Tarski, undefinability of truth, category of intepretations / discussion

0:00 Yes, this meeting, or the group that got it going, I don't want to make a great speech about that, but it's a lot of curiosity. A group of us, John Paulston in Bolivar, and it was like Gabe in Salzburg, and Volker Halbach, at that time in Constance, I met at, first of all, I met Walker at a conference, I became involved in certain questions, which was of technical, mathematical nature, concerning frames for possible world semantics, and Hannes was a joint author of a venture of a paper there, subsequently I met Leon, and it became clear to us all that we had common interests in various kinds of axiomatic and analytical methods. as applied to various issues in philosophical order. And we subsequently had a couple of formal meetings, first in Salzburg, second in Goettingham, and out of that kind of grew a desire to perhaps be a little bit more public in our discussions and we might perhaps have a series of workshops of an informal nature which we could invite people to come and give us some outside input. So this is the result of a supported conference from the British Academy, the first of what may be several other similar workshops, and I won't intrude too much on Albert's time, But just to say that we'll have just, I have here a slightly revised timetable, not much revised from that which you may have seen on the web. Today we're scheduled to finish at 5.30, after which we'll have some drinks in the Mathematics Department of Common Room, which is on this floor, I'll show you where that is. And tomorrow evening at 8 o'clock, I've got some tables at Sands, which is a Lebanese restaurant. Again, walking distance from here between here and the hotel, for those staying at the Rodney Hotel.

2:30 I'll just assume everybody's coming unless you tell me in the break after this that you don't want to come. As I said, we've been fairly informal, we've got a very relaxed schedule, speakers have been allotted 90 minutes each, they may or may not wish to use all of that 90 minutes, 30 minute breaks between talks. We could have a short break after the speaker's finished, and if they wish, we could then have a slightly more formal discussion session. On the other hand, if we don't wish to do that, or want to lend itself to a discussion, then we can admit. Very good, I'll bring this to the next door. I'll be better then on definability of truth than the category of truth. Okay, well, I definitely thank the organizers for having quite a great honor to open this nice little conference. I'm going to speak mostly about the category of interpretations as applied to definability. So for the people who are not into category theory, I think that beyond being able to read diagrams or nothing, it's supposed to be a very low-tech category theory. Okay, so maybe some... well, let's say something about the background. I've been interested for a number of years in things like degree theory for interpretations, and also logics for interpretations, but these things all have certain disadvantages. which is, for example, these vertex for interpretations, only German have lots of coding available. For example, the degree theory is not able to make any distinctions between different kinds of interpretations, etc. It was quite obvious that one should look at something called the category of interpretations, where you do not forget the specific interpretation. But, so, as soon as you think that, of course, it's very strange that it's such an obvious idea, so why has it not been done?

5:00 And I think there's a kind of clear answer. From the standpoint of category theory, this is probably not the most interesting category, so it's not a progress or otherwise. There are lots of interesting problems. Secondly, from the standpoint of model theory, it's in some sense too inflexible enough. So a model theorist wants more constructions than you may have in this category. And in fact, if you look at the fat hodges, so it seems that page after page he argues actually that simply such a category that I'm studying here is not the same. So in some sense it falls, the subject falls between modal theory and category theory, and for those groups it's probably not interesting enough to develop. On the other hand, I think there are some reasons, a kind of battery of reasons to still be interested in it. Okay, so here we have some motivations. So, of course, the first thing is that, well, maybe interpretation among everything in Latinx, They are everywhere, right? And we all know, so please ask your favorite example of interpretations here. So if you look at my list, then you see that also their uses are for quite different purposes, right? Interpretation of arithmetic and sense theory is, for example, it would be part of a foundational program, right? We replace numbers by sets for reasons of ontological parsimile. The interpretation of hyperbolic geometry and Euclidean geometry was there for the famous equiconsistency proof. The interpretation of elementary syntax in arithmetic was used by Goebel as a technical tool to prove this incompleteness theorem. And this is also, by the way, a very interesting example. If you look at the proof, lots of further features of the interpretation are exploited.

7:30 So it's not just an interpretation that he used, one with several interesting properties. There's a famous interpretation by Tarski of true arithmetic in a non-abelian group. So that was used by Tarski to prove that group theory is undecidable. So it's yet another use of interpretations. Okay, now secondly, there are some lovely metamathematical or philosophy of mathematics motivations. namely so they are simply the question what is a first or a theory and of course there's one kind of view in which the signature is really very essential thing right so morphisms for example if you think about it well the theory are very often structural morphisms and the inclusions and the signature but for many reasons you would want to say that we want to abstract away from the precise signature. For example, we have a series of partial orders, and so if anybody who has ever presented that in class says, well, we can formulate it with a strict order, and then you define the weak ordering using identity and vice versa. If you start with the weak order, you can define the strict order, but But of course it's all the same theory. And so there's a notion of sameness modulo interpretation that would be interesting to study. Well, maybe you have the illusion that that's simple, but it turns out that I think I found at least six answers to the notion of sameness. That's only limited by my lack of imagination. Certainly, this category can be used as a tool for conceptual analysis. So the example of Tarski's theory, Tarski's ideas on tools, is really an illustration of that. But there are other things, right? For example, what is conceptual analysis, what is the notion of scheme? So what is an extra scheme? For example, we say that Z-Coe satisfies food induction. That's a bit strange because there are no numbers in Z-Coe and so what do we mean when we say that Z-Coe satisfies food induction?

10:00 And, of course, you have to formulate that modal and appropriate interpretation, and so the kind of general notion of scheme, of axiom scheme, should involve, somehow, the notion of interpretation, etc. Okay, so specifically, of course, I want to look with you at the object meta-language distinction in the classical case, and it's clear if you think about examples. For example, ZFC defines rules for piano arithmetic that this is also not faithful to signature, right? So we simply switch signatures and we switch. And so, to understand Tarski's Triterium, we really should do that moly-loving. And also there are philosophical arguments involving translations and truths, and so all this stuff around deflationism, etc. There's lots of stuff there. for example. There's a paper by Volcker and Shapiro's Boudreaux, and then there's discussion of interpretations back and forth between sexy and rhythm. Also in philosophy, in philosophical logic, these things will occur. Okay, so I'm going to talk about that, and there's a kind a simple way of summarizing what I'm going to say is simply namely that the well-known pair of object language, meta-language is, so I will argue that we should analyze that as simply one arrow in this category of interpretation. The object-meta pair is simply an arrow or an interpretation. Okay, related ideas. So, Panu Rathikainen was thinking about similar things, but I think he put a slightly different direction in his research. Okay, there's another

12:30 kind of necessary, boring part to my talk, mainly I'm not supposing that we all know more precisely what an interpretation is, and it's also important to have some precision. Ok, so first step is reading, I'm thinking only of first-order languages, and when I'm thinking of series, RE series with sufficient recital actions, etc., I'm not trying to go for the most general thing, and I think that's, okay, there's a mythological remark here, I think it's not a very good idea to do that, because very often you need things to get out of hand when you try immediately to do the most possible. So what is an interpretation, so it's really built up in steps, right? Before you can say what an interpretation is, we have to say what a translation is, and a translation is that relates one signature of theta to another signature of theta. And so the thing is given as a pair of delta f. So delta is some formula of the target signature, so theta. So delta is the formula that tells us what the domain of the interpretation is. And f really does the translating, so f associates to any relation symbol m of rdn some formula in the target signature, f are with variables among p0, p, n minus. Now, if you really want to execute this whole thing, you'll have to be extremely careful because there's lots of occurrences with alpha on immersions that are simply implicit, etc. I'm not going to say anything about this, but there's some slight technical details. Okay, so what we do is now, so given that we have such a translation, we really can use it to translate and so the idea is, of course, that atomic formulas are translated to their images under F, which is under substitution of the relevant variables. The interpretation commutes with the propositional connectives, and it also commutes with the quantifiers, but here you see in a little detail,

15:00 And very often you can eliminate this relativization to a domain, but that's sometimes possible not. Okay, translations can be composed, so if we translate one theorem U to a theorem V, and if we translate V to W we can simply compose the translations and the composition of them. So, it's of course very simple, so the translation delta on the tau new is delta on the tau new and then relativize the objects to the external translation and similarly for durations. Okay, so that's simply it. Now, these interpretations are extremely syntactical things, and you never think of these in a syntactical way, right? So, the way mathematicians think of these things is as an internal model construct. So, what you do by specifying all this is specifying an internal model of the source signature inside of an external model in the target. Are the objects in this category languages or axiom systems? It's coming, yeah, so I have not said what an arrow is, right? So the translations are just, the translation by itself is not yet an arrow. So of course the idea would be that we want arrows between series, so action systems. But the translation by itself is not yet dead, right? Because there's two problems, well, one is does it work between the series and the other one is so when are two of these things the same?

17:30 But for the intuition, right, so if we have a model of the target signature theta, then the translation will give us an internal model of the source signature sigma, right. and note that this mapping from models to internal models goes the other way around, right, it's a contra-variant, it will give us a contra-variant filter. Okay, so the next stage is, the next stage is that now we go up from, given that we have relative interpretation and that's simply so we want to have a source theory and a target theory of the appropriate signatures and we want to have the translation really translates the actions of the source theory to theorems of the target theory right so we say So, such a triple is in relative interpretation, if it's so that for all sentences A, if U proves A, then V proves the translation. There's also a formulation with three variables, but that's very cool. You can do it, it's sufficient to continue. Okay, now there's all kinds of theories about, so when are two of these mappings the same? And it's not good to simply take these triples, because then you don't have any good properties. For example, you never have, you never can invert an arrow, etc., right? Because the thing will not present what you get in appropriate constructions, it's not really identity, something that is close to identity. So you have to have some demand. So the minimal demand I've been looking at is simply we say that two of these things are the same. When the target theory proves that their domains are the same and that as long as you take your objects from the domain, then the translations are equally relevant in V. So this notation, V arrow, column, delta, means that it's really a big conjunction, saying that the VIs are in delta.

20:00 So I think it's quite pleasant to write in this conclusion. Okay, so now we have a category, right? The object are the series, and the arrows are the relative interpretation of the mode around this identity. And I call this category int for interpretation. So the main fact of life here, and that's also, I think, an extremely important heuristic because it's very hard to think really syntactically, and it's very pleasant to think in terms of objects, so the human mind craves objects, I think. So, associated here, you have the multiplicator, right? So, given that we have u and v, and we have an interpretation k of u in v, we have the set of models of u, and we have the set of v, sorry, and the set of models of u, and we have this mapping that sends models of v to models of u. and so that's called a variant because the interpretation goes from U to V and the mapping goes from V to U. And so the important heuristic is really that the theory is something like a generic model, it's a kind of uniform way, so the best thing is if you think of the theory to view the models and so when the theory varies, then the interpretation is a uniform way of thinking of internal models, right? So if you vibrate the models, you get in a uniform way other models of the source theory. And so that's, so if you have, for example, the interpretation of PA in ZF, so with every model of ZF, you have an associated model of PA, right? Interesting question, do you get all the models of PA, well, that's trivial, that is not so.

22:30 So, if you look at all the ericmetical serums of the PAA, do you then get all the models of that type of serums? No, because for the non-standard models of ZF, you always get the rest of the saturates. So that's a kind of stronger restriction there. And one of my favorites, so that's hyperbolic geometry, Euclidean geometry. So for every model of Euclidean geometry you construct an internal hyperbolic geometry. Okay, so this... I'm not going to do too much about IMAS. that it would be nicer without a hyphen, but I'm not committed to always use a capital here, and then it looks strange again. So, strangely enough, there's a second way, right? So between models we have structural morphisms, and the models can be identified with modulo-isomorphisms and structural morphisms, right? And so it's to be expected that something will correspond to this. And so suppose that we have two interpretations, K and M of U in V. And suppose here we have a model of V, right? Now, that gives us internal models, let's say, K Oh, that's not... Let's say, P and Q Internal models of U here, right? Corresponding to these two interpretations And so, of course, there could be structural morphisms a structural morphism f from p to q. But now, of course, that is not syntactically meaningful, but in some cases you can, of course, completely, explicitly describe your structural morphism in terms of the theory v, as have been stated. And so between these So these arrows K and M from U to V, we can have a structural morphism F from K to M.

25:00 So that gives us a kind of cat. So here we have U and V, and here we have K and M, and here we have a structural morphism from M to K. So that gives you also a category of the arrows of your units. Okay, so now you can write down conditions, so I'm not going to tell you what is unmeasurable. I hope that you are convinced if you look at this that you can quietly sit down and simply write out what it is for a formula, f in the system of public variables to represent such a structural morphism between the internal models, right? So you, for example, say that it's total. Every x in the domain here, there is a y in the domain of the other ones such as x sub y. Okay, now you can play around with these things in various ways, but the most important thing is this one, and I'm also going to do that in the picture, and not have you worry about I'm just going to see if there's an eraser. I think the box doesn't come through. Yeah. Or shall I rather write here? Yeah. Okay. So here we have... So here we have our... model, let's say, M of V, right? And internally we have here models of U, and so we have an appropriate structural morphism between us. Now, of course, if you look at this model

27:30 M of V as internal to some model N of W, then, model of transitivity, these things are also internal models of U inside this model of W, and this mapping that is formulated in language the specification of it can be translated into the language of W, and of course it's also going to be a mapping of these things consider us as internal power subtilies. That's what the serum says, right? So you can compose this mapping with M. Yeah, it's all in some sense trivialities, but it's still work to Now, the other way around, unfortunately, does not work. So, okay, this is the other way around, right? So we have two models. So here we have a model of W. Now we have models of B sitting inside it. And we have one fixed model of U sitting inside it. more present from this one to that one, say, right? Now... Oh, yeah, yeah, yeah, the same. Yeah, so this is supposed to be the same internal model, right? Now the problem here is that, regrettably, we would have a nice two-category if this would lift, But it doesn't. Because the point is, of course, that this thing is amorphism with respect to the signature of V. But there's no reason at all why it should also be amorphism with respect to the signature of V, right? Because it doesn't preserve arbitrary definitional extensions. And so only in very specific cases will this be preserved, right? So either you will have to put restrictions on your COEU, or, well, that's the thing, or, so, well, you can formulate various further conditions to make this morphism here, this, two morphism, three, these two things.

30:00 So the only thing that I'm interested in here is that it has an isomorphism, and of course, as we all know, at least isomorphisms preserve divinable eternal knowledge, so that they do this, and that's the only thing that we want. Okay, so now I have enough material to at least define two notions of identity of series, namely, you and we are synonymous or definitionally equivalent if they are isomorphic in this category of interpretations. And that's the classical notion of the move here. So that's falsified models in the books. And there's a second notion, so if you look at our category int, now so what we have defined is between arrows in int we can have isomorphisms. And you can divide them out, right? And so there's a second notion of maybe bi-interpretability that says that U and V are biointerpeatable even only if they are isomorphic in this extent. Now, it's easy to... I still do not have a not-done example separating these two notions. It's easy enough to provide all kinds of interesting examples of biointerpeatable things where it's not clear at all that they are synonymous, but then try to prove that two series are not synonymous, right? So there's no good technology to prove A synonymy. A good example is if you have ZF, So, then we can add a set of poor elements and make some internal construction. And inside this, inside this, one of these poor elements we again have the pure sets.

32:30 And you prove that the pure sets of the internal model are isomorphic to the original model, right? And so it's completely clear from this standard construction that CF and CF plus URF plus a set of UR elements, or a structure of UR elements if you wish, are bi-interpretable, so in some sense they are simply the same theory. So this was one of the first examples I looked at and I really hoped that that would be an example, you would not have to know me, but Benedict Leuwe, after some discussions that took a long time back and forth to make completely clear what kind of thing I wanted to see. and then he produced a counter-example, and he produced an interpretation that made these two series synonymous. So that's somewhat surprising. So this synonymy is still quite wide, but there are still cases of it that are unexpected. Okay, so far for the whole setting, and so you can go on developing this and prove several properties and characterize notions in terms of the interpretations of morphisms and the the structural work as he began between interpretations, but I want to go on to the notion of scheme. Now, you can define some notion of axiom scheme in a rather simple way, but we are only interested in schemes of quite simple kinds, schemes. And so what is an E-scheme? Well, an E-scheme is in some sense simply a pattern that we recognize in an interpretation. So suppose we have an interpretation of U in

35:00 And so we want to recognize some patterns in it and this pattern is simply given by some arrow between U and blue prime. And to see that the pattern is there you simply have to ask that these things can be used so that you have this. And to make it all uniform, we add that always when you have an arrow, so this v, this phi is going to be depending on you, so if you have a phi for u and you have a phi for v, we ask that you find and finish this idea. So it's not a natural transformation because I cannot see how you could make this function. Okay, I will rush to my example because you cannot possibly understand this, I think, without the example how to do that. Okay, I will perhaps postpone understanding a little bit, but I first explain to you some properties. So simply, this is some definition, right? Simply take the notion of face value, and of course that's the theory of truth, so the meta object thing will be the application. So why, how this idea works will get clear when we get to the example. So, in some sense, in this example, the meta-object pair will be this arrow M, and so the success of V as a meta-language for you will be measured by, so there will be a fixed theory blue prime, as it were, the standard meta theory for you, And so the fact that you can finish the diagram means that this pattern, maybe that there's a truce predicate for you, can be recognized. Right, so that's the idea. Okay, let's first look at, before making it more clear, let's first look at some false properties. So the advantage of this way of looking at it is that all kinds of serums that are very perspicuous now become certainly very simple. For example, suppose that we have the composition of two interpretations, right?

37:30 So one, when does it satisfy the scheme? And so it's trivial to see that if one of the two satisfies the scheme, then the composition also, that's a quite strong property. And so in the first case of course simple, so we have this, and simply now take the composition of k0 and this one to find that these are line from V0 to U2. And here, so we ask for this uniformity, so take this K0 here, take the K0, K1 here, and I'll compose them here after these are narrow, like that's one of those trivialities as soon as you look at it in a diagrammatic way. Okay, so now how are we going to interpret the object-meta distinction? If you look at Karski's popular paper that I have to teach each year, the first year philosophy students, And in some sense it's hopeless, right? Because it seems to be very much confused between a language, a theory and a model, right? in fact he says about the object of the meta-languages, it seems that in one, sometimes you should read model, sometimes you should read theory, and sometimes you should read simply language. But so what can we do, so what is the right choice? Well since in some sense models could be taken to be special cases of theories, the theory would be a good notion. I think is supposed to prove certain things, it could be a theory about the language. I think the object language could still be a language, right, so the object language is not supposed to prove anything, so there could be a kind of reconstruction saying the object language is a language and the meter language is a series, but I think for uniformity I think it's quite nice to say that they're both theories, so the object theory is interpreted Now also we want to do that modular translation, right, because we want to say that CF has beta series for VA. And so we have to add an interpretation.

40:00 So here is the way I would like to do it. Okay, so, you do it in some sense very naively, start with you, right, so what we, now, what we, what you first do, so, uh, of course, you can mean anything, right, it's just not It needs to not have names for sentences or whatever, right? So how can we do it? Also, it doesn't need to have a truth predict or anything. So what we add, what you do is really... So first we add simply some syntactic machinery to talk about sentences and the operation of in the ocean, etc. So for that I chose my favorite CoE cube, so arguably there's even weaker CoE, R, as maybe the trip I asked myself about it. So let's take it in. And let's extend also this CoE with... Okay, but we still don't have any action. We don't have any action. Now here you see the notion of sum, and that's a quite interesting question, because this is sum in the categorical sense, so it is a sum in int. So it's simply of course that you make the signatures of this one and that one disjoint, in a big language have two different domains for one signature and for the other and then scroll everything in with the obvious kind of excerpts. And now, so really our new meta is of course simply this theory plus the excerpts from Tarski's convention T.

42:30 Right, so that's all. So that's, that's it. So, okay, so that's that. And, uh... Sorry, what role is T playing in Q? Is it just sitting... It's simply added as one extra unary credit. Okay. Yeah, so in that... No, no, no. Well, I'm sorry. So it's intended to apply to U, right? Okay, so what you do is... Okay, let me say again what the sum means. So I have here Q. Okay, I have a theory of U in the language extended with kind of an active predicate. And it's supposed to apply to sentences in U. It's supposed to apply to sentences in U. So they're... But that action that says that it's only applied so that regulated could be here in new retail, right? So we have to say a little bit to make this over. But so I think because we're here at the sub, so let's simply take this through, let's take this simply as one added precedent. And now we extend it with some axioms, so the minimal set of axioms that you can think of to regulate this. So mainly what I want is simply to ask this convention, T. Of course you could go for much stronger things and you could go for stronger series here, and then you lose a bit and you gain a bit right away. You put more things on one side and that's what you do. Now, if we go to V, we can go to V-meta, so that's V-meta. So what we have to do is of course to find this G, to find some G-dita here, right? So we might just get this enough uniformity to be useful.

45:00 Well what you do is of course, in some sense the obvious thing what you do is you translate. So, because these things are so neatly disjoined, you can translate the U part to the V part using K, right? Now, the arithmetic part can be translated by the identity, because we want the same numbers and the same insights about these numbers. So the only real thing for creativity is what to do with truth. Now we are safe because interpretations are such extremely simple objects. They can be easily arithmetized, right? So the interpretation K can be arithmetized by some arithmetical function, Tama. And you can really write down the formula that does it. Okay. I hope I like signatures. Sorry? I hope I like signatures. Oh, yeah, yeah. Sorry. That's the report. I'm assuming that the signatures are fine. I suppose they're getting close. Explodes. Yeah. Thank you. Okay. So as soon as you have that idea, of course, we can see in Vmita that really the Tuscian axioms really translate well, right? So true in Kmita of the girdle number of A now is by definition true of the girdle number of A, but then translated under K, right? by definition, so now this is a concrete computation because this is an actual, this is not a non-standard, it's not an internal variable but an external variable. We can really compute this thing and so Q can do it. Simply take the steps. and so it translates out to the number of ak.

47:30 And now we use the fact that true is the true's predicate in V Vita for the language of V. So you get AK, and since K and B coincide on the language of you, then St. Green and St. Peter. And so you see that we really have translated the task in action. I hope this works. Okay, so now, uh, okay, yeah, so, I'm rushing to get the end of my story. Okay, so now we can see what it means that V is a meta language for you, right? Interpretation K is a meta object where. This really means that if you go from U to meta that we can finish this in such a way that we do not touch the interpretation of U. but just give an interpretation of the natural numbers, some interpretation, plus that we give the interpretation of the truth spreader. Right, so if you can finish this diagram, you have provided a way that we have substantiated that we need to use the Tarskyan actions of the Tarskyan Convention Team for you under the presentation. And so my idea is to call such an error if it's a successful object meta pair, so that means that if you definitely have to finish the diagram, to call such an error a restricted And so, successful object-meta pairs are really, in my framework, restricted arrows, and interestingly, so, well, something that was always, that I think comes up now and then.

50:00 I see that I made a typo here, right? If k is restricted, then so are m's, m composed with k, and if that's composable, and k composed with m, if that's composable, these should be different, right? So if we have two arrows, and if one of them is restricted, then the composition is restricted. So, these have good properties. Second theorem is, if a composition is restricted and one of the translations is surjective, That simply means that every proposition, every sentence of the target language is equivalent to an interpretation of the sentence of the source language. So for surjective morphisms we have that if the composition is restricted and the first one is surjective, then the second one is restrictive. So in certain circumstances you can subtract things. So let's see what we hope when we can. So it looks like this, right? Now this one is surjective and this one is restricted. Then we can subtract the surjective one and do the distance. Ah, okay. Somehow I've missed a small serum, but it's still important. So Tarski's famous serum is simply that the identity of any... no identity marker can be used. That's simply Tarski's serum. But now it follows that, given all the serums that we have, you can conclude that lots of

52:30 things cannot be restricted, so for example, an important notion is retraction. So here, it's KM. So retraction looks like this, right? So we have two series that go back and forth between them. And one of them, the composition of KM has this kind of retraction. So this thing is called the retract, so this one is called the retraction, or the split apomorphism, and this one is called co-retraction, or split monomorphism. Okay, so what you can see, for example, that some very important class of interpretation are split monomorphisms, but what you see, of course, is that no split monomorphism can be restricted. Right, now I should tease you, but it's in fact trivial, right? Suppose k is restricted, then the composition of these things is restricted, and then the identity is restricted. And, ta-da, that's just the same, but that's the thing to be the k. Albert, can you explain again why subjective k cannot be restricted? Yeah, I didn't do it well. You said it again. Yeah, try to be polite. No, no, I think you want to stay in this, because it's not difficult, but I think it's too much technical. But let's simply carefully, because you really... Oh, oh, you mean the consequence? Yeah. Oh, okay, okay, sorry. Yeah, the consequence is... I can't do that wrong. That's no technical detail. So the point is... So the theorem tells us that... Let's see. The theorem tells us that we have this thing, right? So, this one is surreactive, and this one is restricted.

55:00 And what we may conclude, ergo, this one is restricted, right? That was the theory. So you can subtract the surreaction if you're only restricted. Ok, now the whole idea was that suppose a surjective thing was restricted. Now take this one, the identity, and then of course the composition is restricted, hence by subtraction the identity would be restricted. And that's it. Okay, yeah, sorry. So that's indeed a trivial consequence of the subtraction theory. Oh, by the way, surreactive is quite an opening notion because it's not categorical. In some of the categories, some of the variants of the categories, the surreactive morphisms are the Ypres morphisms. but not always. It depends on your identification of the interpretations. Okay, now, there's a... I always thought that philosophically somewhat interesting... a somewhat interesting phenomenon, maybe the origin. And the point of the origin phenomenon, philosophically speaking, is this. So, you have certain foundational programs, for example Nelson's, where you build up stronger and stronger theories by starting with a weak theory and into seemingly stronger theories, and into creating seemingly stronger and stronger theories in it, right? So that's a kind of extension program. And so the very interesting thing, the interest is in this, right, where we interpret V in U, let's say that we interpret U in V, and what we want is that U is a strict extension of V, right, so that's interesting. So you could hope, maybe, so let's accumulate this, and you'll guess.

57:30 But just as with Hilbert's idea that consistency was truth, so the objection is, if consistency is truth, then goes A and not A can be true. And here in this case, at least you can show that sometimes you can do it but it's not unique. The ORI phenomenon says that it's possible that you have certain sentences, the ORI sentences of the theory, such that both U plus O and U plus not O are most interpretable. So, what I want to show is that you can get the aura phenomenon from restrictiveness of an arrow. And the interesting thing is as follows. So, we have seen that the identity on a Siri is not restricted, right? And we can, as usual, confuse a Siri with its identity arrow and so on. So, Darcy's theorem tells us no Siri is restricted. But the interesting thing is that if we widen our scope a little bit and do not just look at identity loops but at hardware loops, So let's say that the theory is weakly restricted when we have a restrictive look. And now the miracle, there's a great miracle that lots of interesting theories are weakly restricted. I specially invented that notion for this talk. It really means that you have a single loop on the theory of an interpretation, such that interpretation is restricted. This means that theory has its own truce predicate, a truce predicate for itself with brutal interpretation. And, in fact, CF is weakly restricted, Piano-Rismatic is weakly restricted, all the series that we non-model series, non-model theorists, know and learn, are, that we, well, totally, yeah, oh well, that may be too strong, but a lot of, any series you think of that has sufficient coding and some current properties is weakly.

1:00:00 primitive recursive charisma. So, a lot of these theories have this property. And so now we can look what happened with the liar paradox. So we can simply take the liar paradox now in the theory. Only the point is, what we get is not that the liar's sentence is of contradiction, but it's equivalent to the interpretation of its own relation. Ok, that's clear so far, everybody seems to understand this. So now it's interesting, so now if we simply add O to U, sorry, if we add O, if we add this sentence, the Ori sentence, to U itself, it's clear that if we have O, then under the interpretation we will have not O, so U plus not O is interpretable in U plus O, and U U plus O is interpretable in U plus not O, and it's simply, of course, the same interpretation K here, so that notationally I am forced to distinguish these things because qua-morphisms They are not the same, but the underlying translation is the same. Okay, now, so what I said is we have an interpretation of u plus o in u plus not o, right? And we also have an interpretation of u plus o in u plus o, maybe the identity. Well, the interesting thing is that in this category int we do have a good notion of Cartesian product. Even that was a surprise for me, to be honest, because the kind of things you expect are sometimes hard to get. But in the case of two series of this form, U plus O and U plus O, the Cartesian product is extremely simple, namely it's simply the original theory of the intersection.

1:02:30 we can uniquely find an arrow that does this. In fact, the idea of this arrow is very simple because it's the arrow that is a kind of promiscuous interpretation that says, well, I wait till I know whether O or not O. If O, then I function like K1. Sorry, if not O, then I function like and if O then I function like the identity. So intentionally in U you can say such, you can do such things. So we can make a promiscuous interpretation that as soon as it gets certain information it becomes this and if it gets other information it becomes that. So we can finish this diagram and so this gives us our interpretation of U plus O and U. And if you look at it, you see that it's even a split monomorphism, right? If you walk like this, you get the identity. And that means that it's not just only an interpretation, but also a faithful interpretation. that means that under the interpretation you can't include more interpretive sentences than humans own, so it's a very beautiful kind of interpretation, and it's precisely the same thing to interpret humans not own, and so in restricted theories you do have boring sentences, and so this kind of strengthening program does not work. Okay, now when I wrote down this question, because this was, you see, the problem is in the afterslows, right, so you can always write down afterslows, but this was, I think I know the answer, so that there are, that I think I can characterize which arithmetic use are restricted. It's a silly question. But this is a far less silly question, so of course this is one construction of horizontal axis, but we do not know of course all the constructions of horizontal axis. So are there any interesting mathematical theories without horizontal axis? Just do not know. It's at least clear that this kind of construction doesn't work. But for example, for Robinson's arithmetic, that is not the restricted theory,

1:05:00 a weakly restricted theory. I do know that it has always been. So in lots of cases we do know. Yeah, so now we have a conversation. Are you all direct in this talk? I'm not going to really do proofs anymore, but simply say, We have to talk a bit of time, so I suggest you to see it. Yeah, okay, we are all ready for it. Okay, so these are... So it's more... The series I'm going to show you are more illustrations, so I could also have chosen that one. To see some of the notions is a little bit more in action, right? Okay, so here we have a theorem that illustrates some notions. So, there are in fact various approaches to, you can take in the category 2 induction. So one approach is to view induction as a scheme, right, so we can use machinery for schemes that I developed. But there's another way, and that looks like this. So suppose we have some basic arithmetic, and I'm never completely sure about Q. So that this act, for example, there are, I'm assuming here there are actions of linear order, etc. So if you look at the details, I always feel slightly uneasy whether you're not silently using these depth of organization. At least that's different, etc. These things can be hopefully proved in queue. So, abstractly, let's look at some weak argumentable theory that's strong enough, that has enough states to make the arguments work. Now, so, one way of looking at full induction is to say, well, we have full, so this is weak argumentable theory.

1:07:30 One way of looking at full induction is as follows, so it's an interpretation jota of the weak arithmetic in the series that is initial in the category of all interpretations, so if we have another interpretation of F here, then we have the unique arrow that's from this one to that one. This model is internally, so look at it like this, you have a model of your app. And now we are looking at lots of internal models of this wee charisma. And so here we have this model, YOTA, and so what we have is that we have a unique embeddings of this YOTA in all these apps. And of course it's plotted and it precisely means that this internal model given by IOTA satisfies full induction. And you can prove that if the thing is initial then it satisfies full induction. And then the theory U satisfies full induction. Conversely you need some extra. So full induction not always implies initiality, so we need a few extra properties to get it Okay, so that's the idea So one possible analysis of satisfying full induction is that there are initial arrows from a weak arithmetic into the given series So, now here's the theorem that I'm not going to prove. Simply take your weak arithmetic. Now we have full induction here and we have full induction here. And suppose that this L is a retraction or a split epimorphism. Then we can prove that this diagram commutes. An interest, yeah, so this uses, it's not completely so, it does not completely work when this is an arbitrary, so you would hope that this is a far more general theorem, but I used one specific property of initiality where these are methodologies, so there's still some work to do to get it right.

1:10:00 So the surprising thing is that if we have this embedding that gives you scroll-induction and project them via such an apomorphism, then we get, then the sync concludes. And so that also means that if we invert these arrows, so this means look at the set of the arithmetical sentences of which the interpretations can be proved in V and U, that in fact, so this, the set here, if we project back here, must be a subset of what we get if we project here, right, so the arithmetical sentences via this interpretation here should be a subset of the ones here, so the U should be the stronger, in some sense the stronger series. And so you get a surprising thing. So suppose that we simply look at series of PA, extension of PA in the language of PA itself. Now the theorem tells us that if U is a retract of V, then V is a sub-serie of U. And so the retracts are simply extensions of the different And so, piano was the original sentence of piano, for example, is a retract of piano, right? And you see that it's not...so you cannot find other retracts than such extensions. And the example shows that the retracts need not be identical to the original. But it also shows that if two series are synonymous, they are simply the same, right? Because an isomorphism is certainly a split monomorphism in both ways. So we get that synonymous series are the same. And I think, in fact, we can improve this even by interpretable theories, I would say. So that's very classic.

1:12:30 But even more interesting is this, and that was one of the questions that I asked myself after seeing Volker's paper on Shapiro's Google. Can an extension of PA be bi-interpretable with an extension of ZF? So, set in a different way is the difference between arithmetic and sensory board and skin-d, right? So is it simply that they, well, we are misled by these projects. It's accidental signatures and that's not the percent they are simply the same. And so the answer is clearly no, and we can see it like this. So suppose here we have some extension of set theory. Let's say ZF or ZZ or C. Doesn't matter. And here we have the ordinary von Neumann interpretation of arithmetic in set theory, right? satisfies full induction, and hence it's initial, because set theory satisfies some other conditions. Well, this condition is simply to state, set theory has a theory for coding arbitrary sequences of sets, so that's the theory that needs to get the prevalence of these notions. So, this is an initial error. And also, clearly, because we have the truce predicate for arithmetic here with V, because there's entity F, it's a restricted error. Okay, here we interpret F in U via the identity interpretation. which is absolutely the same sentences to themselves, and this is so, this is the mathematical sequence. Now suppose we would have a retraction from V to U, then by the earlier theorem this whole diagram commutes, but then this one is restricted and this one, oh well, then the composition is restricted, so this one is restricted. But, on the other hand, particularly this one is surjective, so it cannot be. And hence, you do not have a retraction from a set theory to a mathematical theory, by the same kind of game, slightly different algorithms.

1:15:00 you can also pull out the retraction from generic medical series to a set series. So as soon as you get the hang of the game, it will be as you expect. Okay, I think that's more or less what I wanted to say. Oh, and maybe I want to say one thing more. There are some goodies here at the end of the day. Thank you very much. Would you like to take some questions? No, or would you like to ask a question? I can take questions, I'm sorry. I have a question with respect to the orisances. So, do you have any arithmetic theories where you doubt that they have orisances, or you just do not know if they have orisances? I should have re-checked all this stuff, so it's completely clear that all essentially reflexive series do have all this stuff, so piano and all these extensions do have it. Essentially reflexive is okay. Okay, and a long time ago, in a master's thesis, Mariana Caldwey proved that finitely axonotized retinical theories do have more resemblance. And so, it is quite clear that... It's important that antitrust is done with essential reflexivity, but simply reflexivity, so PRA also has a reason. So it should be, what you are looking for is a non-reflexic, non-finite reaction guide theory. So, all theories that we know, all well-known mathematical theories, are either primarily actually guys, or reflexive, or, again, so it's very hard to, but probably the examples

1:17:30 should be rather artificial. Oh, I have an example for such a theory. Okay. But I think, I'm not sure, what about theories with restricted induction scheme, but without There is a paper by Bexler Mischoff that they are not highly eczema-tizable, but I think they shouldn't be reflective. Ah, okay, great. We should think about this, because the point is, they can be sententially reflexive, right? So, I'm not completely sure whether for the arguments you need, maybe substantial inflexivity is already enough. But still, the example I have invented is by adding some, um, have all, uh, potability six points. Okay. Uh, iteratively, but I have no idea if they have origins. Ah, but you only have to, it's not clear at all. Okay, well, let's just look at that. My guess would be that it's, that if there are examples, it should be rather concise. Okay. But still it's somehow kind of annoying that we have some of the voices. Oh, okay. He's good. So, we have this very nice way of going now for instead of the task in hierarchy. What if you want to do on many methodologies? How do you do? How far? OK, yeah. Of course we can compose these things, but that's not very informative. The question is really how to describe hierarchies in the framework. Okay, well, I think it's a good question, but I don't have a ready answer. So what do you got? So how would you want to say it at all, right? where we have at least, in fact, several objects in the definition of a kind of chain of objects that has some part of what we say. Well, you can say, well, we have a chain of objects,

1:20:00 all the chain, we stick with arrows, but that's not very informative. Well, yeah, there should be something to do with it. Will this stuff work for higher-order theories? Yeah, I... Say second-order, just say second-order. I think this notion of interpretation is so... I don't know what you mean by this, right, but the whole setup that I sketched in the beginning, so I looked at something very specific, right? First order series, RE, simply extramatized, I looked at interpretations that are not multidimensional and do not have parameters, So I'm pretty sure, but the work has to be done right to really make it. Everything lives in much more general cases. So higher order, I don't see a problem. So as long as you have an interpretation, a notion of interpretation, in higher order sequence. Yeah, I think that's okay. But do you know the paper where this has been developed or carried out The model series are the most obvious suspects, right, so if you look at model series papers there's all kinds of, there's interpretations everywhere, but never anything systematic about interpretations, right, so I'm sure that there will be some specific interpretations for higher artisans, so it really worked out seary a little bit. Yeah, so for those people who take the paper for the next month, I think that was, so, I made an error in starting, namely, I started developing the whole stuff for parameter-free interpretations, but that's it.

1:22:30 When the paper was nearly finished, I realized that this phenomenon of f in parameters is in some sense so fundamental that you really need to add it, so that's one thing that one should go through the whole paper again to look at the specific points where parameters So nearly always the theory works for grammatists too, so there are always some points of data. Because, for example, if you think of the kind of implementation of hyperbolic geometry and euclidean geometry, It's clear that you take some circle, right, but in this circle you cannot specify a specific point or a specific radius. So you say you take two points, and there is a sort of blah blah blah, the interpretation is dramatizing, and lots of things would have been nicer to develop that, but in the tradition, people side-step this, right? If you look at the famous Karski-Robertson work, what did they do? They sidestep the problem of parameters by always adding constants to them. So that's an honest story we would say. Can you think about the interpretations between theories having different logic? Okay, yes, I did a little bit, so for the constructive case, this is very, it's quite difficult, right, because, let's see how to explain, so if you look, so for my main interest, we're in art mathematical theories, And if you look at something like Heidens arithmetic, so there's very, very few, it's very difficult to write a non-trivial interpretation of Heidens arithmetic into itself. And the reason, there's a very deep reason there, it's mainly that we cannot prove, so the completeness theorem is not plausible in structure.

1:25:00 In fact, the ordinary model is here at the completeness theorem. In fact, it's much stronger, so if you have a constructive meta theory and you add Markov's principle, those churches ceases, then we can prove that arithmetic is categorical. So the completeness here fails in a spectacular way. And so the whole point is that the kind of interpretations that you can produce between arithmetic and arithmetic constructively are, in a sense, really lacking, because they will more or less be something like alternative models, which you cannot expect in some strong sense to exhibit such things in constructive arithmetic. And so if you look at all the interpretations that are constructively used, they're much more like enforcing, for example, realizability, etc., right? So they don't respect implications in circles, etc. They don't commute with all the logical ones, so you really get a different game. So of course people try to find a wide concept that will encompass forcing and realisability, etc. But then very easily you lose a notion, you're not studying a notion with mathematical content anymore. So in an earlier stage I did write some kind of white channel that I'm a little ashamed of now because I've got a very general notion and you don't ask for commutation or anything, etc. I think it's wiser to simply take some specific class of translations or whatever and still be dead. So if this kind of project is successful, then try to generalize later. But anyway, now this time I made a mistake in the other direction, because I should have done this in those parameters.

1:27:30 It's difficult to find precisely the right choice. You can only know it. No further questions, then let's thank Robert again. I believe there should be some coffee or tea now. Thank you.