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5:00 I'm not going to go through them all, but I'll go through them in a systematic way, and then we'll look at this article. So, Giuseppe had this idea, saying that Einstein knew Gödel, that he knew him in Germany, that all this was still in a time when these ideas of art... And so the idea, indeed, was to, without foreshadowing the final result, to try to present these two, Gödel and the particle, together. The mathematics of quantum mechanics are simple, everyone knows what a matrix is, everyone knows the multiplication of numbers, we know the multiplication of matrices, but on the other hand, of course, there are a lot of supermercats, that's what I know, and whose exposés will be, towards the end, perhaps a little more technical, but they would be in mind with simple technique and above all, I think, I can make a conclusion as well on the side of...

7:30 I have tried to evacuate from these two issues, which have given rise to a lot of literature, I have tried to evacuate everything that is not strictly in scientific order and in technical order. I have printed a few programs. Here they are. You can have a look. Otherwise, it's on the web. On the web, in addition, I have just inserted, this morning, some notes from the class. At least, I think that Thierry will soon have some notes from the class. In a bibliographic course, you will find in it a lot more details than you can imagine. It's a bit of an exercise that we're going to surf on the subject of passing over time, sometimes cheating a little, but by saying more or less. Of course, in the case of complex literatures, we don't cheat, it's the text that is in a French version. It is also necessary, as a rule of inference, so that I am used to the notation, the inferential notation.

12:30 I'm going to use the word meta. This is a meta-linguistic deduction. In language, it is a writing of the consequence, while this one is entirely within the language of arithmetic, since here we have the signs of efficiency, or rather, it can be derived by negation. So this writing is the one at the bottom of arithmetic. We are going to make a convention by AX. I say that I highlight the possible occurrence of a book variable. Obviously, when we have photons, we have negation, we have existential psilocybe. I tried to create something like that with a language view to show that A is in the place of X, Y is in A. Abuses that I would like to do all the time. Obviously, I will note by the signs N the operation of the successor on Z. What is the logic behind it? Well, I have said that it is not my name, but A allows me to say A.

15:00 This will be the linguistic notation, which is quite equivalent to saying that A and A allow to deduce B. There is also the possibility of having a special sign that is not used, but it is convenient. To use a sign for the contradiction, this is the absurd, that is to say, it is quite equivalent to saying A and not A. Or, if you will, zero equals one, since we put it in the axiom that it is false. We can use these special signs, or any contradiction, or zero equals one. And the Roman ancestors said that, then, from false, we can deduce anything. You know that a theory is contradictory when it generates the absurd, that is to say, in a number theory. Why? If you think that in the 6th century, Pythagoras, roughly the same. It is said that everything is a number, since we can actually associate everything in the measure, in the field of Pythagoras, with the ratio of the ratios.

17:30 Well, no, if you take a square, if we demonstrate the coherence of arithmetic, we end up with the sign.

27:30 What are the functions of the general recursive recursion? It is a separate class of the general recursive recursion, such that there is the constant function equal to or equal to zero, there is obviously the successor function on the whole, and there is an identity. This class of functions is closed by composition and closed by the recursive recursion. What does that mean? h of x0 equals h of xy plus 1 is equal to g of xh, which is called the positive recursion. You see, it's a kind of positive recursion. It's the modulo iteration in plus 1. And then also by minimization. What is minimization? Of course, you understand that this is also effective as long as f is effective, in particular with torsion-duction, it is simply a matter of starting to give f.

30:00 The only problem of representation of the function that I call this class if a is equal to 1 is only if the piano shows the formula a. I just wanted to tell you that I am going to write down by a the numbers of 2. When I underline, it is a number, it is a number. How to demonstrate this theorem? Well, the idea is that Goethe writes inside arithmetic the functions that he is going to demonstrate. He does the formal work and on the whole. He denies, such as the number of Goethe. Inside arithmetic, he can demonstrate that it is the equivalent, or the number of Goethe, of the negation. So, inside arithmetic, he writes a function that does the work of a continuum.

32:30 When we write an arithmetic function, if we take two arguments, the numbers of a and b, a times b, a sub-function such that when we give it a term b and a formula in which it appears to be ethical, the number of the formula in which we have subtracted b, the function of substitution, inside the arithmetical equation. This is the first part of the presentation. I will go into detail on these three programmers of history. They program an assembler, they prepare a machine language with the zero and one, the induction of the zero and one, and they write these functions. This function here is a representation of, again, not like that, it's less, it's minimal, for all recursive partial functions, Such as n equals m, it is only if the arithmetic shows a, f, n, n. For once, and much less than that, there is no need to enunciate with the terminology being used.

35:00 But for once, it is an incredible work of programming, since it takes into account the place of recursive-primitive functions. In fact, we can also do recursive-partial. You see, it's a little bit the opposite here. That is to say, here I represent as a function the proof, and here I represent as a proof what is the basis of a function. The third representation lemma, which in fact is a corollary. There is, there is a formula, P-E, in which appears X-line, the formula of the arithmetic of P-A. Earth, Goethe's number of A. You see, this is the planet TH as its equilateral. Again, here, I have particularly cheated in the sense that the implication of the left side is indeed the consequence of the case of A by Goethe. And from right to left, we have to add an important hypothesis called Omega Coherence. What does Omega Coherence mean? If, when it shows an a-n, a whole a-n, or a whole a-n, it doesn't show that there is an x, an a-x, what does that mean?

37:30 Well, I say that I would never say a model semantic model, but let's make it quick. And that means that, you see, if the computer is omega-coherent, it is also coherent, since... Meta-theory is a contradictory theory. It shows anything. So if we have a hypothesis like that, it means that it is coherent. In addition, being non-coherent means that it does not show something that is true elsewhere, outside, in a non-scientific model. But to tell you that I have this regular correspondence. Meta-theory deduction is theoretical. It is a theoretical one. When there is a proof, it is a whole, it is encoded by a whole. If this has been verified, it is only on a standard model.

40:00 And the unitary metatheory of mathematics is interiorism. What does that mean? If I have Earth A, Earth B, Earth B, the history of wonders, what is it? It is reflection, that is to say, something is interior. All that has been done. We will also internalize the more meta-torems of meta-torems.

42:30 Ah yes, yes, yes. But let's go back to one of the masterpieces among these two representatives. They give this equal to b instead of x. And why is it important? First of all, for the technicality. And then, since we will see that the calculus takes the starting point, we will need it. This point of arrival, this is a big problem. The calculation of the calculation of the calculation of the calculation of the calculation of the calculation of the calculation of the calculation of the calculation

45:00 This combinator is born before the lambda-calculator, before Goethe himself, and we are going to demonstrate something like that in a system that, like Goethe's system, replaces with Trigger a term in place of Goethe. I told you that for the first time there were implementations that replaced Goethe. Let's finish here. The expression g of x is identical, identical, it's the same sequence as before. Not the third, not the third, g of x is not the third.

47:30 But then, M is a formula. Gx is a formula. So it has the number 2 of M. But then, if I replace Xm, I would have the identity North, Earth, M, M. Everything is formal here. I replaced something identical. M. Well, M is the number of cubes, so when it appears there, it's exactly the same thing. If I put Gdx, Gdx, Thea shows that Gm is equal to what? Well, look what's written there. If I put the sum in place, it means an equality. For that reason, there is no identity, but the demonstrable equivalent. I put it in place. The crucial sequence of these definitions is that Gm is demonstrably equivalent.

50:00 All of these are synthetic games. If the arithmetic is coherent, we cannot show G, Gm, P1 does not show us Gm. Yes, yes, sorry, but I had sent it up there, because we use it from right to left, indeed. From left to right, it's totally useless. From right to left, at least at this level, in this test, it's useless. And omega is coherent. So the proof, let's assume that P.A. shows G.M., the representation lemma, but also the definition as a fixed point by the theorem of a fixed point that shows a non-fair G.M.

52:30 Yes, there are more than one. I would like to point out that the only proof is intuitionist and some points are not necessary. The B-note is called counterpoint. The intuitionist point is quite acceptable.

55:00 A typical arithmetic formula, which is neither aid nor negation, as said by Iguedelin once again in this text, leads to astonishing results. What is the other most remarkable thing? And now that I see it, I can define in arithmetic even this property so metaphorical that it says it is coherence. What is the coherence of arithmetic? It is our father. Or, if you will, our father, zero and zero equal one. It is written in arithmetic, but it does not prove it.

57:30 They do not show zero to all mathematics, they do not show the same to art and to arithmetic language, and it's easy. It's not corpus theor, which is not easy. The result is the result of coherence, yes, identical. I do not understand, that is to say, to feel excluded, it's because of the post-post theorem. At the level at which I put myself, no. We can minimize things, there is a lot of work that sticks a little, but it's not very strong, but it makes no difference. And finally, the next thing, dpa, is equivalent to gm. Let's see this application, which is the most interesting one. So, arithmetic shows the pair gm by the g of the interiorizations, but we know that our a, and that by double contrast, our pair gm is our g.

1:00:00 The term GM, since, of course, the term GM can be seen somewhere, this interiorization, ONA, the term GM, by contrast, in this term, and therefore the incoherence, the term, I used in the past, in contrast, I used the interiorization that allows the terms GM and ONA, so I have the absurd, by contrast, I have the term GM.

1:02:30 And then you saw that the game of interiorization that led to speculation of the arithmetic of the pianos, if we add the coherence inside the terrain of incompleteness of the pianos, if we add the formalization of the omega-coherence, which I will not give you, shows our term, our g. So if we add to arithmetic the formula of coherence, It internalizes the terrain of complexity. It says inside of itself that the Earth is not a terrain, that a G is not a terrain. And if we add a formalization of the omega-coherence, it shows inside of itself that our gene is not a terrain either. That is to say, it is an internalization of the supermeta-terrain.

1:05:00 But no, considering the essential... That is to say, genes, our different genes. In addition, this one, this one is... Yes, and they are equivalent. Ah yes, we see. So, you see what we did? Now, let's write with this ambiguity, metalinguage, language, in a synthetic way, what we did. We did the following thing. We said, the coherence, and I started with the accent, it's metalinguage, the coherence of the arithmetic of P.A. implies, the big arrows too, it's metalinguage, implies that P.A. does not show G, but it also implies E, for metalinguage. I do not show you our ideas. This is the first field. In the metacognition, I talk about probability, I talk about coherence, and I do not talk about contradictions. Goodell, after having done this, crossed the line, which is a very delicate point, without which the first field has no meaning. In fact, there is no meaning, there is no strength, there is no visibility. But the meaning is understood if we take into account at least the second field. This is the first field. He says that arithmetic, from the formalized coherence of g, sorry I said g, I wanted to say g, you see the masterpieces, the extreme finesse, the columbus of Goethe, he says, if I suppose the coherence of arithmetic, I can't say anything about g. If I write the coherence of arithmetic, I deduce g. It's an extreme finesse, that's how you can see it. Meta-theoretical coherence doesn't allow me to say anything, concretely, I can't say anything about g.

1:07:30 All of this without ever mentioning the words truth or false. It appears then that the remark that GM signifies itself that it is not demonstrable that GM is true, since in fact it is undemonstrable. Thus, the proposition in the BA system is undecidable of mathematical considerations. There are some surprising results regarding the demonstration of non-contradictions of the formal system. Results that we will discuss in more detail in the 11th Turin. I will demonstrate, not by mathematical considerations. I will demonstrate, I don't know if I erased the proof. Ah yes, I erased the proof. This article, in addition, does not speak of its article. It is the mathematician who looks. There are enunciations. He says he is a demonstrator. No, beware, a hypothesis of coherence. Ah yes, because you saw in the book, a hypothesis of coherence. And when he starts to say, ah yes, a hypothesis of coherence, he says that it is undemonstrable. So, he is making a proof. I said attentively, this proof, this is hand-waving, a future of G. A mathematical proposition, it's by the proof. It's indescribable. And this story has given rise to incredible discoveries. People who have not read the second one, who demonstrate G.

1:10:00 From coherence, exactly by the formal proof of the usual hand-waving, which means that the answer is true. This means a poorly done proof, which is a bad copy. And it is said, since it says, I will be demonstrated by considerations. The most amusing theorem is the one of Goethe's theorem of the necessity of solfeggio. It's incredible, as a great mathematician, to find out that there is a secret that is not demonstrable in coherence,

1:12:30 and how is it that it is true? The other thing is that the distinction between language and mathematics is very common. It's like coloring in red a note, marking a text in black. That's mathematics, we don't put it in the language. The problem is that by the school of Bertier and the foundation, Hilbert in 1904 thinks he has demonstrated the coherence by induction of a theory in the main axiom of mathematics. At the beginning of the 1920s, he gave himself a metatheoric induction, as shown by an induction. It does not prevent us from keeping this distinction, but it has no fundamental character. And why does she say that? Well, because there is the interiorization, all coding. Your metatheory, as much as it is finite, therefore codable. I have the theory, the R3, the definitive. Everything you do in the metatheory, I do it in the theory.

1:15:00 You see, the theory of the life has internalized itself, the incompleteness. The metatheory is a subsystem of the theory, so it has no character. Do you understand? It's like a... Not only in the subsystem, it is a different subsystem, because the theory of arithmetic allows us to define the A of coherence, while the metatheory does not allow us to define the A of distinction, but has no foundational value. The mathematical analysis does not correspond to the mathematical analysis. But I'm going to demonstrate it even more strongly by giving you the axioms in the system's theorem, where not only, like Goethe, but in the proof there is an essential mixture of theory and mathematical theory. And we can't do without it. That is to say, we demonstrate its demonstrability And so on and so forth, since Sontereine, who is quite a bit of a physicist, is a bit of a consumerist, since he was doing that. Another big contribution is this story of comparison, but not allowing this sharing of the world in half. What are the paradoxes of the sharing of the world in half? It is to share in half the world.

1:17:30 And even for democracy, it was to share the math in half, the propositions that are demonstrable and the ones that are demonstrable in negation. No, we can't. I would like to highlight a few remarks. I will announce the philosophies against Hilbert. The first is to write a very long summary, 20-30 pages long, of the absolutely extraordinary book by Hilbert on the fundamentals of geometry. It is a great philosophy, but it is said that philosophy does not really please, since this idea of the potential mechanism of geometry, it is Hilbert who insists on it, it is necessary to mechanize the math to have certainty. Since we have lost the certainty in space, since we have lost the relationship between geometry and space, it is the finitude of the reasoning that is represented by mathematics that is the foundation of the theory. And then he continues, but if Hilbert thinks that mathematics... Ah, well, first of all, Jamison's resonators, apparently it was a crank piano from the 19th century that was widely used in the soil for mechanical reasoning. But if Hilbert thinks that mathematics is like the Chicago sausage machine, that too, maybe it was something... And we have known them from the start and they come out of the ground. I don't know if it's my friend Umberto Bottazzini who has reversed the gold and made the chiasma. In any case, it is reported as it will be. It's for science, the genius of science.

1:20:00 And many times, that's very good too, many times already we may have solved all the problems, or at least made the inventory of what contains a solution. And then the essence of the word solution has expanded. Insoluble problems have become the most interesting of all. And other problems have occurred to which we had not thought. And that was the moment, the history of the non ignoratibus. To demonstrate where the question is. Knowledge is not a question of mathematics, it is a question of the theory of knowledge. And then Hermann Weil, the best student, in 1818, at the end of paragraph 3 of Das Kontin, it is not clear since he hesitates and says that it is possible. I'm not sure, maybe, there may be rational sequels that converge into a reality so quickly that we can't code it in arithmetic, write it in arithmetic, I don't know, maybe we can, well, he hesitates, but still he concedes, it's absolutely incredible, and then he will say, I was a lone wolf, a lone bull, since everyone worked on the program, but especially in his text several times, this story of mechanization of math, it took him by surprise. And then Wittgenstein. I don't know if he knew Witten in his 30s. I don't think so. He thinks that mathematics, mathematics, metamathematics will turn out to be a disguise. He appreciates the disguise of the disguise. A metamathematical proof should be based on entirely different principles with respect to those of the proof of a proposition. In no essential way there may be a metamathematical proof. I may play chess according to certain rules. But they may also invent a game where they play with the rules themselves. The pieces of the game are the rules of chess, and the rules of the game are, say, the rules of logic.

1:22:30 Let's say that the merit of Del and Kierkegaard and Péan are extraordinary, after the improvisation of arithmetic, these important attempts at the time to get out of the delirium, as Kierkegaard said, of geometry, are at the heart of an absolutely remarkable development of teleological mathematics and arithmetic machines. We have to change the world. So it's not at all that the interest is complete. It doesn't mean that it is genetic. Let's reason with an example that I have to give. There are a number of construction principles that allow objections to have an induction like this. This is what has been going on for a long time, since then, since the work is technically remarkable. It's enormous. It's about taking the blow to stay strong.

1:25:00 The sum of the numbers that match the meter. The sum, but we have said that there is a limit to the meter of Gauss. He's going to help him, he's going to follow him, he'll do two or three more little tricks like that, and he's going to be in a peasant family, his parents want to take him to his studies, he's the master. The little tricks are remarkable, they are remarkable for several things. This proof is not just any n, it gives a geometric structure. Of course, if you avoid the sum of n, you give this formula to anyone and you don't have it. Where is the foundation of this theory? How is it that the foundation of this theory exists? Yes, of course, apostasy, I read it in the formula, apostasy, I read it. But what is the foundation of this theory? The elements that I consider to be the courage of Peligron, is to see the rest of the people who are in his head,

1:27:30 as you see them all, to see the rest of the people and to reverse it. What is incredible audacity. These are the mathematics that see this symmetry by reflex. And he does that. Speakers include a generic element, n, and a structure. This is a major challenge, and my colleagues, for example, demonstrate this formalist myth that to demonstrate a theorem is to demonstrate a formula. That's not it. Demonstrating a theorem is to answer questions, it is to correlate structures with automatic notation. They know very well that not only is it the difficult passage, but also that if the proof is not the same, the choice is the same. In a theorem, it is very rare that we demonstrate the formula we know by induction. By a notion on the formula itself, normally, even in automatic demonstration, if we give something a little difficult to demonstrate by a notion, the machine thinks that the symmetries are so important. For example, if you look at the fold with your eyes, you see that these are only the symmetries, the axes.

1:30:00 What are the two points? Tracing is the most important. It is the situation that is most important. It is a game of ideas. What is a game of ideas? Well, Geo-physics, we can say conservation, is the end of the earth. These are symmetries. Trace the circle. Take a part and put a part of the earth on the floor. One and only, it is the situation that generates more symmetries. The symmetries that give Geo-physics is that to have an optimality, their symmetries are not only that of Geo-physics as a property of experience, as symmetries. The order that allows us to talk about them are these constitutive principles of conceptual construction that give us these useful concepts. But what is this work? It is a prelude to the announcement of the Count, which is very ancient, very human. If you know the book of Stanislav Dehane, The Creatures of Animals, these judgments of work, of organization by counting, obviously we add language, which gives us interesting information in the same subject. Brauer says differently, Brauer is fascinating. The name of the course is the fact that temporal spans are shared in two. It is the discrete temporal sequence that is the judgment on which it centers. It is true that it is necessary to add several acts of experience, active experience, of the practice of the world, the launch, the counting, the consciousness of time,

1:32:30 to have the variants, the variants and the concepts that transfer from one to the other. The analysis will focus on this problem in the sense that I'm going to try to show the evidence by basing it on the history of the order of the students. You see, it's a good order. What does it mean? The sequence is discrete. It is posed by language as the generalization of the order and the counting. You see it well. If one is sufficiently mathematical, one generally sees the points after the other in a discrete way. For mathematicians, a theorem of a theorem, if one cites what is said between them, Enquiries include, therefore, students and members.