Syntax & Semantics (contd.) / FW Lawvere: What mathematical culture should be taught? (& others)

0:00 Mathematical theory does not reduce the need for inspiration, and it has been done very well so far, but perhaps it is not possible to explain it very well. In what direction can we expand existing syntax and discover a new theory? To be efficient, those advertisements, those extensions, are pretty rare. You hardly ever happen to find out something like the non-steered-wood analysis, or a predictive arithmetic method, which was a few years ago, and that's actually a major achievement of people. If you think of intelligent mathematics, you have plots in which you look for motivation in the world of ideas. So, thinking about the future in particular, I would like to know about the possibilities to continue investigating forms which later we have. Well, my answer has to be a bit more technical. In mathematics there is the famous principle of Lascius, which says that each characteristic of zero is practically closed, that two such changes are fundamentally equivalent. Such fields are, as I said, a technical point that basically includes all the elementary theorems in one field, also including the other fields as well, and this proved to be very useful in geometry for quantum mathematics.

2:30 For the complex numbers fields, they do have an ability to prove theorems, and if the theorems are of an elementary character that's principal states, they are also true for a different field, a purely algebraic field. Instead of trying to use logic, it would be much easier to have a new form, a new type of function, which is not a classical function, but let's say a progressive function, which Establishes isomorphism between one field and another. So I hope I answered your question. It's a theorem that every closed... It is demonstrably not the case that every formula with three variables is equivalent.

5:00 Now I'm using the word concept to give... The formula was pre-variable. That sounds very unexciting. Concept, though. A force of change or an inconsistency to a discourse. After a day of semantic agreement. I do not maintain semantics. I simply meant The notion of class is really a semantic notion, and in the semantics of Peano and his school, there is a complete confusion between the semantics of class and the semantics of school.

7:30 There is some confusion between semantic and syntax. I'm not saying that semantic has no role to play, that's absolutely not what I'm saying, but I believe this is a question and answer time for all of the morning speakers, right? Something you think is more subjective and quantitative to you, this is the way you think about it. You can't really tell the values of the quantities of this. You can't just tell us the size of the volume and the quantity. It's all in parallel.

10:00 Theoretically, theoretically, theoretically, theoretically, theoretically,

12:30 There is a new, new, new, new, new, new, new, new, new, new, new, new, new, new, new, new, new, new, new, new, Mathematical physics is an independent practice of demonstrations, since the fundamental thesis of Einstein was that there would be a mathematical theory that would be able to demonstrate it. It is a great achievement of the connoisseurs of classical mathematics to believe that mathematical theories are possible.

15:00 It is important to understand that the mathematics sense is inscriptive and that it tells us that rather they are purely prescriptive, they are syntactical rules showing how to use them, they are purely grammatical. What is your position? The syntax of the codificator is that there is a progressive function. How much is there in it?

17:30 Mental infinity will always give rise to consensus, so many will be left with semantics. So, if, by, if we say that some things function, we're saying that we need the existence of a mathematician. In the sense in which it was used earlier in Professor Nelson's lecture, I think that Professor Nelson could... If you want to stay on the syntactic level, like Professor Nelson, you can answer him. It is clear that in a different interpretation, the author can say that he is able to solve many problems that are not explicative. There are various definitions. This word is very simple, you have to look at the context. In the context of Nelson, I think he wants to say that it is full of theories. If one invents them, instead of discovering them, they will never be missing. If you invent them, rather. This is not a direct answer. It is useful to ask something. My perception is that semantics is not part of mathematics itself.

20:00 Cultural fact, as an important thing for science. The questions of significance and semantics are very, very important. Does the edict depicted by Professor Nilsson this more change the attitude of mathematics towards other sciences such as physics? What is the role played by mathematics within the framework of the physics environment? The question is short but the answer is not short. It is very important and also mysterious to try to answer the question of the role of mathematics and physics.

22:30 I think no one has a result of this defined by this. This is mathematically. It can, even though it might be interesting, choices aren't made. I think it is the concept of relevance that has been sort of rendered here, but in my opinion, the concept of relevance is extremely important. I'd like to add that you cannot know in advance whether a computer will be relevant or not, but this is an awesome thing to do, and I think it's surprising to what you have called the Lefschetz principle.

25:00 Two algebraically closed fields with the zeroes their characteristic are only isomorphs if their transcendence level is the same. And this is just to point that out. I have a question about the issue of semantics. In an attempt to better understand what you mean by semantics, so you didn't give any definition of it, if you understood it correctly, semantics tends to play in any interpretation of mathematics, maybe it has some other problem to play. If that's the case, could you please expand on that? And then I have another question. Don't you think semantics might be about intuition as well? And I'm referring to intuitive of infinitesimal. Intuitive concepts of infinitesimals have been related to mathematics and with the semantic sense, sorry, isn't that enough to justify the use and the role of semantics? Well, the first remark is about the algebraically flowed field. I'd like to have the term isomorphism as part of the function establishing isomorphism even when the degree of transcendence is not the same. As far as math is concerned, it serves as an interpretation, it serves as an inspiration for the application of mathematics, but in my opinion, it is not a part of mathematics.

27:30 An appreciation is an appreciation, it is only an answer, it is not a real description. The reality is that it was correct. It is a more fine, logical concept, but it was correct in the end.

30:00 It's a major contradiction that must be resolved. So how can we respond to that contradiction? Of course there are many aspects such as the funding and so on. I'm not going to go into detail. Concepcioni due all'attentimenti va in matematica, va in sinagoga, in all'attentimenti va in matematica, va in sinagoga, in all'attentimenti va in matematica, va in sinagoga, in all'attentimenti va in sinagoga, in all'attentimenti va in sinagoga, in all'attentimenti va in sinagoga, in all'attentimenti va in sinagoga, in all'attentimenti va in sinagoga, in all'attentimenti va in sinagoga, in all'attentimenti va in sinagoga,

32:30 The two people must defend their knowledge against all others, since their knowledge is so in-game. The opposite attitude is that you can't write each method. Actually, no one can distinguish between the precise or specific concepts, so that's the gist of my talk. The need to explicitly teach those concepts which we consider important, so that's another topic for discussion. The ones that are most important, technically, they're the... Our idea is that those concepts have to be made explicit. The other hand, for instance, the government of the United States is paying billions and billions of dollars to organize, to disseminate the opposite idea, that is to say that geometric concepts, for instance, Or a kind of evil, something that hurts students. It is unbelievable, but it's true. So that's the general approach I wish to stand against.

35:00 Now culture, especially from last century, we have this idea, it comes at that some way or another, mathematics is The reasonable conjecture seems to have a counter-example. That's what was in our experience. So going against that culture also means going eventually against the improvement of the teaching of mathematics. The improvement of the teaching also means improving mathematics itself to some extent. My experience is that there are a lot of publications that are supposed to be popular, such as scientific dissemination books and magazines like Le Scienze, for example. In America, you buy these magazines or boots or cows in the drugstores. But my experience and the people's experience is that you can teach science by home here. And the students are really impressed when they first find out that there's something happening there, but they still have absolutely zero capacity to take it forward. This popularization works. Do not allow readers to receive anything that can help them go forward in science, be it physics or mathematics and so on.

37:30 It's a pretty... It's a proficient culture, there's no real capacity of going up, and to take this capability forward, we need to explore further, even though just a few people will eventually take it forward. I just wanted to mention a few examples which I think are fundamental, but In teaching, in education, as far as I know at least, these are hardly ever taken into account. The first is homomorphism. It's quite important to understand not just that, but... Square preserves the product and the logarithm and applies the product and the actions and actions and so on. It is also important to say that all these are examples of a general concept, which is homomorphism. As we understand, there is one known example, which is trigonometry. Measurement of angles. We shouldn't just ditch those examples, but we should refer back to this concept, which is simple, but many times it's simply not done by teachers. Another very important concept, I think, is the idea of functional. And I wish to elaborate a bit on this. Our practical experience is that when studying rational analysis, we find it very difficult, so we don't really want to think about that too much. It's a complicated concept.

40:00 It is true that the concepts are complicated, theorems are complicated, but the concept itself is not that complicated in the end, and it's quite useful to try and study the relationship between, for instance, if x is a pre-infinite space and we have an integral as a functional field. There are also systems to perform integration or the remount system. These are all examples of functions such as that the domain is a space that consists of variable quantity or variable amount. So why is it useful to communicate that? Because we can... Notice that these are approximations of each other. We can actually compare different functions to each other because we can also compare the function just like you would compare points. All of those concepts actually have an Italian origin. And, in particular, Volterra. 120 years ago, worked by using this concept, the cohesion, the possibility for continuous, differentiable, analytical movement within the space of a function, and within the space of the function of a function. As we understand, the word functional is what's first created by Hadamard, Volterra's enemy,

42:30 It could be used to fight against the very strict viewpoints in the set theory we have been taught as a preparation for the discourse of what he called Line functions or firm functions. Volterra remarked the elements of a space of X are not just points. Actually, points are elements, but there are others that should be termed elements as well. Curve, surface. Curve in X, surface in X, functions of points are important, functions of those other higher-level elements are important. Should we say C is the time or co-time occurred, then x to the power of C is a space such as that the points of that space are the elements with the C form in x. So in general the elements of a certain form of x are also points in a different space. The functions with that domain are called functional.

45:00 In general, the points do not render a faithful image of space, and somewhere that's a major mistake for set theory in general, the concept that elements are always points, or at least that points are always points. In general, the elements of maybe three or four little number of forms and not just points are a faithful representation of space. In any case, there are different types of elements. Functional has always existed in practice and mathematics, especially in statistics. Statistics is really tied to functionals. So these are ways to transform variable quantities into constant qualities like the average and so on and so forth. A third fundamental concept which I think is worth expressing is the concept of the tangent. Tangente is the fundamental concept for differential calculation, meaning that for each space x there is another space of tangente. Each morphism, each transformation of spaces would lead to a transformation and induce the transformation between the tangent spaces.

47:30 And that being a user, it simply preserves conditions, so if you take a transformation condition, also derivatives, that's to say the induced morphism, on tangents, behaves like the whole composition. If the tangent space varies in that way, that actually means that this rule is substantially two-thirds of the famous chain rule of derivative. Because z to the t or phi to the t is actually a derivative of t to a certain extent. Foundations tell us that the Newton-like methods are key for the tangent function, as we can understand the formally expressed both are necessary, and that can be formally and conceptually expressed as tangent function. In fact, I talked about spaces and the transformation of spaces, indeed there are many different categories of spaces, like differential analytical continuum and many others, polynomials, algebraical functions and so on, but all this apart from pure continuity And it is necessary to study those categories, you can't confine yourself to only one category, and we need to study transformations between different categories, that's another important principle.

50:00 Concretely, with this wide variety of categories, the point related to the tangent function is honestly very, very similar. If you think of the Euneza Grothendieck theorem, of course it would be Cayley or Dedican. We got the essence of the results formalized by Juneta and Gertendieck in a slightly more general context. So having situated the concept of a representable function, Amongst the categories, there are morphisms that are called factors, that is to say, the transformation for each object in this category into an object of the other category, for each morphism between objects, to a morphism between corresponding objects, in such a way as to preserve the composition like that. Representable functions are pretty peculiar, like the functions of x versus x. So that there is a space c that f of x equals function space of x to the c.

52:30 If this is true, we actually say that f is the representable function of c. Theorem or tautology, maybe, is if f is representable by c and also by d, for example, then c equals d, in the sense of... Which represents any function provided it is representable, because it must be representable. So whenever you have a representable function, the object that does it is absolutely unique. There is no deduct as to the foundations. We can say that the tangent function is presentable. I have already written an example of the fact that the tangential space of x is nothing but the space of a function. The tangential space of x is nothing but the space of a function. One, which I call T, that is one point, only one point, which has to be given that T is isomorphic with one point, which is one point in general, not two points in general, not two points in general, not two points in general, this is an object with only one point.

55:00 Such that the space of the human body is large enough to form the tangent space of T itself, this endomorphism contains, as a result, a mobility, and in fact this proponent is the model given to this kind of light light light light light light light light light light So, rather than having a bottom-up approach, like Rinaldo Dedecky did in an attempt to find the real, by measuring it with a top-down approach, we can see that the endomorphism, the mode of No, sorry. Because tangent to x is what I said is... So, if we take the value of x and this unit point, we can find the point x, which usually lies on the tangent of x, zero, so this zero itself is the tangent space of x. This X is a tiny fragment of time. T, I'd like to stress once again, is a very simple object, a unique object. It is a tiny interval. I'm not talking about measurement.

57:30 Conceptually, if we take such a composition, such an operation, the x, on the tangency to x, x lambda, x lambda is another moment that passes the same point of movement, x zero. So the effect of lambda is that of accelerating or decelerating the motion that we call x. And this forms an isomorphous system, or ideally it should be isomorphous, a reale, a isomorphous, a reale, or a multiplication, a multiplication of the reale. And that occurs by the multiplication of the real things going on like that, we can say. Art is unique for every category where there is this object representing the tangent and where there are these functions by Voltaire. The functional space, this is R, is a supplemental space, so there is a functional space of preserving the operation of R.

1:00:00 According to Witten, these are the measures of R, and so we come to a theory without numerability or accountability. Are there any questions you prefer for translation purposes? Sorry, we missed the first part of the question. You presented viewpoints based on which this new way to look And things should become a part of the basic span of the field and together with some other I have several models that I have really sketched but we did not use the classical set of the same kind of other

1:02:30 This is the category of the categories of abstract systems that has the same potential in the sense, however, to bring up the opinion that both are foundations for teaching the sense of being, the direction to follow being a formal system that is based on... There are various categories, but it is necessary to describe categories such as space of functions, objects, tangents, etc., which in itself is important for all mathematics, because it can be defined by using all of them. There are many spaces, but there are many very particular ones, such as this one here. And what is that? It is the inclusion of constant morphisms among all the different morphisms that vary around T. These spaces form a subset of the power categories in which we... And this plays the role of the category of sets. Sets are discrete spaces. So, in a general theory of spaces, there's an extreme case of discrete spaces that automatically enjoy all of the properties. From this point of view, the theory is based on a differential calculus, but that in a nutshell is what I was actually forgetting to say.

1:05:00 Presented at the end, these tangential concepts can be regarded as a generalization of the concept of a tangential vector in a geometry, in the algebraic sense of the word. So a tangential vector can be seen as an idea of a square form. In your theory, this is actually the first part of Taylor's description. For example, T2 represents not just a linear approximation, but an approximation too. We can create an equivalence between the pictorial permutations of the group. This is not just a specimen, it is a video session, exactly represented by the hour and the minute, so the both an internal and an external. Excuse me, I want to talk about concessioni di tangente. Every curve is a straight line regarding an extremely tiny space. It is not just an approximate. It is real when the space is as small as it is. The tangent and the curve are the same in the t-interval.

1:07:30 The tangent, the curve and the straight line are one and the same in the t-interval, in a nutshell, is what I wanted to say. One of the most interesting functions, especially for physics and practical applications, is distributions, a plain encryption problem. However, distributions are used to derive continuous functions. At least one of their most important goals is to derive continuous functions. The most important object for these important functions is missing here. It is not true that what you say about distribution is a statement that points of confusion and distribution are intensive rather than intense. This contrast has been found. In physics in general and in thermodynamics in particular, intensive and extensive quantities are still referred to. It is like a dimensional analysis and there cannot be an equality between them. There is an intense and extensive quantity of energy that is extensive. They have a volume in area and in area alike, but density, temperature are immense.

1:10:00 150 years ago, Brassmann gave his famous book, the Austenus letter. He said that unfortunately, even if these extensive quantities can be found everywhere, there has always been a tendency in the mathematics in the 18th century to an extent. He wanted them to be exactly the opposite, of course. This distinction can be found in pure mathematics in everyday life. Functions are models of intensive quantities, as in the case of differential forces. We also have Currant, Curranti, and Gerard, which are essences. So distributions are extensive quantities. Currant and Calderon, just to mention a couple of great mathematicians. I don't know what I'm talking about, but I'm talking about how this distribution does not behave like functions, because you cannot make a description of a distribution because generally we are going into a whole space and we're going to have a different sort of difference. In fact, they are generalized, but as a matter of fact, in its behavior is more effective.

1:12:30 Obviously, it's because representation of a distribution, applied to F, is somehow integral. Where phi could be a function, but not necessarily the play function, even if everything exists in categories of a very high level. And adjectives of mathematicalities are linear, but it might be a continuous and representative function. We need to go back to the basics, functional, on morphism, tangents, and extents, and still find them in our everyday life.

1:15:00 Professor Petito is going to talk about the continent Africa. I wish to thank the University of Medrisio and Professor Sergio Ferreiro and Professor Fabio Minetti for their invitation. I will try and speak Italian, but I can't write Italian, so I'll just write in French. But with my talk about the possibility to build a different... I have brought with me a pretty exhaustive bibliography, so if you are interested, we can have it xeroxed. I think the discipline is a fundamental problem for its classical version and conception. The classical version of the conception is an issue that has become pretty marginal from the mathematical viewpoint because major progress has been achieved on one side with the theory of algorithms and calculative science, on the other side within the... The framework of the category theory that was just mentioned by Professor Aguirre right now, also it became somehow marginal because there have been recent developments in cognitive sciences, cognitive psychology and cognitive norm sciences, and all of those works, and I was trying to understand the very places of high level cognitive activities, i.e.

1:17:30 The mathematical activity, for instance, works on vision, calculus, video, and, pretty deeply, our understanding of the foundational issues of geometry. The famous discussion between Ries and Ramos on the foundations of geometry has now finally become an issue that pertains to our national interest as well. Nevertheless, the classical issue of foundations remains very interesting despite that, very interesting from the philosophical point of view, because it's still at the very heart of the discussion on the foundations of geometry. I don't want to give you long answers. We didn't talk about the issue of mathematics, but there were huge debates that opposed mathematical fundamentalists like Hattie Potter in his work, for instance, or the cognitive anti-platonists like Friedrich Kircher, for those who are involved in mathematics, this is just a symbolic approach to science. I just had an exam on reality. On the other hand, we still have realists, like Jean Dauquies, for whom the question of foundations is an empirical problem that pertains to cognitive sciences. There are Platonists who are somehow cognitivists as well, such as Penelope Manley. There are some other structuralists. Plasmids like Michel de Ristig, whose thoughts are from Bourbaki and the philosophy of mathematics, from Steintmüller to Shapiro and philosophers. At that point, eventually, he drew a philosophical conclusion from the fact that mathematical entities are not objects in the classical sense of this word, but more than structures. Hence, there is a huge discussion at the same time where we find, as usual, all possible metaphysical positions. If we look behind, there is this question of the type of existence of mathematical ideology and the acceptability or non-acceptability of abstract entities.

1:20:00 Now, the real issue is that the ontological issue, so to speak, that can be considered as not to be possible, There is an epistemic dimension which is particularly important and which concerns the cognitive accessibility to abstract entities. Professor Lally this morning has talked so well about all these issues. The traditional issue of Platonism is formulated by Crispin Wright as well as the The traditional Platonist answer is that the true conditions of pure mathematics statements are constituted by the properties of certain mind-independent abstract objects, mind-independent abstract objects, the even proper reflection of mathematical studies. Y-numbers can believably be applied to half the field. It's a discussion, a discussion, a program that we want to use for the country. No, but there is a special... There is a kind of the... Transcendental dialectics. That mathematically includes a thesis to an antithesis. The thesis is semantic and epistemological. It says that mathematics is a descriptive science that describes a certain type of object. So they can actually describe some objects that are abstract entities, mind independent, that exist in the platonic world, the third world proper. And the antithesis from Wittgenstein, conventionalist, syntactic. Nelson, we should say, started today. That says that mathematics is prescriptive and not descriptive. They do not concern the issue of truth like natural sciences, but as Wittgenstein said, the grammar rules for the use of concepts and the existence of ontological sense, as Professor Nelson pointed out, is not an issue because it's a...

1:22:30 The only criterion for the existence of mathematics is the proof and we should understand that the demonstration of the existence of mathematics is the introduction of new. But all of those discussions went on because there are real fundamental issues behind them not just for the sake of philosophy, gratuitous philosophy. I believe there are absolutely fundamental technical issues that lie at the very basis of the difficulty in solving that problem. The basic part, which is always behind discussions, is the statement of infinity in the set theory and especially the power of continuum, the infinite continuum. So this picture comes from Cantor and it was taken up by Cantor in 1947. The problem was that to work out a satisfactory theory for continuum, we need, and this is going to be the topic of my presentation, we need a set theory with axioms that have to be strong, and those axioms have to be strong. So, either we start off from predicative, constructivist viewpoints and we see what kind of continuum theory we can start off with. For instance, Solomon Fitzgerald, in his 1999 article on mathematics, asked the question,

1:25:00 Philosophical topics in financial mathematics is necessary. Starts off from the intuitionist reflection of Kerman Wright in the famous Past Continuum of 1918. The subtitle of Past Continuum was a criticism in St. Bunge under the Goethe analysis and Pfeffermann imposed the same style and tried to elaborate Continuum. There is that are deflationary in terms of the axioms. Other logics, especially Gödel, suggested a totally different approach, which is about studying without any bias. Or deflation of bias to begin with. The infinite axioms, the higher infinite, that are necessary in order to work out satisfactory theory of containment. And I would like to provide you with an overview of the works in the 1970s. Del continuo della teoria descrittiva The theory of continuum in the descriptive set theory, and I think these words are pretty interesting because they reached, more or less, the conclusion of the Cantorian problem of continuum. As far as philosophy is concerned, I agree with what Professor Edelman said on syntax. In terms of the belief that the semantic of infinite was a major evil for the philosophy of mathematics, and because of those irresolvable metaphysical problems of the ontology of infinite, we ended up being limited to deflationary models of the set theory. But I think we have to adopt a layman's view to this, on the question of math, and so if we need to have new ways of proving, if we need very heavy axioms for the existence of higher order infinites, that's a shame, that's a pity!

1:27:30 So let me now focus my attention on the question of the definability of content in this theory and the difficulties that emerge from the completeness theorem. To Gödel, this was one of the fundamental aspects in the... There is a very beautiful article in Inferenzia that synthesizes all this issue, actually it's a double article by Penelope Maddy, you'll find it in the bibliography. I'll try not to be too technical. I'll begin with demonstrations. Results are absolutely equivalent. So we wish to study this culture substantially in a moment. And we begin by studying the hierarchy of quantum mechanics.

1:30:00 We've talked about it in camp already, but we begin with a hierarchy, so we start off with open sense, and we iterate the complementation, countable, in the first hierarchy, which can be written down with tradition. P0 and N are the open sets, whereas P is the complement of sigmas, so P and 1 are the closed sets. In general, P0 and N are the complements of sigma0N. You have the definition of sigma 0 n plus 1, which is countable union of P0 n. For example, sigma 0 2 is the countable union of P0 n. There is a hierarchy, delta, zero, n symbolizes the intersection of the subsets which can be described in two different but equivalent ways that are at the same time sigmas and s's. Delta zero n's are strictly included within sigma zero n's, and sigma zero n's are strictly included within delta zero n's, and plus one and so on. This is the first hierarchy of mathematics.

1:32:30 Maths are complicated, but there is a construction rule there so they can be complicated. But today can be mastered, it can be managed thanks to the knowledge of principles. It's just a question of considering the irregularity properties of these subsets and I'll go back to that in a minute. And so on and on and on and on and on and on. All the closing terms are not only complicated, they are more complex, more iterative, and the information is not the same until you generate another hierarchy as you project to them. And this is very hard to do by a computer.

1:35:00 There is also a representation, as we remember, by the sequence, and the sequence, and the sequence, and the sequence, and the sequence, and the sequence, It is said that the property of a perfect set is set to have the property of a perfect set if it is countable or if it holds a subsystem that is closed without island points, or if the regularity

1:37:30 Properly introduced, they are able to be applied to be approximately the same. They are approximately done with an opening so that there is a very slight negligible difference between the two of them. It can be multiplied uniquely into a countable plus a perfect set. So, why does this theorem say that the power of a closed set, a continuous hypothesis, is true? Either it is countable, or it contains a perfect set. So, there are intermediate cardinals. For closed set, the immutable and the sub-set has the property of being a perfect set that has a distribution of regularity.

1:40:00 The continuum is always true. The analytical subset of the 19th century of the 19th century of the 19th century of the 19th century of the 19th century of the 19th century

1:42:30 So the key is that there are some natural properties of these words, the projectivists, that are independent of the concept of theory, and objects. Sorry, but the speaker is speaking French more and more, and it's very difficult for the interpreters to keep up with him. So here you see the problems. The issues of the SFC, in the sense of the future of humanity, are constrained, constrained, constrained. So the issues of the SFC, in the sense of the future of humanity, are constrained, constrained, constrained.

1:45:00 So you can satisfy that, as you like, you have a very ample latitude to choose from, and this is a mathematical, logical, graphical factor to set you free. The arithmetic of the coordinates is completely sub-determined by the length of the Zermelov-Renkel. I apologize for this handwriting. I'm not sure you can read it. But anyway, we have to take a look at the different models of the SAT theory. What happened in the SAT theory is basically what happened in geometry. There is no such thing as a single model of the SAT theory. There are quite a few of them.

1:47:30 So we have to look at the different models and classify them. Hence, a dual strategy. What is very interesting from the philosophical field, especially in the history of mathematics, is that Baudel himself spotted both strategies that are totally opposite to each other. The first strategy is a depressionary, minimal, constructivist theory, which states, let's try and limit the universe of seven theories as much as possible. And that's what... See what continuum is in theorem. It's the strategy of constructive universes. The other and opposite strategy states. We want the projective hierarchy to keep going in good properties over and beyond sigma 1, 1 and 0. We want not just brilliance, not just analytical... But all the rest in the objective hierarchy of the good properties in terms of regularity, measurability, the property of the perfect set of properties and bounds. The discovery is that there is a price to pay for that. The price being axioms of infinite superiors. So there is a second strategy which is inflationary and not deflationary that was also promoted by Gödel himself. B equals to all is the language we know in literature. V being the universe of the set theory of Newton. L being the sub-universe of constructible systems. V equals L means that the sub-universe is equal to the universe, so that the universe takes, in the universe we consider only the constructible systems.

1:50:00 I'm not going to give you the definition of constructibles, but the idea is that virtually all issues in the sense of theory come from the axiom of the set of parts. If you have a set of axioms, you have a set of parts over x. The issue is that there are too many parts, at least normally. The idea is that our family is limited in parts and size which can be explicitly defined by means of formula. So, technically, Axe is a spirit. It replates the set of parts T of Axe to the T of Axe, so the set of parts of Y of Axe, which can be defined by a first-order form belonging in this structure, where we have Axe for Axe, the symbol of belonging, and the symbol, which is the object stating that the set belongs to Axe. We have a vivid structure. We can look at all the formulas in the first order that can be constructed like that. And you look at the parts of that which can be defined in this manner. Then you can perform a transfinite induction exactly like it happens in the classical set construction. It's a transfinite iteration of those set constructions. You place in p of x with d of x. That way you get to the universe of sands where all sands are constructible. Gooden did that between 1930 and 1938 and 40, and the main theorem is that to have the equal cell constructability axiom, the continuum and the choice axiom are all in the same set.

1:52:30 Further, this maxim of choice implies there is a good order on R, and it leads to a good order that has the type of complexity of just one term. That said, delta 1, 2 cannot be measurable because following the well-known theorem by Fubini, a good order on r cannot be measurable. Hence, in this model, r is constant. You have immediately at the very beginning of the productive hierarchy, delta 1, 2, virtually have immediately, as I say, immediately after sigma 1, 1, you have sounds that are not regular, that are not measurable. So either you find it's better to have constrictability than having delta ones which are not measurable is not serious, or you may want to consider that having a delta one two which is not measurable is a serious problem and not having constrictability is not that serious, but that's part of ideological construction. The thing is you can't have those. The fundamental problem is that you cannot have at the same time the properties of the regularity of the inductive hierarchy as complete and of constructability at the same time. Now this is a fact, this is a fundamental theory, which I believe... Renews the whole issue of foundations. This is a quotation from Patrick Dernoir, specialist of these problems at the Burbecki Seminary in 1989.

1:55:00 These were conducted at a small in terms of cardinality and in terms of the minimum operations of sats to go to the set of bonds that are necessary for their construction. In other words, those small sats are connected to other properties that involve cubed terms but are very distant. From those same viewpoints, because of the incompatibility theorem to get the regularity of projectors starting from 0 to 1, 2, you have to introduce axioms of infinite superiority or higher infinity. What is even more interesting, if you want to have not just good properties of regularities and properties, but also the strength of your vision, so if you want to improve, what can be demonstrated? In the set theory, you would eventually realize that you always have to introduce axioms that are equivalent to the axioms of introduction of higher infinity. That's a given. It's merely an experimental fact. Now to conclude, I will not be able to give you the explicit definitions of the great cardinals, I will work in science more, but I make reference to some theorems showing that with this kind of supplementary axioms, there are a few

1:57:30 So, proof that allow to demonstrate the regularity of conduct is over and beyond testamentary. One of the first textured results using the forcing of this type is the Solovey-Fierce Ripper of 1969 that says if you add to CFC the sum of infinite, there is a measurable coordinate. And I tell you what it is in a minute. So if there is a measurable coordinate, which is a very large coordinate, then sigma12 is regular. It has a mere property, the measurability property, and the property of perfect sense. And this is the typical result. And then there is the Friedmann result saying that we need... This is an absolutely typical example of the measurable problem of information which was first introduced by Glulam in the 1930s. Before I mentioned the cardinals, we have inaccessible cardinals as defined by Zerbaloy in 1931.

2:00:00 The situation was not as per its semiotics. It was the son of the famous mathematician, Benjamin Peres, who criticized the theory of real by Cantor and Etiquette. And introduce the idea that cardinality there is extremely large. And Kielce said that cardinality was inaccessible with this idea is not. The particle is inaccessible if it's regular and if lambda is lower than k and two powers of lambda are below k. So you can take sets of parts as you like, but you'll never get to achieve this infinite. There is also a cardinal which is much, much larger than an inaccessible cardinal. Actually, the idea, and I will trust your intuition, is that it's a cardinal where, within its uncountable infinites, there is a partition which is somehow equivalent to the finite in finite opposition. There are infinite and finite points, so the infinites as meant for cardinals can be cut in two between small, infinite and large subsets that have a number measure and others that have a measure of one, either one or nothing. So these are cardinals that are huge, absolutely huge. The theorem of Solovey says that with this we can prove the regularity of sigmas 1, 2, 3.

2:02:30 To conclude, let me switch to another concept of regularity, which is the concept of determination. There are very interesting results about that concept. It's a more powerful concept, the concept of regularity, which is more powerful than the three concepts I have used so far. And there is, for instance, the theorem of Pristina, and I'd like to say that if you have a measurable cardinal, it can be... the one-two is determined, so it's ultra-regular, more than regular. The theorem of Friedman and the assumption of a measurable... well, the assumption of a measurable cardinal is... In the 1960s and 70s there was a huge effort being made to understand exactly what the price of proving regularity would mean. If we want subjectives to be regular up to that level, what's the price we have to pay in terms of infinites? Let me give you the result without an exponent definition. I think that will give you a flavor. Just an example, the statement is quite clear here. The theorem by Martin and Steele, 1985, so it is quite recent. If there are n cardinals of 15, that's a certain type of large cardinals going beyond the measure of infinity. So if there are n cardinals of Woodin plus a measurable cardinal, you can prove the determination of this strong regularity of p1n plus 1.

2:05:00 It tells you exactly the price to be paid in terms of platonic axioms of existence and the price to be paid for regularity. You can even go beyond this and say, what if I want all of them to be regular? Is that possible or not? Is it possible for all of them to be determined? Is it possible that all are to be measured, for example? On a similar note, with the axiom of choice I can't, because I can create a system that is unmeasurable, but they are quite weak actually, even if they can't be ordered, they may be weak because there is a system that is unmeasurable. To the end of this lecture, and this is a finding by Shannon, stating that if there is a super compact cardinal that is measurable, which is even larger than the inaccessible, then the continuum theory is crystallized spinning tanto with With the forcing techniques you can no longer change the continuum. Behind all philosophical debate there is a key difficulty. If you want good properties for the classes of the continuum substance, you have to introduce axioms. And the more regularity you want, the more infinity you need to use. Most like the infinitesimal, since these infinites are monsters from an ontological development, but I think that the existence for the negative metaphysical connotation has no sense whatsoever.

2:07:30 Behind that, there's the mere fact that, through the regularity property, you need such an assumption about it if you want. If you have a copy of the professor's bibliography, you can find that on the desk at the back. Any questions? The choice of a model depends on the consistency of the model. It depends on the consistency of the model. It depends on the consistency of the model. It depends on the consistency of the model. It depends on the consistency of the model. It depends on the consistency of the model. It depends on the consistency of the model. It depends on the consistency of the model. Could you comment on this, because I think about this audience in general.

2:10:00 On a technical level, you can decide what you want, because you have absolute freedom of choice. The only point I wanted to make is that, in my opinion, these results should be part of the philosophical issue of the foundations. In addition, other findings, such as the Paris-Harrington theorem or the Finitist version of Kruskal theorem, where you have strictly combinatory and Finitist theorems that involve incompleteness. Which, as Friedman said, marks the beginning of a new dimension in the problem of foundations, and the only point I wanted to make is that behind the philosophical issue, where you have Platonism and so on and so forth, there are some... There are real and actual difficulties, and one of them was absolutely not evident at Kantor's time, and that's that you cannot have at the same time a good regularity property and a good constructability property. There is a conflict between the two. What I've shown is actually a result, it is a mathematical result, it has no philosophical value whatsoever, but I think it can be valuable as far as the foundations are concerned. Then you can really do what you please. Constructive theories are absolutely enough. As such, you don't really need all these constructions to do mathematics. It's like with the infinitesimals.

2:12:30 And now the floor goes to Professor Royben Hirsch, who is a professor at the New Mexico University at Albuquerque. He has dealt with the political calculus and he is well known to the large public. I'm so sorry, I'll turn it down further. I'll keep this low as possible. I need it. Excuse me, excuse me. Excuse me, excuse me. I'm sorry, it's only in English, so please change it. Number one. Excuse me. I understand. I want to thank the organizers for inviting me to visit this beautiful room. I'm very happy to be here. I'm actually a medical worker down at this school. A friend of mine, his writing is important, but he inherited it. She's in California. Her name is Elena Marchistrato.