe of constructible sets of Gooden. There is also another conception of the universe that is given by Grotem Dick as the set of infinite sets that are closed by all means, UF, to construct sets in the framework of ZFC. It means that this universe is complete in the whole operations of set theory that are needed to make ordinary mathematics, let's say. But also, categories can also be interpreted as universes, because much of mathematics can be taught of a stake in place in the category of sets. So, that's it for the... There are also universe conceptions in type theory, but so that's a very broad term that is used in mathematics. It's a short question, not very important. I just wonder... It's a long one. How is Grotem Dick? How do you avoid, I think, like a quantum paradox? It's not very clear. It seems a bit naive, you know, to set the infinite set. Is such a set exist? I just wonder. Yeah, okay, for Grotem Dick, I think that, yeah, the Grotem Dick universe exists and is well-founded. I don't know precisely which axiom and at which place you can avoid to have that, but that's for sure that it's a well-founded universe. So, it changed the definition of set according to von Neumann minimum? That's a set that is made of infinite sets. No, I don't think it changes the definition of set, but that's the way it's building its universe that is properly made. So, the multiviruses in mathematics, so one can define the model of a theory as an interpretation of the language of the theory in such a way that you satisfy the axioms of the theory. And you have the forcing technique in mathematics which is a very powerful technique that has been developed to build models of set FC. It was invented by Cohen to prove the independence of the continuum of the thesis. And he starts from a given model that is satisfying the theory and then is building an extension of that model by joining a new element that is outside of this wrong model. It's a little bit you can consider G as a kind of imaginary set. And it is building a way that is what we call a generic set which decides the forcing conditions that are true. So the forcing conditions are the conditions that allow you to make that CH as a given value. And the result of this construction is so that the extended model, so M to G, is itself a model of the theory. So you end up with a new structure which is like the smallest possible collection of sets that is including your starting model that also contains G, but it's satisfying set FC. And using that technique, it could show that the continuum hypothesis is one of the undecidable statements of set FC. Because from Godel, if you assume that set FC is consistent, then you have a model, you can have a model of it and it can show that set FC and the continuum hypothesis is consistent. So to prove this, he invented the method of inner models and he showed that the continuum hypothesis is true in his smallest inner model, which is the unifrance of constructible sets. And then later on, Cohen did the same, but he showed that set FC is consistent with the negation of the continuum hypothesis. And in order to show that he used the method of the outer model, so that's the construction of MDG that he developed to show that. And so as a result, if you assume the consistency of set FC, as you can have both see it's true and false that is consistent with it, it could show that it was one of the undecidable proposal of the theory. And so from using that technique, people applied it and there were many other statements that were shown to be independent in mathematics. So that was called a kind of the independence phenomena in mathematics. And as a result, you have a kind of proliferation of conflicting models of the theory. And the multiverse was introduced to represent these conflicting models because they can be used to perform mathematics. And so the main philosophical question that arises from this is, do these forcing extension of the universe really exist? It means that are there really universes outside of the traditional cumulative here of your sets? And then in order to clarify a little bit the discussion and the different conceptions of multivorces that were developed, the reddit I will show, we can classify mathematical statements in let's say four types. So the ones of which we know the truth value, the ones for which there is research going on, but we will probably know their truth value. The third type is the ones for which we cannot know the truth value. And there is one more category. These are the statements for which a truth value does not even exist. So that's a possibility that is probably not universally accepted among mathematicians. So it could be a category that is totally empty. But from the philosophical point of view, that's interesting to consider also that kind of statement. So now I present the three main multiverse formulations that seem the most important and the most discussed. The first one is the one of Amgenes. There are two aims to the approach of Amgenes. That's to justify the set theoretical practice and to give it a formal framework. Because as I mentioned that you can build conflicting models of the theory. It means that you can work in one model and someone else in a conflicting model and developing all your mathematics. But then how do we represent the fact that it's something that is done in practice in mathematics? So that's a formal tool to try to justify that. And that's also a way to justify a conception in which models of set theory are considered to exist really in a strong petonic sense. So it means that it's really a realist approach. So these models are considered to exist really. And so we need a way to represent them and their coexistence. With respect to the settling of undecidable proposal, there are words in which the continuum hypothesis is true. In other words, when it's false. So if we want to deal with such a statement, we need, of course, a multiverse conception because we will deal with that kind of model. And for him, there is no need to settle the true value of CAH by introducing new axioms. You just work in the framework. It is the most convenient for what you want to do. A result, as a consequence of this point of view, is that you have many different concepts of set. Each of these sets can be instantiated in a corresponding set theoretic universe. So it means that you do not have just one single universe of set theory, but you have a multiverse of set conceptions that are all legitimate. There is some of set conceptions are preferable to others because they are not used. These are the standard ones that are used. But there is no reason to dismiss others. So none of them can be said to be the true universe. So that's really what we call the pluralist point of view with really the widest possible multiverse because all set theoretical conception can be taken on board. But the result is that there is not a clear background set theory conception, but just an interpretation of the multiverse that is showing the fact that a model of set of C can be considered as twofold. From the internal point of view of the theory, it's just a model. But then if you leave just a theory and that you adjourn between these model things you need to perform your mathematics, you have that universe point of view. So the second conception of the multiverse is the one of Van Allen. And the one was introduced, the idea that there could be, as he says, absolutely undecidable, there are statements in mathematics. So it means that there are undecidable statements that we will never be able to solve by any means just because they do not have a true value. And it's aim with developing this multiverse view is just to give a formal framework to such statements in case they exist. But he never says that he thinks that this kind of statement exists, they're just giving a formal framework in case you have that kind of statements in mathematics. But yes, it shows that by considering with the development of the forcing the classical meaning of negation has changed. So if you have two models of the theory, so two realities, and if you have one statement, for example, the continental hypothesis that is true in one model but false in another one, you cannot say that your sentence is true because it's false in one of your universes. But you can neither say that it's false because of course it's true in the first one. So as a result, it's neither true nor false. So if you now take the negation of that sentence as it is not true, can you say that the contrary is true? If you were saying that the negation of something that is false should be true, you cannot because you have two models and one of them where it's false, so as a result, the traditional meaning of negation is not true. It has lost that classical meaning. So the conception of the multiverse of Nanan is that you have a multitude of possible universes. And to have the truth in the whole multiverse means that you need to have the truth in each of the universes that are making the multiverse separately. So that's one of his conditions on the truth because of course the truth is a big issue in the materialist conception. He also, so as I mentioned, the negation lost the usual meaning of not true but it keeps the law of the excluded middle and the traditional principle of classical logic that's still valid in this universe. So for him, he introduces what he calls absolutely undecidable statements. So you have undecidable statements which are the ones that are true in some universes and false in some other universes and the ones that are neither true nor false are just simply because they lack a true value. So that's not that you don't have access to the true value but it's simply not possible to access because it doesn't exist. That's the fourth type of statements. And so as to deal with the issues I mentioned so for the issue of the negation and to keep the laws of classical logic he introduces, he develops a new logical approach that is based on what he develops which is the teams of team semantics. And the idea of this universe is it's not that every model of the actions of sect theory can generate a universe in the universe because if you would just have a collection of possible models depending on the axioms you choose you would not need the multivall structure. So you would just have talk about the collection of axioms. So for him, it's more than that. And it allows him to represent to give really a formal framework to this kind of the mathematical statement for which no true value exists in case these mathematical statements exist somewhere. So what he wants to have it's really a collection of universes not a collection of models. And then he wants to make sense of them from the internal point of view of the theory. So the members of this collection are not models but they are really independent universes. For the logical aspects in this multivall so the variables of sect theory range over all parallel universes of the multivall simultaneously. And he says that all these multivallances are by signal but they only differ at the edges. So they are really similar, but these are some small characteristics that they are different. The third conception of the multivall that I consider ill it's the generic multivallance of within. The multivallance of within it's a collection of possible universes of set. And it's a generic multivallance. So it's a multivallance that is generated from each universe of the collection by closing under the generic extensions. So that's really the smallest set of the countable transitive sets of set FC. So you consider the generic extensions only of these countable transitive sets of the theory. And the aim of within it's to find among all the collection of possible universe and all allowed structure the right one, the right universe or set theory where you can settle all set theoretic problem because for him every mathematical statement must have a definitive and right solution. So his aim is really to find that definitive solution. And for him that's something that's really important that's to really to keep the classical notion of set theory that should remain fixed. So a sentence is true if and only if it holds in each universe of the generic multivallance. Hello, in fact he developed that generic multivallance and to show that the multivallance view is not satisfying. And there are too many reasons for that. The first one is that he wants to keep a true notion that is fixed and that comes from the classical set theory. And then the main issue for him is that if you allow to have this proliferation of models of set FC, it's not compatible with the foundational role of set theory because there is not one, there are different models and each model cannot have that foundational role where to find the right one. So as he argues again that generic multivallance view he proposed the universe view in set theory and he argues for that view. So for him first of all there is just one unique absolute background concept of set. And this absolute concept of set is instantiated in an absolute set theoretic universe which is the classical cumulative hierarchy of all sets in which every set theory assertion should have a different truth value. So in practice you would be working in the different universe when you can find the right one where you can fix the continuum hypothesis for example and give it a definitive answer. So in order to better understand the meaning and also to be able to compare these conceptions, I've tried to compare them from different standpoints. So I will compare the two multivallous view of Amkin-Sanfa-Alanen and the universe view of Woodin and I will compare them from different criteria. The first one is the status of the notion of set. Is it really central? Is it the generic notion of set that is central or is it the model? Is there a background motion of set or is it any possible motion of set that is on which each model is based? Is there a shift from the set conception to the model conception and the basis element of the universe and of the theory? The kind of commitment you have for a mathematical object. So really the ontological status of the mathematical object, the notion of truth, the foundational status of the multiverse of the universe, what's the philosophical background and how is reality conceived? So first of all for the notion of set, so for Amkin's there is an array, there is not one defined set conception, but that's rather an array of equivalent set conceptions and many such concepts are for Amkin's they are sufficiently closely enough related to be analyzed from the perspective of a single set conception. For Vananen there is no real commitment. I would say there is no, it is really working on the level of, suppose that this exists then what happens? It will not commit to a particular notion of set, but this multiverse is constructive on the iterative concept of set. And for the universe view of within, there is certainly no proliferation of different concepts that would give a proliferation model. For him the right set concept will be the one that will allow you to fix and to give a definitive answer to given mathematical problems and find the right universe that is generated by that right set concept. I have a small question for clarification. You used for Amkin's the word equivalence so set as an array of equivalent concepts what's the meaning equivalent? Is it just like equally good or something like that? Or is it something deeper? That's not a mathematical equivalence they are all of the same status there is no reason to prefer one conception to another. That's what I thought about. Everything is possible. That makes perfect. Then so usually classical set theory is based on the concept of set, but for the multiverse there is a shift from the concept of set to the concept of model. So for Amkin's that's clearly the element that is really on which is a multiverse is built it's really on the model, the different models that are possible. For Van Anand he said that clearly these are not models but really independent universes that are the basis element of his multiverse. While in the universe conception of Amkin's of within, sorry, there is no shift to a motion of model he really sticks to a one universe conception. Then how are the, what's the type of commitment with respect to a mathematical object? Amkin's says a lot about the ontology. He talks a lot about the different sets concepts but he term is not a clear ontology, a clear element that is really at the basis because that's really all possible models that are set generated, which is something really that is very schematic and you cannot really compare from a formal point of view with that loose definition of the mathematical objects set that are at the basis of the multiverse. For Van Anand there is never a precise commitment for anything that's really the formal framework that he wants to give in case, as he says, absolutely undecidable proposal exists. There is no particular ontology. It's based on the iterative concept of set which is the common ground but he doesn't say that all mathematical objects need to be iterative sets. While in the universe view of within the commitment is very strong and very clear each statement has definitive and right true value in the correct universe. A small question, what is it the type for what statement? I just want to make sure the one that I wanted to have. What type? So you remember I introduced the four types of statements that we can consider. So the ones he calls the absolutely undecidable. Absolutely undecidable. So it means these are the ones that do not have true value. You can go back to the slide because it went quite fast. Yeah, it was when it was not possible to say true or false in all universe, for instance, or when it was true in one universe and false in another universe. So that's the type three. That's the continuum hypothesis. So these are, I would say, the simply undecidable ones. So the ones for which you, I'm sorry, I have to... You've lost that super... It's this one. I remember. I was trying to say that. Yes, I see. Okay, thank you. So for now, the comparison for the notion of truth. So for Rankine's, there are different set... The different set relative truth. And so there is a link with this concept, with this multiverse conception with model logic. So that's considering the possibility or the potential true value of statements. For Vananen, as I said, so it's a real problem for him that negation loses its classical meaning. So for him, there is just one notion of truth, and so he needs another type of logic that he provides, and that is based on what he calls the team semantics. And for Wooding, it's clear that the notion of truth is fixed from first order logic. Then, from the point of view of the foundational status of a set theory, for Rankine's, from the point of view, it's really difficult to reconcile the different notion of sets, because these are different notions that do not necessarily have common ground. And so it allows you to have different foundational models. So the question would be, if it could be, that there is a meta-theoretic notion of set that would be more the background and more primitive that all the other notions that are possible to generate the different models. For Vananen, it doesn't say anything about the foundational status of the role of the metaphors on that respect. For Wooding, there is just one foundational model of set theory, and the foundational role of theory is really critical and very strong in the sense that every mathematical model will have to find its definitive answer. And for the philosophical background, these three conceptions are realist conceptions of the materials. There are other conceptions of the materials, and these are ones that are conceptualized, but I did not consider them. So for the three, these are realist points of view. For Hamkins, it's a very strong realist and epitomistic point of view in the notion of the existence of models. But when it comes to the set theoretical notion that's really a relativism because there is not a clear set theoretical notion, in the background. The role of mathematical practice is really important. I think that's also one of the main motivations. For him, for designing these multiple roles is that all set theory conceptions that are used in mathematics can be integrated in the duty roles. For Anand, that's also clearly a realist point of view to represent different things that could be possible in reality. And for Wooding, that's really semantic realism. It means that the truth value, that's realism in terms of truth values of the mathematical statement, and the last criteria for comparison, as I said, they are all realist point of view, but they have a slightly different point of view on what mathematical reality is. For Hamkins, reality is really unique, but there are different facets. You cannot approach all these facets at the same time, but you have different mathematical tools that allow you to access one of these facets of reality at a time. So that's really the way these multiple roles are conceived. For Van Anand, that's very different. The reality for me is not at all multifaceted, it's stated in pen text, but it's as if reality don't fix everything because these absolutely undecidable statements are possible. And there is not a well-determined reality of mathematical objects, but rather multiple sub-realities that are represented in all the parallel universes that are part of these multivores. So the aim of the multivore is to represent the full reality with each of the parallel universes that correspond to a sub-reality. In the universe view of Woodin, the set theoretic universe is really the universe of all sets. So the universe is the reality and it must represent all the existing sets. The conception is that you have to find the right universe of set theory that represents the reality and in that view, in that right universe, everything would have a definitive answer, but the problem then is to have the way to discover the final truth when you have just one universe that represents all the reality. So the whole truth and chance, so I focused on three main multivores-realist conceptions. I think that all of them deal with different design, to deal with different type of mathematical statement. For Woodin, that's the universe view, so let's find the right universe so that all mathematical statements would have a definitive truth value that we would know. And his aim is even to mark the third type, so the statements of the CH type as statements that we are on our way to know the truth value. And the multivores of rankings represents really the type of mathematical statements for which we determine decidable. And for Van Anen, that's one step further, so the ones for which we don't know, there is no truth value. The multivores of Van Anen is very promising and very interesting because it gives a formal framework for the multivores from inside the theory with keeping the laws of classical logic, but he needs to adapt the logic and the notion of truth. So the question now is could the other types of mathematical statement fit into his formal context? So it would give a very satisfying formal framework for all types of mathematical statements that could exist. But of course, the strongest commitment and the most clear conception of the mathematics is of course even by the universal view because the truth as a very clear notion sets a very clear notion. But of course there is no, it doesn't say how to find these final truths. And also from the foundational point of view, of course the universal view is the one that is the most attractive. So there are still perspectives of course to this work. There is no answer to the main question, is to know if these passing expressions really exist. And also the kind of corollary question to this one, is the multiverse really a mathematical object in itself? Or is it just something that's a tool that we use now that will have a temporary life, that we use now to represent mathematics as we know them at this time of the development? Or will it really, will we always need this kind of view for considering mathematics? Following the criteria, the universal view is of course the most satisfying. The main reason, which is the foundational role of set theory, but there is still the big issue of how to find these right universe and how to identify. So anyway at this stage the multiverse remains a useful tool to account for the variety of models that are generated by the forcing technique. And so I was wondering if we could perhaps, but I don't know, make a comparison with the multiverse that is used in quantum mechanics. So there was the many more interpretation of quantum mechanics. So in quantum mechanics from that point of view, while that's the cat of running, we can be both dead and alive, and once you perform the experiment, you put an end to this conflicting coexisting state. So the multiverse would be seen as a tool that allows us to interpret our experiments, the one in which the ones that give you the result are dead and the ones that give you the result are alive. And I see the parallel between the two. So in mathematics there would be the existence of conflicting models of set theory. And the multiverse would be a tool or a structure that would allow to interpret and to represent this coexistence. But I think that the conception of the many words interpretation is very different because I think that in that conception the universe represents everything that exists. So it really represents everything. And it's inside the universe that there are these many words. While in mathematics that's rather the contrary. We have the multiverse with the parallel universes that are these many words. So that's just a guess. I don't know if it makes sense to compare these two approaches. If it represents something in physics and mathematics the fact that we cannot access the reality fully. I think that's it. Maybe five minutes. Let's have a five minutes break. Yeah, that's better. And then we shuffle the order. No, you can go. I just had a question about this slide. Okay, we'll go now. No, no. No, no. I took your break now. You had to explain your work. Okay, if you're ready. Yes, thank you. Yeah. First, I was expecting you to show a booklet where we could vote for Fabrice. I would vote for Vahana. You would vote for Vahana. Yeah, Vahana. Maybe because you presented it in such a good way. You mean there. Yeah. So my question is about this latest slide. I just wonder what is the... If there is a cause, a common cause for the reason that we consider multiverse in quantum mechanics and mathematics. For instance, you know, in quantum mechanics maybe we have a concept of set in somewhere which leads to indecidable statements. And for mathematical reason in quantum mechanics we start thinking about multiverse. But I guess it's not a set theory. But maybe we have some mathematical tool similar. So your question is, are there similar mathematical tools for quantum mechanics that lead to that conception? Yeah, yeah. You know, if the reason is not mathematical in quantum mechanics. I'm not a specialist, but I do not think that it's for a mathematical reason. But I think it was just to explain something that seemed well in reality that you would have coexisting states. To solve the measurement problem. It's to solve the measurement problem. Yeah, that's really just... The tool, as you said, to interpret the experiment. But I know that there are also other interpretations like the boom interpretation that allows you to get rid of this kind of conflicting representation of the wave function. But that interpretation is also some weak bones or weaknesses. Oh, they all have. Because each time you win something on one side you lose it on the other side. But if I can add something, if there is an analogy to do with what you explained the multiverse it's probably not the multi-world Everett version but the Wallace version. So because in the Everett there's perhaps of the branches but in the Wallace version, the dominant one today all the branches exist forever. They all exist and they use branching logic to define truth in the trees of the possible trees. So of course it's not exactly the same as you because you win parallel universe. But there is parallel branch with different values. Something could be false there and true there. The same statement. So if there is an analogy, which I'm not sure of, if there is one, it's not the Everett one. It's what we do with Everett today. It's the Dutch Wallace Saunders version of the multi-world because they still coexist. There's no collapse. All the branch, infinite number of branches, they still exist. The total system is the universe. But when you are at the branch, you are in isolation. Something true for you could be false in the other branch. So you have to continue to think that this is true for you but false in the other branch. The branching thing might actually also work for these approaches. It's not what people do but I find it an interesting thing to think about because you can extend universes like these outer models. So you can go look at which is my universe, in which a bigger universe can you draw your universe. So we are at this universe but you might think that there is somebody living outside of there who sees you as just one of the sets. And you could say that that is one branch to go out, one super-universe, one branch. And then another super-universe is another branch, something like that. I don't know. It's... Honestly, the question is because I have a feeling it's not only about the experiment program that we talk about multiverse and physics. When I listen to some talks it's also related to the fact that we can change some physical constants. For instance, we change the light speed and you have a kind of other universe. So I just wonder if we could have at some point the competition between different worlds. What kind of multiverse is this? It's another... There's the many worlds, which is a multiverse. And there's the multiverse in the string theory. Where in one you have a certain constant but in another one you have a different constant. Yeah, because I wasn't... They are not related. Sometimes it's mixed to two, but they are not the same. Yeah, such elements seem very different. But maybe this one is closer. It's a bit closer to the universe of banana. We have action on a different painting theory and maybe we try to find a common... If you have different physical ones, it looks like different models. Awesome. Yeah, it's not related to your topic. Yeah, no. Yeah, and also the conception of banana is of parallel worlds but another point of view on the multiverse is that there are more those that are intricate. So that's another conception that is probably a better interpretation for our routines. I think that... So there are different interpretations and different types of multiverse in mathematics and perhaps also in physics just not... I don't know a parallel world but intricate worlds. I don't know if it's... But I don't know to which point it would make sense. It seems as if... That actually gets to a question that I had. It's a nice transition. One thing that occurs to me... And again, I know some set theory but I don't know any of the literature you were talking about so I have a pretty fundamental question. But when I see how different... I think your comparisons are great and they show really nicely that these different concepts are really not the same. These are radically different ideas of what this is. So I wonder is there... Might we just respond to this and say... Well, there's not just one notion. We're using the word multiverse to refer to just too many different kinds of things, right? Maybe there's a sort of equivocation here. Like why... I guess maybe a way to put it is a question. So what makes it desirable to study all of these different theories as sort of being about the same thing as opposed to just saying well, look, these people have radically different goals. They are building radically different metaphysics. So yeah, I mean, what is it that makes them all multiverses? I guess it's maybe one of them. Your question is what do they have in common? Yeah, give it all the differences that you showed. Maybe the right answers to say that maybe only one of these should count as being really about the mathematical multiverse and the others are just confused. They're calling it the wrong thing or we should call it something else. I don't think I can answer from the ontological point of view really to do everything as in common. But I think that the idea is that to have a structure that would explain that it would allow to represent some conflicting things and then the structure you put with the multiverse gives you, I would say that it enriches your point of view on reality. For example, for developing other kinds of logic that would explain that. So that's a little bit to say that there is that it's more than a juxtaposition. There is not just one conception, but yeah, that I would say that it allows you to have a richer interpretation of one technique which is the forcing. That's all for thanks. Pierre? Thanks a lot. This is very interesting. I am not familiar with this literature. I've been interested and willing to study it but I didn't have time and strength to do it. I have a question though about Woodin's approach and the sense in which it is realist. Because you said that they all qualified as being realists in some sense, but it wasn't clear to me how Woodin was really. I think you talked about semantic random and it's not exactly... So it's a clarificatory question and perhaps another question which is related and probably gives a bit more indication of what I don't really understand. If I'm not sure correctly, so Woodin has more constrained notion of multiverse in the sense that he wants only considers possible models of sex theory that are generated in a certain way. Which means that... So one way to approach the question of realism but I'm really not sure that's the way to ask it but one way to do it is to say well, you have all these... You have a number of possible models of sex theory and by possible you mean anything that can be described in some technical way. And then the question is should we be realist about all these descriptions? Do all these descriptions capture real mathematical, or really possible mathematical universe? And here it seems that Woodin's position may be described as saying well, only a subset of those possible models will actually capture real mathematical reality and others they are just like... meaningless or like words also. I mean I suppose that I think some of them that are not generated are generally those that don't have the generosity property will not be considered as like genuinely describing sort of reality. And it is in case perhaps there is a way to compare within... Say that Woodin is actually a realist about those specific models. But I don't know if he would agree with this. I don't know if that makes sense but I'm right and if I am I know if he would agree with this way. That's true that the generic multiverse of Woodin it's really the smallest multiverse possible and I think that he would design it as really the smallest one and the universe of the outer model of ZFC for the continuum for example. So I think that he would design the smallest one and then already show with such the smallest conception he ends up already with issues for the truth which are really for him he cannot live with that let's say and so that's a way to reject and to show that even with the smaller multiverse you already end up in trouble with... So you said that it's a realist the point of view of Woodin it's really the semantic realism really in the truth value for him so the idea is that every mathematical model should be settled and so that's just to improve and to increase the research to find for example links between some undecidable proposal if you make the assumption with it absolutely if you fix that other statement and that gives you the right universe and the mathematical reality where you can work. So I don't know if I answer your question with that But really it's because you think every the others are wrong and we capture the reality and there's a... The thing that might be the problem here is that what you're realist about so he's not realist about the multiverse of course and that should be bullshit however the truth value has to be determined that's purely a realist tool that mathematicians use to get to the actual universe so he's... and that's what you mean probably that it's realist about universe not about the multiverse as being containing something more than just one universe does that make sense as well as the solution to this problem that we're confused about what he's realist about I guess so why is it so clear that if the claim of semantics is real it's just the claim that every mathematical statement has a truth value then of course I completely understand what it means but when people... let me make an analogy with the metaphysics of relativity because I think this is relevant and this is what I had in mind the metaphysics of relativity we'll talk about possible worlds in order to define the notion of possibility and electricity and then there's a question whether we regard those possible worlds as real things or just as assets that we use when we make calculations about possibilities and it seems that so you have this view of whether or not realism or realism about possible worlds which is the view that those things are there they are part of reality and so when you ask the question of realism with respect to the multiversal set theory I take you that this is a kind of like analogical question whether you consider all those possible models of set theory and you wonder whether they are part of mathematical reality or not and perhaps like realisms can be classified as more or less like strong depending on how much of those models it encounters like the really part of mathematical reality and if you approach realism in this way then it seems like within like it's very low there's a lot of possible models that you would not have a realistic attitude to unlike for example having kids and unlike also a little bit more being in that's what the kind of things I wanted to I mean it's more like to check that I kind of understand the point about realism because I just, you know wow this is not too much in any sense if she does then I'm happy yeah thank you I have a close related question about because it's I have difficulty to understand so so we start from mathematical universe has very loosely domain of discourse ontological domains of mathematics something like that we discover all these bizarre results the first two you present seems to give us a variation of the idea of domain of discourse ontological domain to make sense cementically of all these new results so oh ok you have this independence of very important it's because it's true there and false there so together cementing interpretation of it and Woodin says everything new is epistemological it's because we are confused every statement are true or false so this guy is clearly a reactionary it's the history of mathematics so we must have a motivation to say something bizarre like that so what is the motivation to just say that every big understandable proof of 20th century is just epistemological it's because we don't have the right one how could he say that then we take it seriously so I'm confused the question is the other way around the other ones are not taking seriously many mathematicians they still have this reactionary view there is among mathematicians they don't have to defend themselves these things have true values and we don't know what they are and there are some limits to our formal systems but we have tools to think about these models like as abstract constructions and that's good enough and maybe she will confirm is that wooden is the mainstream yeah in mass in non-traditional math to the extent that satire ever is non-traditional but yeah yes so do you agree with that so in fact wooden does not even have to worry because it's the main position every mathematical statement has a true value that's classical logic it's classical logic yeah it's that's what it matters no I found the result of I found the result of I found the result of co-ed was among and there's many of them in 20th century that people said okay even ZFC not the good, the weird ZFC push us towards that kind of things so sorry but we are cutting the server yeah but wooden is a satirist and is making research on the large cardinal actions and I think that from that point of view if you are not convinced that you can go forward to settling values of the mathematical statement which is working it would be useless to do to do what it does no so so I think that it's really the hope of being able to yet to further link things that are undecidable but that can be related and to complete the universe view that is the mathematical reality but this reactionary position is it just I'm asking a sociological question is it mostly satirist because in category theory quite sure people would accept that multiverse thing is complicated but satirism is true if you work on something in some of the site no I don't know because I'm surprised because I thought the lesson was coming from geometry we thought that everything has a true value oh there are some of the other kind of geometry things becomes very complicated everything becomes contextual you have a proof there suddenly it doesn't work there or another domain another and the multiverse that the two first you presented seems to be the same movement is that ok we have this result let's build a semantic where it makes sense it won't be anti-realistic and just everything goes will be something that will be sophisticated with reference with true value for me the within I don't understand the guy is just saying everything has a value so every proof that we did of understanding are all insufficient actually I know that I had the same reaction like oh this is really I'm trying to understand where this is coming from but maybe it's coming from satirism yes the best satirist around is he's a satirist he's like the best and satir is better than he is and satir is always been a pluralistic enterprise I mean even the most like anti-realist mathematician if they use the word sat they don't have this idea that there might be several kinds of sat like take the set of natural numbers that these set of natural numbers take the omega there is not like something like that or the set of real numbers these are these branches that are not algebraic and so it's as long as these satirist results they don't say anything about problematic nature of such hardcore realism they just say at the theories we have the extrematic approach does not give us any will not always give us an answer but we have these constructions of universes like constructible universe of Gödel that gives an answer to continuum hypothesis for example whether that's true or not so we have other ways than the extrematic things that are I think intuitively giving us access to that reality whatever that is and so there is there has never been a real deep push towards like in geometry these things seem to be equally okay but in satirity there is like deep discussions about whether C.H. is true or not and Boudin himself has changed his mind and gave very deep arguments why you'd either go for C.H. or not C.H. so this sort of relativism I think is very threatening to and very unnatural to somebody working with satirity and you're right that in category theory they found a way around it and they think that that's probably reason why it's superior as a method of mathematics but then there's problems like based around collections and collections are not well defined I'm not saying that they are superior I'm just saying that it's culture that seems to have other prejudice yes yes yes very different prejudice so like in these kind of proofs you know they use model logic so in these proofs there's also like boxes in squares like in model logic or how should I see this model aspect of these things they're investigating they talk about being truth is being true in all universes and all these kinds of things how do they how does it look like on a paper such a proof just to have an idea for the for Amkin's it's the framework it's the formalism of model logic for Vernenan that's more the formalism of classical logic I think really with the main I think with these teams and Amkin's it's just introducing just not the fact that you use a sort of ensemble of statements that would represent the statement in different universes and that has implications on its definition of the distinction but for the rest the framework I think that's really classical logic for Vernenan but in model logic for Amkin's with the paper of Benedict Lowe so with the possible quantifiers so for Amkin's indeed they have worked a real model logic with the boxes and everything and it's purely a semantic thing like the quantifier for all does not mean for all in a universe but something like it's true for all objects and all universes and the negation is something weird you have to do some the model you don't have directly not A being defined in terms of A but not and B for example being the same as not A or not B see the model law applies so this kind of recursive semantics that just looks a little bit like a traditional semantics but it's a bit more complicated but there's no explicit model connectives or something used but of course under the hood it has a model of A or B so I'm just trying to make a little bit sense of what kind of mathematics this is so what these research papers are about so this is it's not like we make some conjectures and we see how we can prove them with what kind of actions it's not like that it's more like provability of systems in terms of you know good little questions because I just tried for myself to make sense of the mathematics that is done these are questions presupposing it's not like a real question about the talk but just in Amkin's paper he shows different things and demonstrates some theorems to argue and to develop his point of view in that's more really that's just for me a logic paper than I would say that in Amkin's it's more like the traditional mathematics theorem propositions, demonstrations but you also have that for in the paper of Al-Naman but in all of these things are really satirical results maybe some mathematicians one day would use this to do their satirical these are mainly philosophical so it's really more like foundation yeah okay so it's really there's many philosophical papers that are built into this not just proofing there are papers yes, yes, yes but it's not really satirical no, I agree but you were pushing them that these guys are doing philosophy no, but they are philosophers and they are on a mathematical level I mean Amkin's is also a great mathematician but I don't I don't think he wants theorems that I don't think he sees his theorems that are related to this to be multiverse as mathematical okay it's not a mathematical concept of multiverse sorry, I can't hear you no, no so I have a question about the the costs of those multiversities if we look at the dialectics as how reactionary you want to be how reactionary you want to be or mainstream you want to be and how revisionary you want to be so you take within view as the most mainstream and then the two other I don't know how to classify them as which one is the most revisionary but to go beyond the within multiverse view it goes with some costs so I mean it's mostly practical to go and see if I understand what I say makes sense so if I understood correctly in the Amkin's case in the Amkin's view the cost of multiversities to have like semantic in the determinacy about the concept of sets I'm not sure anymore about what set means can mean several stuff and we have to accept that that can be seen as a kind of a cost we lose that understanding the impression of having an understanding I think this is an important cost and that cost for the other foolish is like we lose learning we lose we need to change learning which is also a big cost very big one I mean usually I'm fine I mean I'm fine with changing classical logic but but the way it's it seems very out of the sense that you have to change it in a certain way to meet the multiverse but it's not, for example you haven't changed classical logic to deal with like a semantic paradox and stuff but the semantic paradox is kind of very well not very well understood there's a structure that is kind of like robust and those are basic phenomena with analogies with sets and stuff kind of like and also you can see how to look at classical logic when you remove the semantic predicates but here it looks like a new like a new it looks like it seems much more analog than the kind of change that you make so do you see those things as costs for the multiverse view or do you see these commitments do you think they are problematic or I don't know if it's really the multiverse view that costs or if it's the fourth thing that costs because the multiverse for me it's certainly a tool I would say that it's a well-founded mathematical tool that allows you just to represent the consequence of a perfectly mathematically correct technique because the cost of changing the logic comes from the fact that starting from a theory from CETFC you with the same language and everything you can build models in which one given statement is true and the same model with the same statement that is false so I think that it comes from that that costs you that you have issues with the negation with the for truth rather than building the multiverse view I talk about each other by multiverse view I mean the philosophical view that takes seriously these multiplicities of models of course take it as a sort of fact that those models established by enforcing technique are there in a sense that we know that those models are possible in some sense we can describe them and we can use them to establish independent results and then there's the next step which is to build a philosophical view to connect those models to the value of mathematical statements which is what is trying to do in some way understood correctly and Hamkins is trying to do it in another way and what I mean by the multiverse view is the result of this second step which I take to incur some extra philosophical commitments on top of just recognizing mathematical results that are not enforcing I don't think that makes sense I think I see a bit Why do you see a big cost that you don't seem to see because why do you see a cost particularly heavy I don't know I suppose I'm not mathematical I'm not a sex theorist if something understood by self and in fact means like several stocks it seems like my epistemic position regarding what I do every day would be much worse so it would be like a bad news for me it would make the epistemology of slightly way much more complicated so I see that as a cost I don't know if this cost is the right way to think about it that's why it's kind of a question but so if you need to change your logic that's definitely a cost yes that's something to argue for you know when you're done branching logic you have this internal external that suddenly you have to think about that you didn't have in classical logic once you pay that cost it's not excessive it's very analogous to branching logic you know in the logic when you're in a branch and the true value can change the other branch but you have to look at the outside of the branching system and talk about the logic but you're talking about the logic of tensile modality logic of tensile modality so it's not a revision of classical logic it's an enrichment it's an enrichment it's just a few I don't think it's attached to the logic I disagree with that because the language of that theory is just in the case of Hamkin's the enrichment view can be defined because it's modality but in the case of Benena I'm not sure it can go that way because it just keeps the same filter the language of of of that theory the same deceptive connectives so it just changes the meaning of negation and of the quantifier of the branching so it's a good old revision of classical logic so there's a cost here which is I think I still don't understand this attachment to self-theoretical stuff probably I don't know since what science is completely stable and does not change even in mathematics I think it can change for good reasons and Hamkin is changing logic there's very good reason there's a lot of undistributability and problems but yeah I don't know if I'd like to it seems like so so you don't have to the reason why I present it as a cost is that you can just decide but who said that ZSC is like the complete truth about the inverse of set stone so that for us we have missed some actions and we just want to find the reasons to add the actions and initially do that we don't have to change logic and we don't have to say well actually we don't really understand what we mean by set you see so that's why I think those philosophical and very interesting I think it feels like so it's a very interesting situation but perhaps it's overkill to solve the problems that are raised by the independence results I think that's that's a view I think that makes sense to me I don't I cannot I cannot talk about mainstream because I'm not even my attention but I suppose it should be around but yeah I agree with you that you have a cost in terms of logic but on the other hand it's an enrichment in terms of interpretative power okay so of course some costs are right so that's a matter of creation like David Lewis he obviously didn't say that there was no cost for having interpretative possible worlds but his main argument was that the cost is right because that's the best way to solve the number of problems that he felt were important that the competitors could then solve so the fact to point out that there is a cost is not to say that the cost is a it's not a reason to reject I don't doubt that well it has a cost it's not for free and you might consider the cost like seriously before you just like going back you draw some cash just buy the stuff to use the capital I have to say don't underestimate the cost of your solution and say oh maybe I can have action to fix that thing I'm going to win okay enough enough all the cost should be all like assessed I think just maybe you could I think it's between the algebras and the centaurs well the centaurs they have this idea of objects and axioms and things and there's another tradition in mathematics that is becoming more and more dominant and morphism and things like that and my impression when I was studying mathematics was that the algebras are winning and the centaurs it's very but they are still the purest and not the problem and these are people working on the prime numbers in the department of mathematics they are like a sacred capital and they that they work on when I was an undergraduate that was John Conway everybody backed around looking at him like no one knows why he does any of the things he studies but it's all really cool so we're not going to tell him no he's very smart he's clearly the smartest man on this university so no one will tell him no about anything and maybe the question of cost depends on which culture you are and I'm going to study the game of life and large cardinals and people are like cool I wanted to add something to the discussion here about banana to defend the mobility so there is this change in logic is a little great cost but it's not intervening with mathematical practice and so you could say that this cost is like epistemically not that great because there is still the possibility that there are no type 4 sentences and then it will just collapse to complete classical logic and as soon as we prove something all the classical anyway all classical technologies and so on they are all old and all these theories they all get the value true so it's only if you look at it from a sort of very broad not that perspective not that perspective but a broader perspective than the usual that satirists would care to look you don't see the distinctions with classical logic but that it's still I kind of get the point at all why is theme semantics just one one of the ways that you could make this work I just want to add that I think it's quite useful to address this I had a question I was when we were studying the Van Aan paper together I did not properly see that he was basically only talking about type 4 that sentences that was the goal why he made this so I'm now wondering and it makes of course total sense like reconstructing why the way I read the paper but if you look at at the continuum hypothesis how does he approach is it a possible type 4 question or is it a 4 sentence or is it doesn't he say whether it is one or he thinks it's not one it's not the kind of thing that his system is supposed to deal with that's something that I should check but as I see it could be anything well known any sign or stuff as I see I have the feeling that for just a C H I would say that it's type 3 so because it's just that's an undecidable statement of ZFC but there could be ways to settle it perhaps from the large gardeners, axioms and all these work in set theory so I think that I would say that it's type 3 and that he considers something even on another level in his paper and when he says that because he really uses the term absolutely undecidable so I don't think that he would consider C H as absolutely undecidable but I don't know I mean it's good that there is nothing that forces him to consider it as not absolutely indecidable right? because we are still debating about this but you think that he would not see it as a that's my feeling but I'm not sure it has a definite truth value in his system C H I don't know if he says that I don't think so but if it's not type 4 sentence then it has to have it must have a definitive truth value somewhere but I have the feeling but perhaps that that the formal framework here first allows also to take into account type 3 and statements I don't see in all the developments any reason even in these semantics or why he wouldn't be able to deal with with the other type of statements I don't know he is able to deal with all of them but it's not the topic of the paper they don't they don't I mean they are mostly more systemic notions that the other distinctions like it's about what we can know type 1 and type 4 has nothing to do with what we can know I mean the distinction between 1, 2, 3 and 4 is not a nobility thing it's about whether it's absolutely decided or not and it's with those absolute that he deals not with these other distinctions of course it's perfectly compatible with the system that I'm sure you are right, type 2 or any model type 1 but you see it it's the distinction that is the system seems to be made for is between 1, 2, 3 and 4 so the reason why he proposed that kind of multiverse because so would the universe of amkens be able to also deal with type 4 statements I yeah but he, I guess amkens wouldn't be able to make a distinction between on our one hand between 1 I guess amkens but I don't know but given the speed that the talk you gave I would say amkens is more like interested in distinction between 1 on the one hand and 2, 3 and 4 on the other hand like there's the ZFC system that is fixed in all the multiverses all the universes in the multiverse and then some ones we can access and so on and the more epistemic constructions are important while I would say after your talk and my quick call with the paper for now and it's more 1, 2, 3 and 4 so it doesn't mean that either of the two cannot deal with 1, 2, 3 or 4 it's just if you're a hardcore universalist like wooden or it cannot deal, it doesn't exist you can deal with the category but there's no such thing as a sentence without truth value but as a proper multiversalist all these four categories make sense I guess so it means that we end up with two types of multiverse tools that are very different but that are able to end up the same kind of issue so what's the right framework between the two of them so I don't know so I don't know but I think that's something that it's just to understand your debate could you explain how and can you type 4 how would you understand type 4 if there's no multiverse it's also type 4 and on kids it's yes so I would say just it doesn't have this category it doesn't matter so much but if there isn't type 4 you can just one of them it will be true in some universe and also in some other universe that's enough it doesn't matter it's unknowable for us or absolutely undecidable that that is something you can this thing you cannot make but it has this distinction between internal and external and kids so from which point of view we would see that the internal and the external so it's something different than the one between so yeah it's true yeah that's true that I said that the concern of Amkin is more on type 2 and 3 and when is introduced yeah that's true the way I understand your talk is that Amkin would just say there's no such thing in my construction so if you wanted to treat it I wasn't asking how because it seems obvious to you I would say that it does not the distinction does not really matter because I think that it doesn't change anything to the construction I thought I'm confused now I thought that for Amkin when you're internal the internal point of view you have a definite value from the external point of view we could say okay it could be true or it could be false okay but this is not absolute in the city building I thought that absolute in the city building was something stronger or something else or I misunderstood absolute on the city building as I see it like from an external perspective but also from an external perspective you have to pick yourself in a universe in a bigger universe where you can look down on the two other universes something like that so how do you know that from the external it's true then false there or false looking at the different context yes I thought that absolute in the city building was the absence of a true value completely yeah it wasn't totally clear to me I had to think there was two definitions in type 4 like first is that there is no truth value and the other was like true in one universe and false in another but maybe I understood from from yeah that's I really introduced it like that that absolutely undecidable for me in Vienna means that that does not even exist a true value to the statement yeah that's that's it's not just that the true value is universe relative no that's it's it's just you will never have access to it whatever happens that's how I understand but in a universe it can have a truth that's I know that's thing yeah I think hackers would just deny this idea of truth simplicity there that's maybe the proper answer to give like you always are in a universe even if you're not aware of it I mean to do proper mathematics you have to imagine there being something surrounding you being a universe then you might consider that the existence of other universes but you can study them but you're always in a universe so it's always universe relative while this notion of truth this sentence is true or not I think Hamkins would not like that and so type 4 in a sense can perfectly happen but it's not I mean to the extent that there is then sort of a truth that is not depending on universe it should be one that is true in all in all the universe considering legitimate and that's basically just in ZFC actions and their consequences that are true I mean you don't have to mention the universe because whatever you will end up in will always be true but CH yeah will be actually clean and then if you see that way I would say then for Hamkins everything that is inside ZFC is going to be type 4 but yeah it's complicated and then we should ask him what he thinks of this type 4 category okay unless we are happy