 enf persuaded is to give an introduction toolve order categorical logic with the aim of explaining the crucial notion of classified octopos or the geometric theory. And especially the way classified octop poposis can be used for um building for discovering connections between different mathematical theories much as in the spirit has nekaj, ki sem udeljena? To je tudi pri vzacom. Ta sem pozhhhwaja, da počevajo vzacom tem, da je učešnja matematika vzac, in vzacom vzacom mese vzacom, kdo nekaj je tudi vzacom. Vzacom, če je nož, vzacom za našelje počešnje, zazet, ker toga ne zmorez, a tudi je to to informalizujezno eletากovati parametrik. A znače je, da se ta izgledaj z Qinouchi, je zelo lepo vzztavno, da se vso de Peše illustrationa počatila tega v praž seat. Prelajno toga vzeljati tega dyga nekaj doživstov, da zač tudi je večje vzelo, da jaz se nekaj zelo lahko vzelo, katera ta zelo deta koment nekaj tudi mladne del, ali tudi je tega najsistva vzela, in vzložiti vzložiti s nekaj različenih teori. Zato, da jih izgledaš, da te rečene v pravnih to popsu in zelo to je tudi, da te rečene je zelo to samo v nekaj rečnih teori. Jih se tudi da je vršo. Vršo je zelo potenšelj, vzložiti in tektivno, ker vseh, ki sem je dobro vzelo v početke, pojeljno raz,... ... pa občasno moj najbolj počeljega je... ... počeljega je od tega in s edmine... ... ko je, da je izgleda... ... kot nekaj, da ne pomembno, da je... ...i se tko, da je... ... gažno zelo. Ok, počelj... ... pa pročem... ... skupnim, kda je... ... počelj... ... kaj je, da je, ta tega... ... počelj... ... začelj, na taj... Ispečno, da je ljubez? Tudi, ki je v tem legoz, ki je naj grupo v logičnih vzup, je stavljena nožno modeli teori, z radočenim v klasi glasnih teori, je se je vzupo teori, v sve, da modeli daj vzupo z vzupim, určenim vzupim vzupim, ki je vzupo vzupan nekši legoz, da vzupim vzupim na strančni matematik, lahko, ne, to je vzup, in je tudi prišlično, da se zelo prišlično v 2 zeločke nene. Zeločno v sematih in sematih. Zeločno je, da sematih je tudi matematika, ki je vse vse in desite. Zato, tudi, kaj je, kaj je, kaj je, kaj je, kaj je. Zeločno matematika vse vse zeločno vse zeločno vse zeločno vse zeločno. Zeločno je vse zeločno vse zeločno, poradno volene, da vsobeli nounčenimi vstahljemi. Ušliščen različnoj se boš nekaj nebezidentno počke, že v sebe boš nekaj nebezidentno počke, in v logiku je neko se takvena odliš approach. I zato seite da s aravenimi vznemovimi vznemovimi vzemi. Logika je vse obačnja, zato malo na corríklja vzelo. Vzač je več boš, da je čljen, nekaj zrog je vse zvali očenja. je ovo ozvrčenju, ozvrčenju, ozvrčenju, ozvrčenju, ozvrčenju. Včasno, tehnikali, vseh želje, je želje, ki se vseh dvečnega, kaj je, vseh, vseh, in, nekaj, da se priče, da je to, priče, priče, simply means that in the axioms over first order theory you are only allowed to quantify over individuals of your structure and not over collections of individuals or collections of collections of individuals so for instance the community GA, property that you analyze in a ring you can formalize in a first order way but for instance if you want to describe the property of the ring real numbers, that every bound is subset as a supremum, this, it is a second order if you want to formalize the real numbers as a set of elements, because you have to refer to collections of such elements. Okay, so this is the basic restrictions that one makes in order to have a well-behaved model theory. In logic, people have also investigated higher order logics, but also if we want to investigate the growth and ectoposis from a logical point of view, we will always remain at the first order level, even though we shall use sites also. And the notion of site that is fundamental for defining toposis, basically you can define a growth and ectoposis as any category which is equivalent to the category of sheeps on a site. Prof. Cartier talked about the sheeps on a topological space, but actually the general notion of topos is this, sheeps on a site. And this generalization from locales, let's say, because one cannot say that this strictly generalized topological space is strictly generalized locales, like open sets. So this generalization is actually very, very, very powerful, because here you are still in the context of topology, while here it is the place where you can realize the marriage between the continuous and the discrete as growth and ectoposis was saying, because on one end. So these sheeps are defined in a completely formal, formally analogous way, as you define sheeps on a topological space, but the resulting level of generality is incomparably broader, because this encompasses, at the same time, this and the discrete categories that Joyal talked about in his course. So this you can see, if you want, as continuous, and this factor here, to get just pre-sheeps, you take the trigger growth and ectopology, and while here the growth and ectopology is by no means trigger, because it is given by the usual covering relation that you have in topology. And so, I mean, you can present growth and ectoposis as a classifying toposis of theories, axiomatize the infrastructure of the logic, but you can also present them through sites, and the notion of site is actually a second order notion, because a site consists in a small category C, whatever category, not just Cartesian, and what is called a growth and ectopology unit, I will just briefly review what it is. The growth and ectopology is a way to assign which object of the category, a collection of sieves from C, what is a sieve, while it is just a collection of arrows with codomain C, which is closed under composition on the right. So, if I have an arrow in the sieve, then I want its composite with whatever arrow also in the sieve. This is because it is useful when you want to define sheeps. Of course it is completely no, because if you have a collection of arrows in the same codomain, you can always close it up like that. You take all the arrows that factors through at least one of the arrows in the page. OK, so you have an assignment that sends to each object of C, a collection of sieves. These are called the J-covering sieves, and one writes S belongs to J of C. We indicate that a sieve is J-covering, that it belongs to the topology. And, of course, one has to require some properties, which correspond to the properties of universal closure operators that Jojala alluded to in his talk. These are just three properties, the maximality condition, which means that the maximal sieve on each object should be covering. So, first is maximality, second is pullback stability. So, this simply means that if you have a sieve, which is covering, then any pullback of this sieve along an arbitrarium should also be covering. And then you have a third condition, which is what is called the sensitivity. This you can formulate it in this way. Imagine that you have many sieves that are covering, and that you can multi-compose them. Well, you want that the result is also covering. Ok, so, you see these are very natural conditions that one can put. And this is what a grotentic topology is. And you see here that it is a second order notion, because you take a collection of sieves, and the sieve is a collection of arrows. So, you see it is a second order, because you both talk about objects and arrows, and you also talk about collections of arrows. So, this is a second order notion, and I mean, it is useful for studying second order notions through toposis, such as the notion of topology best place itself. Ok, this was just a digression from the main line, because now I want to talk about first order languages. Now, first order languages consist, as I have written, into three basic ingredients. Source, terms and formulas. Now, sorts are things that are meant to specify the kinds of individuals, the kinds of elements you want to talk about. For instance, imagine that you want to formalize the notion of category. When you present the notion of category, what do you say? You say we have a collection of objects and a collection of arrows. Of course, if you want, you can avoid talking about objects, because you can identify them with the identity arrows on them. But it doesn't matter, you still want, I mean, it is natural to talk about objects and arrows to distinguish them. And so, for instance, you could choose two sorts. One for objects and one for arrows to formalize this notion. So, sorts are meant to represent the kinds, let's say, kinds of individuals. Terms are meant to represent individuals themselves. And formulas are meant to make assertions to these individuals. And where do you put operations, for instance? Well, operations is when you build a formula, you start with some basic formulas, which are called atomic formulas. And then, from those, you build more complicated formulas by using all these logical operators, which are known. I didn't mean that. I mean, what I meant is that when you say kinds of individuals and all that, do you consider, for instance, that the product is part of that? For instance, if I define, I don't know, a group. Yes. Then you have just one sort in that case. You have one sort? Yes. Where is the product? Ah, the product is, I mean, it's a term. Basically, a product, you have an operation. But I will give the precise definition. This is just the basic idea. But I will give the precise definition later. Formally, it is an operation. It is a function symbol. The product is a function symbol. And starting from function symbols, you can formulate the terms. Because the terms are just what you obtain starting from variables or constants by applying function symbols to them a finite number of times. So, for instance, in the theory of groups, you will have a binary operation. You can say like this. So, just one sort, one binary operation. And, of course, you can consider, well, this is really... Ah, but the binary operation is a sort. No, no, no. The sort is one. I mean, you just want one set. Take the classical set priority foundation. You just want... I mean, when you consider... Normally, one doesn't need to have many sorts. The reason why I am presenting this in full generality is that when you talk about the internal language of a topos, then, actually, the natural thing to do is to take one sort for each object. So, for groups, you have one sort. Just one sort. Then you have a binary operation, which is the multiplication, say, if you think of a multiplicative group. So, a binary operation is a term? No. This is a function symbol. I will define... So, you start, do you have another thing? No, but I will give the definition in... the formal definition, as I have said. I am giving the formal definition later. This is just the idea, the idea of the language. So, then starting from... Then you can consider, of course, this. This is another way of writing like that. And, of course, you can carry on this. You can multiply, also, with another thing, z, for instance, and go on, like that. And all of these are terms. And as you can see, these denote individuals, because x and y are individuals. And if I apply function symbols to them, I still am inside the individual. So, why? For formulas... I don't think of them as elements of a set. No, you can think of them as generalized elements. When you work in categorical semantics, you should replace elements with generalized elements. Anyway, this was just meant to be, I mean, the general philosophy, but now I will give the precise definition. Now, formal languages, you can always interpret them in the classical set theoretic foundation. And this is known, at least, since Tarski. But what I'm going to tell you today is categorical semantics, which is the interpretation of first-order languages in categories possessing enough categorical structure for one to be able to meaningfully interpret all the connectives and quantifiers that one wants to talk about. So, formally, we have to define the notion of first-order signature. It is, like, in most books, this finitary, only finitary operations, but the other calls of periphery is spoke about things with infinite... Yeah, of course. Normally, when you talk about a first-order, you can also take infinitary disjunctions or infinitary conjunctions. I will specify exactly the fragment of logic that we are going to investigate because of its link with grotendic topologies, which is called the geometric logic. In geometric logic, when you form your formulas, which are called the geometric formulas, you are allowed to use finitary conjunctions, infinitary disjunctions, and existential gentrification. And the axioms of a geometric theory, which I see, they are all of this form, phi entails y in a given context. The context is a finite list of distinct variables, which contains all the variables of these formulas. And you require this to be geometric. Geometric means that they are built from atomic formulas by only using this, this, and this. And here you don't have any bound on the cardinality. The only thing that you need is a set, just a set. So it is an infinitary logic, if you want. But do you allow, like in most cases, to have propositional sayings, and usually they are both quantifiers, exist and for every and also you can... No, at the point... No, the number of free variables should always remain finite. I mean, in principle, when you use an infinitary disjunction like that, you might, a priori, end up with an infinite number of free variables. You don't want that. You don't want that. Geometrically you can understand it immediately because if you allow this, then the interpretation will be a sub-object of an infinite product. And infinite products are not necessarily preserved in the images of geometric morphisms. So you need to impose this condition. But I will come to that. OK, so... So what is a first order signature? Well, we have a set of... So we use this... Signature is a synonym of language. I mean, it's a technical term for that, used by logicians. So a first order signature consists of three sets, set of sorts that are usually denoted with these capital letters, like that. Then a set of function symbols. And each function symbol is equipped with entrance sorts and output sorts. I mean, normally we require a finite... We require a finite list of input sorts in just one output sort for each function symbol. And then we have a relation symbol. Relation symbols, as well, are equipped with an indication of the sorts on which they act. So to come back to the situation of groups, in groups you have one sort and you have one constant. A constant can be treated as a zero in function symbols. Here, if you take n equals zero then you get a constant. What is called a constant. So for groups you have binary function symbol. So here, of course, I can repeat the same sort. So I don't need this a1, an to be the instant, so I will have my binary operation. One sort and no relation symbols because I don't need, in the case of groups, more generally, when you want to formalize... When you study structures that are formalized in universal algebra sense, then the axioms of your theory will be of this kind. Equalities between terms. And so you will not need... You see associativity and these are all of this kind. Yes, if you want, you can do that, yes. Yes, because otherwise you have to use an existential quantification and... Yeah, I mean, there are... There are various ways to do this, but yes, it's natural to introduce a unary operation to formalize the inverse. Okay. Yeah, of course, if you want to formalize the notion of preordered set, so what is a preordered set is just the set with an order relation, and in this case you see it is not a function, because you see here I am talking at a completely syntactic level, so everything I am saying here does not make any sense yet. It will make sense later, of course, otherwise. But for the moment it is completely formal, but one should already keep in mind this is why I put the first... I brought on the first blackboard, the intended interpretation, because the intended interpretation of function symbols are functions or arrows. So you see here you don't have a function, you have a subset. I mean, you can take... This is formally... This is a subset of the product of X with itself. And so this can be formalized with a relation symbol. Now, starting from all of this, what you do is, first of all, you build the terms. In terms, I have already said how you do. You start with either... So you suppose that for each sort you have an infinite stock of variables, normally to indicate that the variable has a sort A, you write like that. And starting from these variables always a finite number of times, a finite number of them, you build terms inductively, applying functions symbols to them, and so on. And so you write formally because it's very clear. Now, formulas. Now, when I define formulas, I come to them. So, formulas. So, first of all, as I said at the beginning, formulas are meant to make assertions about individuals. And individuals are meant to be represented by terms. So, formulas, the simplest formulas that you can think of are of this kind. You can either have inequality between two terms of the appropriate sorts because, of course, it doesn't make sense if the two terms don't have the same sort. It doesn't make sense to write something like that. So, of course, you require them because of course. This is because I didn't give the formal definition, but of course since function symbols come always equipped with sorts and terms are built just inductively starting from function symbols, well, they will also be equipped with a sort. Yeah, yeah, I mean, the variables are always all equipped with a sort themselves by definition. You have infinitely many. No, no, it's not because normally you don't need infinitely many. Yeah, I mean, this is a problem that you might encounter in this. Yeah, if you want, yeah. Okay, so the simplest kinds of formulas that you can consider are these on one end equalities between terms and if you have relations symbols, you can also consider things of this kind. Always provided that everything is composable, I mean that everything has the right sorts. So, these are called atomic formulas. Okay, now, of course, you can see this as a particular case of that if you take R to be the equality relation, but of course, in all languages that we consider, the equality is always so we don't even treat it formally as a relation symbol, but, I mean, morally it is. Okay, so these are the simplest formulas that you can construct and for this reason they are called atomic formulas and now to build the full first order logic you can operate on these formulas to build more complicated formulas by using various things. So, on one end, you have the logical connectives, which are these the end, the or, the implication, the negation. You might also want to have false formulas for false and formulas for true. These you can also see formally, I mean true, you can see as an empty an empty conjunction and false you can see as an empty disjunction if you want, but I mean, I just prefer to write them explicitly because they are very important for example here, you see. So, you can either use these say, operators or you have the quantifiers that you can also use the existential and the universal quantifier. And you can operate on starting from these atomic formulas you can build first order formulas by applying a finite number of times all of these things. As I have already mentioned we also allow, I mean when we speak of infinitary first order logic we can allow also both infinitary conjunctions and infinitary disjunctions we shall particularly need infinitary disjunctions when investigating geometric logic so bear in mind that this is by no means restricted to be finite except that when you define formulas you simultaneously define the three variables of these formulas so you have to define the notion of free or bound variable and this simply means a variable in a certain formula is said to be free if it appears in the formula as not quantified. So for instance consider this formula and bound otherwise so you can consider this formula and you see that y is free in this formula but not x x is bound because you have that so it's important to make this distinction because as we said in the case of infinitary conjunctions or disjunctions a priori the number of free variables might be infinite and instead we require in terms just compositions of functions in that term OK, so now we know about free and bound variables so we can define the notion of context for a formula so a context for a formula is just context for formula will be just a finite list of distinct variables that contain all the free variables of the formula which contain this is very important that you don't just take all the free variables of the formula but you allow all the finite set of distinct variables which contain this this is important for technical reasons because if you had only to consider the minimal context the theory would become much more cumbersome for instance in this situation in appropriate context for this might be Y, Z or Y, Z and W OK so how shall we present our theories well since we want to put certain restrictions on the kind of formulas to use we use the sequence notation well a sequence is just a formal expression here as I have said no meaning yet so take things just because we have not yet talked about the semantics so it is just a formal expression of this kind where you have a non-desident and the conclusion and these are formulas in the context in the same context here I used the vectorial notation this is supposed to be something like that a finite list of distinct variables like that so in the same context and the intended meaning of this expression is that basically if you are used to the logic is just that for all x, phi entails psi so it means that for any values attribute to the variables in this context if this variable is satisfied then they will also satisfy x I said why no the context here OK the context is always finite by definition of context always finite but you can always enlarge it yes exactly but just to be the finite number OK because as we shall see this is important because we shall always consider formulas in a given context and when talking about the interpretation because of course the context will determine the interpretation a formula in a given context x like that will be interpreted in a sub-object so suppose that you have a structure m in which you can interpret everything then the interpretation of this formula this is the notation for the interpretation in m of this formula will be a sub-object of this Cartesian problem and so you see this would be infinite if you allow the infinite number of this will be the interpretation of the sorts in m so each sort we shall define no it is just m when you have a structure for a certain signature you need to interpret each sort of that signature and so m of a is just the interpretation of the sort a in the structure n OK so this is the sequence notation to formulate our theories so what is a theory well it is just a set of sequence here I work in a meta theory in which I make a distinction between sets and proper classes set of sequence and of course sequence within the given fragment of logic I mean a fragment is just the choice of allowing only certain types of formulas and not other formulas so now I shall define the various fragments that we shall use over over a fixed signature OK so let us define different fragments of logic so as we had already mentioned the notion of algebraic logic so algebraic theories so the axioms are of this form just the qualities between then we can consider regular theories these are theories whose axioms can be presented like in this form where one requires this to be regular and what does it mean regular means obtained from atomic formulas by only using finite conjunctions and existential quantification then we have the notion of coherent theory but you mean no negation for instance no, no I will explain later what negations in arbitrary context as the complement the problem is that in an arbitrary topos you don't have complements unless the topos is Boolean and so the way you do is to use the pseudo complement the I think pseudo complement but then this will not be a true complement so you can interpret all the connectives that I have written there in a topos, you can do it the problem is the presentation models by morphisms of toposes because in order to have classifying toposes what you want is to have representing objects for certain semantic punctures and in order to have these punctures just to be able to define these punctures these punctures should send models of the theory in a certain category to models of the theory in another category and if you allow the axioms of your theory to contain negations of morphism given by the notion of geometric morphism then you have that you have to you have to use these punctures here the inverse images of geometric morphisms and these inverse images as it has been said they only preserve in general finite limits and arbitrary limits and they will not in general preserve pseudo complements and so you have to exclude these negations from the axioms but this is not a big problem because you can make a trick you can if you just are interested for instance in the set based models of the theory what you can do is you made this trick instead of doing this so you know that you cannot talk about that but then you can introduce a predicate and then you can which you can call the say n of this formula so you can put some coherent axiom some geometric axiom which ensure that this is a real complement to your formula and in this way you end up with a geometric theory and this is a process that in full generality is called the Morleyization of a geometric theory and it is a very powerful means because it allows to study any kind of finitary first order theory but this in an arbitrary topos will be different because this will be a real complement this will force the model to contain a real complement while in general this complement might not exist so in general the models will be different but in the topos of sense they will be the same and so I mean if you are just interested in classical model theory you can live with that geometric logic doesn't contain negation so I'm not caring at all about negation and here I'm just suggesting that what you can do if you want to treat negation or a sort of negation using geometric logic we are really forced to not to have negations because of this reason unless we change the morphisms between the toposis that we want to consider I mean nobody forces us to take geometric morphisms but there are advantages for doing that because other kinds of morphisms would be sort of less frequent also in mathematics less frequently arising you can consider for instance the notion of logical factor between two toposis this is something that you can do but there are very few of these these funtors at least they are rarely arising in geometric situations the reason why it's called geometric morphisms is that it comes from geometry because any continuous map of topological spaces or locals more generally gives rise to geometric morphism between the associated toposis so here you have the direct image and here you have the inverse image so this is really the notion of morphism between toposis to consider when you work with grotin dictoposis for many other reasons as well that we shall see so I mean negations yes of course I will come to that no no of course yeah but I'm getting a bit slower because of all the questions so sorry it also creates a great series of capital T it's like Julia's and Rotated T which I don't know what this means no just on the very right yes see ah no true the true implies this because you necessarily have to do something here you have to do something here true implies if I had to put false this would be really true because Rotated T also is like different attention on place in atomic form no no no this is false this is false and then that one is true no but in the middle in the middle or the implies and or implies negation the trans tile no the trans tile well no no no no this is the the sequence notation this is just a sort of the idea of consequence this idea of consequence ok it's just sequence notation ok I was writing coherent theories ok so you can define the fragment of coherent logic so in coherent logic you are allowed axioms of this form here you require the two formulas to be coherent coherent means that so here you only allow this and coherent you allow this finite disjunctions and existential quantification and finally we arrive at geometric logic maybe you should have the false assemble for false but it is I didn't mention because it is an empty disjunction so it's not the the symbol is not the binary disjunction no this is a arbitrary finite disjunction including false including false and also the finite it's not binary well you can say binary plus false to take into into consideration the empty case as well geometric theories geometric theories as we already said you require these two formulas to be geometric so these are built from atomic formulas by only using finite conjunctions possibly infinite disjunctions and existential quantification and of course also we arrive at the case ok so these are the main fragments of logic that you might want to consider and of course now that we have defined the notion of theory we are still at the syntactic level so we should talk about probability in a certain theory what does it mean because of course a theory is not a static object a theory is meant to be just the beginning of a mathematical activity so these sequences that you start with are called axioms these are called the axioms of the theory and what you are supposed to do is starting from the axioms to deduce other sequence from these axioms and the way you do this is governed by what is called deduction systems so you have deduction systems now here I have waited until after writing these different fragments before introducing deduction systems of course each fragment has its own deduction system and these deduction systems are not the same even though there is a compatibility given by the fact that now I don't have the time to write the details of these deduction systems they are just rules that are quite intuitive for instance just to give you an idea you have rules of this kind so this is the cut rule or you have the rule for finite conjunctions things like this this is cancer type yes yes so so you have inference rules so this means that from this you can deduce that when I put this line and if you put a double line it means that you can also go in the other direction ok so you have deduction systems for each of these fragments of logic that I have introduced here but as I was saying there are compatibilities between them because of course imagine that you have a regular theory for instance and you want to consider it as a coherent theory and then you have a notional probability here and a notional probability there and for regular sequence and so you wonder if they are the same yes the answer is they are the same it's easy to see we shall briefly see it as a simple application of the double construction of the classical entopos that this gives naturally a rise to anyway so we have a notion of probability inside a theory so we can probability of sequence that a certain sequence is provable in a given theory if there exist a derivation a derivation is just the idea that you can have of a formal proof you apply these inference rules and axioms and what you get at the end as conclusion you can call it a theory inside your theory ok so now we have talked about let me check I'm not forgetting anything yes exactly I mean when you work in regular logic you only talk about regular sequence which means sequence of this form when you work in geometric logic you only consider that yes yes yes yes yes I mean the older rules that you could intuitively come up with there is an important exception that is the rule of excluded middle you don't yes yes yes but they are presented in a slightly different way since we are presented them in terms of sequence so they might not be recognizable easily or I mean really the treatment via sequence is yeah I mean they are presented in this way so I give references to that in the program so you can that's a very good question that's a very good question so Goedel's completeness theorem is something that refers to the set based models of a certain theory and it requires the axiom of choice here we have a great advantage over Goedel which is the fact that we can consider models of our theories not just in sets but in arbitrary categories these categories possess enough categorical structure for us to be to interpret all the connectives and quantifiers that arise in our fragment this allows us to prove a strong completeness result which is even stronger than Goedel's result in a fully constructive way through the notion of syntactic category that I'm going to introduce because the notion of syntactic category maybe it is a bit premature to talk about this because I haven't even introduced the categorical semantics but suppose that I have then you otherwise I don't answer the question but suppose that you can interpret theories in a certain category with an appropriate structure then the notion of syntactic category is very nice because it allows you to build a universal model of your theory inside this syntactic category which we can denote like that so we will have a universal model what do I mean by universal well it is universal in many senses first of all it is universal because as we shall see you will be able to obtain all the other models of the theory by pulling back this along certain functors but from the point of view of completeness it is important because in this model satisfies the property that whatever is valid here is provable in the theory and so it is a model that by itself realizes completeness and this model is constructed in a completely, say, totological way like the Lindemontarsky algebra of a propositional theory you see when you have a propositional theory you can take the algebra consisting of the equivalence classes of propositional formulas of its language and so on so it is a model of the theory in which what is valid is precisely what is provable in the theory and so this realizes completeness and you see the difference between what you have in Gödel's framework because in Gödel's framework you have to take all the set-based models of the theory and moreover you have to to suppose the axiom of choice here we allow ourselves to consider models in arbitrary categories where we find this model and then this is the model in which the syntax and semantics of the theory meet and thanks to this meeting to this unification you can get back to provability in the theory this is one of the main advantages of factorial semantics it is a perfect illustration of the fact that looking at what happens just in sets is very limited because you don't get a fateful image of the theory even in the you see it for finite theories you need the axiom of choice and well, with the axiom of choice you can do it but imagine if you have an infinitary theory then even the axiom of choice will not suffice to reconstruct the theory from its set-based models you will be lost if you want to study infinitary theories with the classical techniques in model theories you are really lost while here even for infinitary theories such as geometric theories you have these things this is called strong completeness because it is not just completeness but it is something more because everything is concentrated in a single model now the formal definitions so what is so given a first order signature sigma what we can do is to consider the notion of sigma structure in a category with finite products now you need just finite products because the only thing you need is to be able to take these finite products if you want to interpret cartesian logic that we shall I can define it now because we have talked about so let me just define it for later this will be useful not today but tomorrow so cartesian so what is a cartesian theory so a cartesian theory here you require these formulas cartesian but relative to the theory so here one is really to refer to the notion of probability inside the theory it means that so a formula is cartesian if it is built from atomic formulas by only using finite conjunctions and existential quantifications which are provably unique in the theory this is the logic of cartesian categories in the sense that if you take a cartesian theory then its cartesian syntactic category will be cartesian and conversely any cartesian category can be seen as the syntactic category of such a theory so this is finite limits then yes this is the terminology that you use yourself in your course otherwise you can say that's the other but I think we use a cartesian that's the standard terminology so suppose that we have a category we find at products C and we want to define an interpretation for all the syntactic gadgets that we have introduced so we have to interpret the sorts function symbols sorts we interpret them as objects of our category C function symbols we interpret them as arrows so suppose we have a function symbol like that this is going to be this is going to be interpreted as an arrow going from this finite product to B this is not it this generalizes what happens in the classical Tarskian semantics in the classical Tarskian semantics sorts are interpreted as sets and here we replace sets with objects because we are going so in the classical case C is equal to sets category of sets and functions between them relation symbols will be interpreted as sub-objects equivalence classes of monomorphisms in our category C no I mean just monomorphism so this will be a monomorphism like that constants will be interpreted as zero array function symbols so here I will have an empty product and I will need to take the terminal object of the category so n equals zero so constants so this will be like that so we know how to interpret all of this everything goes very smoothly from from it is a very smooth generalization from what happens in sets to what you can do in an arbitrary category with final products but here I notice that we have just interpreted the very basic things because we have just interpreted sorts function symbols and relation symbols of course in general if you want to go on and interpret complicated formulas we need to require some categorical structure present on C in order to be able to do this because suppose for instance that you want to interpret a conjunction of two formulas well here finite products in general will not suffice anymore because I would like to take the intersection of the interpretation of this with the interpretation of that and such intersections formally they are pullbacks in a category and so finite products will not in general suffice and you will have to require at least the fact that the category is Cartesian and of course the more complicated the formulas are for instance you might want to interpret things like this with a lot of quantifiers and this also is possible to do in particular in every topos you can interpret these things but then you need to use a particular structure in your category to be able to do this you cannot do it in every category and as we shall see each of these fragments has a corresponding class of categories in which it can be interpreted but let's go with order so once we have defined these interpretations of basic ingredients we we describe before talking about formulas we have to talk about terms how to interpret the terms well this is completely straight forward because you see function symbols are interpreted in this way terms are just things that you start with variables function symbols applied to them so it is completely clear how you interpret them as arrows they will be interpreted as arrows the terms and this you don't need anything more than finite products to do this now for instead if you want to interpret atomic formulas no this only if it is a constant if it is a variable it will be a projection a projection or just the identity it depends if the context is larger it will be a projection it will be just the identity so now terms we know how to do now consider for instance these formulas of this form imagine for instance that you want to interpret a formula of this kind or a formula more generally you might want to interpret a formula of this kind well you necessarily need more than finite products take for instance this formula then you will have that the interpretation of these two terms these are these will be two arrows towards certain thing and the interpretation of this will be the equalizer of these two arrows so again you need that your category possesses these equalizers and here you will need pullbacks in order to you will make a pullback where you consider the interpretation of R which is a sub-object and then these terms will give you an arrow and then you will form a pullback and this will be the interpretation of your formula now after you have clarified how to interpret atomic formulas all the others are formed from then by using these logical connectives excuse me or quantifiers and how to do this well let's start so algebraic theories you don't need anything more than finite limits so here I can like that so this means here I take just well actually just already finite products as a suffice because all the axels are of this form so here you just need the categories with finite products here suppose that you want to interpret both finite conjunctions and quantifications so for sure finite conjunctions you need to be able to take pullbacks and you also need a terminal object so it means that you need a Cartesian category but it doesn't suffice because in a Cartesian category you might not have the possibility of interpreting this existential quantifications in a meaningful way now you realize when you when you wonder about this you just try to reinterpret categorically what is the interpretation of an existential quantification and so for instance what you consider is when you have an arrow or just a function and when you consider its image this Andrei has talked about that in his course this operation is very important this operation of factorizing this as an image as a subjection followed by a monomorphism because it is the key to know how to interpret this existential quantifier because if you look at how this is defined you see that there is a sort of existential quantification here so there is an element you see ok, like that it is the set of elements such that there exist something which is sent by f to it so basically here you need the notion of regular categories so regular categories are categories of Cartesian and in which moreover every arrow has a factorization as an image the image in general is characterized as the list sub-object for which the arrow factors and you require moreover these images to be stable under pullback the stability under pullback is related to the fact that in your fragment of logic you want the you need also this pullback stability condition now for coherent theories you need more because you see you have this thing here so you need what is called coherent categories now coherent categories are regular categories in which moreover you have finite unions of sub-objects and these finite unions are again stable under pullback a disjunction how do I do? suppose I know how to interpret finite disjunction because here we are in coherent logic and I know that the interpretation of each formula will be a sub-object of my my finite product like that so all of these will be sub-objects and how would you interpret the disjunction of these? well, you would take the union of these sub-objects but again, this is something that you can always make in set but not generally in an arbitrary category so you need to impose the existence of these unions these finite unions of sub-objects and you moreover have to require them to be stable under pullback and this gives the notion of coherent category now the infinitary notion of that is the notion of geometric category and what is particularly important for us is that growth and diktoposis are geometric categories and so we shall be able to interpret any kind of geometric theory in them so I have 10 minutes before ok, excellent ok, so well, if you want no, here you don't get any negation you just have the false in what? in coherent no, no, no, you don't get negation but I mean what you can get in coherent theory is the following thing you can write things like this something entails false this you can write but this is the only kind of negation that you can write you cannot write something more complicated ok, now sequence, you mean no, no, it works the only problem is the intuitionistic or classical nature of your system because the usual systems for first order logic they are classical so they have the law of excluded middle and this is the only thing that that doesn't work so this law here it doesn't work in an arbitrary in an arbitrary category in general because in order for this to be true well, this means that the interpretation of this is a complemented sub-object and in general you don't have even in a nighting category which is what you would need to interpret all these negations and implications in a meaningful way in a nighting category unless this category is boolean and you can have boolean categories but they are not very frequent and in particular the majority of toposes are not boolean so if you want to interpret this in arbitrary toposes you don't want the law of excluded middle ah, yes no, no, thanks for this suggestion this is sort of what I suggested to do earlier when a person asked me what can I do if I want to introduce just one particular negation and if I want to introduce the negation of equality so if I want to introduce this I can do it while remaining inside coherent logic the way I do is simply that I call this let's call it N for negation of equality so this will be a binary relation symbol and what I will be able to do is to put axions which ensure that this will be the negation but the negation in the sense of real complement of the diagonal which is the interpretation yeah, of course, this is just that and the other one is when you take the intersection of the two you should get the false so you see these are completely coherent axions because here I have this junction here I have the false and this is coherent so we are in coherent logic and this is an operation that we are going to talk about it later so it's good that you ask me to anticipate because tomorrow this will play a role because it is a very natural thing to do when you have a theory to un-injectify it in the sense that you want the category of models to remain the same except for the fact that you want the arrows to become monomorphisms and this is the way to do if you add to the signature of your theory this thing then automatically you will get I mean, in terms of set based models the models will be the same but the arrows will become the injective homomorphisms for your field of course without of course, you can always do what I said before concerning what I said earlier concerning the modelization is a general procedure that doesn't look the theory specifically in phase it works for any theory in a blind way and so of course you don't look in most situations you do this way you just take a specific formula it's not geometric you try to make it geometric that's what you do and of course here we have seen an illustration of this in the case of negation but you can see the implication it is more general than that because the negation can be seen as implication to false and you can make the same trick as well for the implication so in particular situations you just do this in order to get the geometric theory you don't need to apply this very heavy machinery of modelization ok so ok, 4 min now what do I cancel ok, so what should I say now I would like just to conclude this first part by talking about soundness and completeness I alluded to completeness earlier ok, yes ok, I need to no, there was something missing actually ok now, ok, we have said what it is a structure a sigma structure in a category with finite products of course we are now ready to say what it is a model of a theory well, I haven't defined the homomorphisms of structure but this is completely straight forward I mean a structure homomorphism is something that has the natural behavior with respect to both function symbols and relation symbols so you can write this is straight forward you can write the definition yourself and it doesn't present any surprise so I can skip this and instead I should certainly give the definition model of a theory so you have a theory t over a certain signature and suppose that in a fragment that the theory is in a fragment that is interpretable interpretable I mean that it has enough categorical structure for one to interpret these things so suppose that this theory is in a fragment that is interpretable in a certain category c then one says that a sigma structure m is a model of t if all the axioms of t which are sequence of this form so this will be things of this kind satisfied in m what does it mean satisfied in m it means that the interpretation of the antecedent is contained as a sub-object into the interpretation of the conclusion this is clear because remember that the intended meaning of a sequence was that for all x phi entails this so this is the same thing as saying that whatever element let's say that is in this interpretation is also in this interpretation so now we know what a model of a theory is and by considering the category whose objects are yes this is what I well actually no, you are right to make this remark but it suffices to check the axioms because we have soundness and I am about to talk about soundness so soundness and completeness the fragment is just a part of the logic in which you make the choice of restricting your attention to certain particular classes of formulas so it means that you choose to consider just some axiom is just when you define a theory you say a theory is a set of sequence these sequence are called the axioms of the theory by definition you don't have anything no, no, no, I mean a theory is really a presentation so whatever way you have to present and just a list of sequence that you call axioms no, no, not at all no, no, no, no, no otherwise it could even be a proper class so no, no, it is a set of sequence and then there is this very important now just a notation notation is this t mod t this is the notation for the category whose objects are the models of t in side of the category c and the arrows are the structure homomorphisms so the notion of morphism here at the level of models or the notion of morphism at the level of structure is exactly the same so you define the category of structures and this is a full subcategory ok, this is just for notation now standardness and completeness is precisely what answers the question soundness means that any sequence which is provable in t is valid in all models of c in categories here I put categories of the appropriate kind relative to the fragment of course of the appropriate now as you can imagine this is a theorem that is very easy to prove because the only thing it's quite straightforward is whenever you have an inference rule if the premise is satisfied then also the valid means that the interpretation of the antecedent is contained into this is the notion of validity in general ok, and completeness is I just write the converse I mean if a sequence is valid in all the models of the theory in categories of the appropriate kind then it is provable in the theory and I have already remarked that here we don't just have completeness but we have strong completeness thanks to the existence of syntactic categories but this will be for after the break