Essential ambiguity of natural number concept discussion (contd.) / Karl-Georg Niebergall: Weak theories & consistency proofs

0:00 I think, so I can, let's see if I can remember this. So we, we, we take the x, you know, so the, the class, so I, I do this classically. Yeah. So you take the set of x, so that for every y is, eh, yeah, oh, the x plus y in, every y in, in this initial segment. Well, no, no, sorry, so it's a set of X, so let's call this thing N, right? It's a set of X, X in N, every Y in N, X is Y in N, right? Something like that. And then you can prove that this set is a subset when it's closed, et cetera, right? So that's the methodology of shortening cuts. So I don't know whether this works. Well, I've not read Nelson as carefully as I should, The thing that worries me about it is I know, for example, we know that there's no constructive description of any non-standard model. Sure, but that doesn't tell us anything about this, because... No, you can repeat these arguments in here. These arguments, for example, if you take the von Neumann and do this kind of argument, an initial statement, it would produce, if you add this action, the negation action, then you would really show that if this construction works, we would see that that's relevant. There's one stand-up number here, right? There's two of some number here. I think it's at least interesting, so maybe your method of expanding and this method of contracting are in some sense the same. Well, I don't know, because I don't know, so practically I would be... The proofs in these cases are just calculations. Yeah, I don't know the detailed connections. I thought I would recommend that you try that. It's a completely obvious question.

2:30 It's very funny that you... But it could very well be that both directions work. No, I, but it struck me as, yeah, there's the, there's the book. So there's a, there's a, what was I going to say? I mean, when I first read, when I first saw the law to the phenomenon, I was surprised for exactly the reason you are, because I was thinking, well, the way you get these that closure properties is to cut back. You say, only take things that are small enough to have this function defined. But this thing is saying, we're making, we're throwing enough extra elements in the term. No, no, no. So it's intuitive, it's quite different. Yeah, it's a, yeah, it's a, it's a, and I never saw it used precisely in this way. The point is, if you, if you do this stuff inside, I mean, what I'm saying, I guess, Forget about all the art and foundations. If you want to study these phenomena, it's better to use elementary arithmetic than it is elementary arithmetic. Because, I mean, basically, so much of the stuff that's weak arithmetic is tied up with just coding. Yeah, there's no coding here. No, no, no, no, no. I sympathize with that, right? It's coding for you. One way to do that. Let's go get a coffee. Still one other remark is... Do you need a... Well, let's go get a coffee. It's simply that there is a theorem in the concept of non-standard models by Alex. Well, thank you very much for the invitation. Sadly, unfortunately enough, I don't have slides. I didn't have, for several reasons, I didn't have time to prepare slides on the finished text. So, the whole thing will be a little bit experimental. Maybe I will lose track sometimes. I don't need this again. I have written some formulas, or most of them, that I need on the blackboard, on the whiteboard already. So, the talk will be about nominalism, kinetism, deducibility, theory proving their own consistency,

5:00 until that's programmed. So, very nicely, what has been going on for me? That's something like nominalism, nobody cares. Okay, so I'm surprised and happy in the last 17 years. Well, the problem... Personally, I think nominalism and kinetism are very plausible, But the problems, the finite system, the nominalist hands are these indispensables to come down to them. But how should he react to him? I think he should survive a nominalistic treatment of terrible things. For example, of certain ways of talk found in ordinary language. I don't address this point, but that's a big area of discussion. There are some formulations in ordinary language where some people think they cannot be nominalistically reconstructed or treated. Well, I don't know if this is so bad. Another problem is using nominalistic treatment or construction of mathematics. Further is of formal syntax and of formal semantics. So, for example, public simplax. Now, if you build up the meta-theory of propositional categories, such principles as this, for x is x, So that's a formula then non-X. So that's a formula then non-X. So non-X is the negation sign concatenated with X itself. What you do there is you talk about this concatenation sign, about a concatenation sign, about this concatenation sign, and you may concatenate the whole thing again with this concatenation sign. So the assumption that you can add the concatenation assumption, the negation sign, you can add the negation sign in front. Unrestrictly often, this is, of course, an infinitistic assumption, as I said, but talking about the negation sign, not about the negation sign in description, but the negation sign

7:30 is more essentially about the type, so it's an anti-nominalistic assumption. Platonism, some sort, and infinitism, some sort, is already built into the meta-theory of the propositional calculus. Of course, the nominalist has to deal with this. Well, if the nominalist can give a nominalistic reconstructional treatment of, say, terminal factor-death theory, he also can give the reconstruction of these things, because he can develop form of syntax, etc. And he also has a nominalistic reconstruction of model So, in some sense, it's enough to reduce or to treat mathematics in school general ZF nominally. And if you want to have finitistic treatment or reconstruction, you also aim at a finitistic treatment of all of ZF. Of course, one could have the idea, it won't work. I cannot give a nominalistic treatment or finance. Well, in other words, let me formulate the reduction or treatment. Now, I've talked about nominalistic reconstructions and treatment. This can mean a lot, that's very vague. I opt now for a more precise formulation, and I don't ask you for it, so it's just another topic, another talk to ask you for this. the discussion. So, in nominalistic reduction, thesis would be mathematics is deducible to a nominalistic theory. And the sinicistic reduction claim would be that medicine is deducible to a sinicistic theory. That's perhaps too quick to formulate the sinicistic reduction thesis claim in this form. Okay, let's replace talk about reconstruction or treatment, or ontological talk in some sense, to talk about theories.

10:00 So that's a game in some sense, but still, we have to say what sustainability means, what mathematics means, and what nominalistic theories are, or financiistic theories. Well, for me, mathematics is theta or theta-3 is the best case. But maybe it won't work with theta-3, so P-A would be good, Q-2, but if we don't get a reducibility of Q to noministic theory. This piece is failed. U is believed we should be able to reduce noministic theory. So let's set some test points here. We had also yesterday another type of statement, with the number of F is M. Okay, this would also be a task, but I think what What I would demand here, in addition, is that the n of the numerals are examples. That there should be a variable. And then, of course, the normalistic strategy of using long-existence statements, so it doesn't work. You could have a variable as index. And, well, we do not only want to claim that for given numerals n and m, they are equal, but we want to have it for variables. So, in some sense, this should be global, immunomalous theory. It isn't global in Q, so Q somehow connected to this point. In Q we have the position for the In the end, it may be that one should think about changing mathematics, somehow, in the first talk. I want to stick to mathematics as we are used to it, but I don't conclude that after a long One such type of reconstruction has been suggested by Michielski. Do you know this? Maybe I just will shortly say about it.

12:30 So, as reducibility, well, I'll say more about reducibility, but you can, may think at this moment as relative inter-precibility, presented in our physics. But I will also look for other types of recusability relations. Of course, this is also, again, a discussion on its own, which inter-theoretic relations deserve to be a recusability relation. So I will opt for some versions, but probably not provide any argument that these inter-theoretic relations . But I will start with nominalistic and finalistic theories. But first of all, even on an informal level, there are different conceptions of modern nominism. So what we had yesterday, for example in Fields as a position, is the nominist avoids the assumption of abstract abstractions. And that's the Quine position. Quine makes it strong. But there are other positions. One is, the nominist avoids the existence assumption of universals, and you should not start universals with abstract objects. So some people who hold these rejections of universal positions do not identify universals with abstract objects. For example, Armstrong. Armstrong is no nominalist, but, of course, his cousin loves the subject, and for him the nominalist may well accept sets, for example. Sets are no universals for Armstrong, and the nominalist rejects universals. But, of course, nominalists, for quiet, can never accept sex as just another type of woman. And the most famous nominalist properly, Nelson Goodman, has still another version of the normalism. Goodman does not reject actual objects or universes, but not individuals. Goodmanian normalism accepts individuals, and individuals are not concrete objects, or not particular. It's again something. The version of nominalism I like most is the one, the quieted one, rejecting abstract objects. And that's also the field position. Now, field doesn't have a reducibility program, but a conservativity program, something else. And I would like to talk about this point later in the discussion.

15:00 you said that the conservativity claim of field is false, and I thought so too, but in the meantime I'm unsecure about it. Right, Burgess and Moskovak has put it out that there's a Godemnian construction and Shapiro wrote a paper show. But perhaps they are not right. We should talk about it, discuss this later. Okay, the reducibility question is rather, again, Aquinian or Goodmanian. So, what are nominalistic theories? Theories whose models only contain concrete objects or individuals or so? Well, what is a concrete object? What is an individual? And how does one manage that an arbitrary model of a theory contains only concrete objects? you can exchange models consisting only of concrete objects, you take one of these concrete objects out and put an abstract one in, and you get an isomorphic structure which is still a model. So these model theoretic explanations do not work, at least they do not know anyone which works. Now what is a number theory, or what is a set theory? A number theory is a theory only about numbers. It doesn't work. Again, model theoretically is the center of I think it's the other way around. We learned number talk, and we abstracted from it to build a specific number theory, like the other is the beginning of the excessive count. I don't know. And this is our paradigmatic example, which we use to restrict admissible explication of what a number theory is. Number theory should be close to PR. PR is a test case for us to explain single number theory. But what close means is very basic. So I think what we can do here, in the case of number theory, in the case of set theories, is to give paradigmatic examples which are accepted for the people working in this field as number theories, as set theories, as novelistic theories, as kinetistic theories. I'll come to kinetistic theories later. been accepted as the paradigm case of a nominalistic theory of what is the calculus of an individual. And I have written down once a version of the calculus of an individual.

17:30 So the question would now be, is the theory like C, C, R, Q, reducible to the calculus of an individual? Now, there is not only one calculus of individuables, but there are many calculus of individuables. You can add core calculus and add some further axioms. This I will consider. So, the calculus is written in a language which has just one two-plasticity. And I will look for extended extensions of the protected catalyst only in this language, for example. So, in part of it, find this Y overlap scene and Y overlap. x, x and y overlap if they have a common part. If you have objects x and y, there is a sum of them. Maxim says, if x does not overlap each object, then if x, an x which overlaps object is the universe, at the same time. So, if it's not the universe, then there is this couple. Okay. Well, let's call this the CI. And now the question would be, is, say, Okay, Robinson arithmetic is reducible in CI. If it works out, the nominalist has given, at least, the nominalistic reduction of Robinson arithmetic, and so on. Okay, now we have to say more precise things about reducibility. Let's start with relative interpolation. So the question is, is you relatively interpret in this theory Ci? And the answer is no. The reason is as follows. This theory has a finite model, and you cannot interpret a theory which has only infinite models, like you, into a theory which has a finite model. So, the reason is simple. If the theory has only infinite models, like Q, it proves there are at least n objects, n is arbitrary.

20:00 Now, the explanation is, there are at least n objects which belong to this additional domain, but then there are at least n objects. So if the theory S is adjusted in T and S has only models of size at least N, T must have models of size. In the argument, but you can use it to defuse a special version of the nominalistic reduction claim. It's only a special version. Now the nominalist goes on and presents other theories. Now what he could do is, there is in this language at least one extension of the space theory, which is strong enough to relatively interpret cues. And as a result, there is no extension, no one at all, that interprets cues, would be a strong reputation of the idea that this sort of nominism is enough to reduce my life. Okay, I do not know if there is any connection. Then what are the rules of the game means we should not extend the language, right? No, right. Because otherwise it's easy. Right. Sure. Now, one can do the following. One adds an axiom of atoms. They feel like atoms. That means for each object there is one which is a part of it and is an atom. SNB has no proper form. This gives the atomic calculus of individuals. And now there is a classical paper by Hodges and Lewis in News 1968, where they show that, so this was CI, atomic calculus of individuals is ACI, and they show that the completion of ACI are these theories, ACIN, for n, this is 1, and ACI infinity. ACIN is the theory ACI, atomic classes of individuals, plus the claim there are n atoms. And this is ACI plus the set

22:30 of sentences there are at least n atoms for each n natural number. And these are the completions of this infinitely many theories here, each of them, and that of ACI, and these are all There are no more completions than those theories in that language. Now, all of these, and this is shown in the paper by quantifier elimination, but you can prove it also with such a categoricity argument, all of these are decidable. So, if Q were interpretable in a consistent extension of ACI, it would be interpretable into one of these. That means it could be interpretable in a decidable theory. And this, again, cannot work. The reason is that Q is essentially undecided. So what we have in the first document is theories with finite models are weak with respect to interpretability. The second document we have is decidable theories. Okay, so as long as you look for extensions of this kind of list, both get to the normalistic reduction. Now, one can instead of asking the atomic theory, asking, one can ask the atom streamer for each x there to y, which is a drop-off part of x. Okay, this is a code for three calculations. The theory is, again, complete and recyclable. Same after that, doesn't work. Now you can look for mixed versions. Mixed versions contain the negation of the atomicity and the negation of the atom-free. So they have atoms, but at some point they have incinerantly descending sequences. And you can add to this theory, MCI, again that there are n atoms or that there are infinitely many. Now these theories are, are, again, complete and decidable. They're both interested in Q. But now the question is open, what are all the completions? It's not known as possible. But I don't know, then I haven't found the paper on it. The guess would be that mci infinity is the only completion depth, then it could again be decidable, and when we start with ci, there would be There is no way to give a relative interpretation of EEQ into this nominalistic domain.

25:00 But, okay, that's a point for research. What are the completions of this theory in that language? By the way, this theory here is complicated, but it's not a less small category. So, I fear that you won't get categoristic of LF-N-Categoristic of the Mink-Bosler here. But somehow, this is the models have as the part models of this here. Okay. Okay. Now, nominalistic reduction in the sense of nominalism and reduction percentage here both work. We can show it. Well, extent of the language and the theory. Now, in which way can the language be extended and still be nominalistic framework? This is how to One thing you can add is concatenation, and we'll talk about this in a little bit. The other thing is there is a paper by Lewis, from 1970. Nominolithic set theories. He has the language, um, . And next to X is next to Y. And she says, footman has it in a spoken structure here and so. You could add second order qualification in terms of this rule. I won't do this. But I think second order. What's the problem? You can add X to the parent of Y. Goodman's and Quine is paper. That's the one constructed nominally. They put a lot of work into designing, in concatenation theory, X is an ancestor of Y by using X as a parent of Y. So, they try to mimic the theta definition of the incestuous, but, of course, you could add axis and accessor of y as a primitive.

27:30 Here, as a primitive, c and c have the same length, or have the same size. If you're doing axiomatic geometry in tasking style, you have A lies between B and C. And the distance from A to B is the same as the one from X to Y. Of course, the last one is cross-experiment, it's not supposed to be about distances. So, these may all be normalistically compatible threads. But, yeah. Okay. So I will talk about a special version, namely about concatenation, concatenation of tokens, that's important. But before that, let me say a few words on finite system theories. It's hard to say what finitistic is. Did you mention the difference between finitistic and specific finitistic? Maybe explain a little bit more later. I have difficulties on bringing together intuitions I would have, understanding the work finitistic, and certain claims about finitistic theories which are accepted as not theories. I would say that a theory is finitistic if it has finite models. Even earlier, a finitistic theory or a finite theory shall not make assumptions of infinity. I would try to explain this by it has finite models. I wouldn't try to explain it by all of its models have to be finite, because I want, for example, first order logic to be finitistic, So I would prefer theory of finacistic if and only if it has finacism. But then in the true theory community, it is accepted that primitive recursive arithmetic is finacistic. That's the example for a finacistic theory that has no finite knowledge. But primitive recursive arithmetic has a special alpha property, namely, if you look at the theorem, which is in a sentence, which is closed, this has a finacism. Because, following Richelstein, the theory is that each of those theorems has a finite model, locally finite, just because of locally finite. So the idea would be to explain T is a finiteistic theory, but T is locally finite.

30:00 But I think this is too far. This allows too much. So maybe finitistic theory is something in between these two challenges. A theory which has finite, well, is finitistic, and a theory which is finitistic has been locally finite. This doesn't give me a definition of finitistic theory. Only wins, no? So I would like to hear your intuition about what finitistic theory is about. A clearer understanding of what noministic theory should be than what finitistic or finite theory should be. Okay, let me prefer to convince the same, some more virtue of refusability. Yes, I'm coming back to the Lewis statement. What Lewis does is, he has this nominalistic language with part of and next to, and he gives definitions of, is an element of, well, member, is a member of. And he lends his definitions to be in such a way that his separate versions are so, that schemaca, which are close to well-known set-theoretic schemaca, like sectionality, comprehension, or separation, turn out to be derivable, giving his nominalistic actions and his definitions of membership. That's his criterion for developing a nominalistic version of set-theoretic. Now, there's a problem. It doesn't give accents for for those and for next two, so one cannot really say, if it were. But apart from this, the idea would be, if a theory proves extensionality and separation, extensionality and separation, it is a set theory. And I don't agree with this. I think for being a set theory, the theory has to be able to prove there are many sets. And this theory with extensionality and separation has this one-element model. So it's just some A, and the epsilon is interpreted by the epsilon relation. But already classical logic gives me such a model, because classical logic doesn't allow the end of the main. So I wouldn't agree with Lewis that the derivability of extensionality and separation is a criterion for being accepted. Now he adds, following, also he doesn't give actions, he wants his models of his theory to be finite.

32:30 Finite areas of objects in our space. So whatever his nominalistic finite set theory is, it is again a theory with finite models, like here. Now, then we, this has finite models, this here too, here we go to infinite models. Now I overextend, theories with finite models and decisable theories are weak with respect to relative interpretability. I want to extend this in one direction. The reasoning was, we see this, the probability of these n-object claims. So what you could do is replace relative multiplicability by a wider reducibility relation. Maybe as a nominalist would be your reaction, maybe a dubious reaction, just widen the reducibility relation to obtain the hoped-for reducibility to a nominalistic set theory, a nominalistic theory which wouldn't go up with relative multiplicability. There are many possible ways, of course, how we could buy it in such a litter, the articulation, and many irrelevant ones, just that some sentence or so, that's not enough, some structure should be preserved. The relative interpretations could preserve quantification of the structure. Now, one natural liberalization would be just preserve the composition of the structure. So the question would be, is Q reducible in this sense into ACI, for example, or into Lewis's theory? So we're looking for mappings, mapping theorem to theorem, which commutes with negation sign and conjunction signs. And these are inter-theoretic relations which are quite wide in the area of the usual, usually meta-mathematically considered theories. There is a paper by Puerl, do you know what it's pronounced? Vulkan Puerl? Puerl. And what's the first name? Vulkan. Ah, yeah, right. Mary Bullcott for L.

35:00 I think it wasn't only for L. And Krypton. From 1967. Fundamentum on the market. And they have the result that, in this way, even a theory like St. Louis Blanco-Sensory, for example, or you can at large, it doesn't matter, it is reducible, So it's non- and translatable to Q, or a little bit more than Q, but Q doesn't do. So if you look for the usually consider theory, this sort of reduction is much too wide. But still, it won't work that you can reduce with this very wide form of reduction Q to ACI, for example, or to ACIN. I don't know if you can reduce it to ACI infinity. So it has nothing to do with the decidability of the goal theory, but with the finite models. If the goal theory has finite models and it's formulated in a relational language with finite signature, it just don't work. Even this wide form of inter-theoretical relation is not wide enough to do the reduction. Well, and now to consternation theory. I have written down the version. You can consternate So, as an example, you do the negation of phi and the negation of the same negation sign for a type, but of course, it's not always the same object on the whiteboard. So what you do is, you do not concatenate the negation sign with the negation sign, but a negation sign with another negation sign. That means a sign, in this case, with one of the same forms, or a congruent sign, same length, or whatever. In usual concatenation theory, you have this concatenation function, and x concatenated with x type concatenation theory. It exists. To use a thing, we need to build existence assumptions into the function of the pages, and this function can be applied to whatever, x concatenated with x. We don't want to have this in solving concatenation. What we want is, maybe, that x is concatenated with a y, and the y has the same length or size as x.

37:30 Now, the aim would be to build a concatenation theory along the line, which means tokens and P-explosible, which is enough to interpret Q or P-A or C-C. And, of course, this theory won't have finite models. We have seen, we cannot interpret a theory like P-A, which has no finite models, into a theory which has finite models. So it will be an infinitistic total concept. There is no way around this. You have to divide the deducibility concepts or to change the mathematics to obtain more. But these are the exits, so to say. So maybe one or two, I'm missing, I have to bring it up. So, this is a Gaussian concatenation theory, and you can interpret, relatively interpret the A. And by a trick, you can also then interpret that I see in another total concatenation theory. Well, let's go to the axiom. The first one is a variant of Tasky's axiom. I think Tasky was the first one to present an axiom system for type concatenation. It's in der Wahrheit. The other page doesn't do much about it. And so concatenation of A with B is more or less the order pair of A and B, but not exactly. If B is the step of the form CC' this should be the same. And this, of course, with order pairs, these are different. And these should be the same as expressed by the first action. And of course, I do not assume, it's not for free, that the concatenation of x and y exists. So I have to add as a premise, if c is the concatenation of x and y, c is the one of a and b, then these are equal if and only if. OK. These are equal, or this is an extension of that by something such that, if you add that to this, you get y.

40:00 The second one is, if you concatenate x with y, but x is itself of the form uv, then it's the same, like concatenating U with A, and A is dead, it's not so far, Vy, so it's somehow the same as the first, but what I do claim here is that this UV, oh, oh sorry, that this UV exists. So it's clear, if it's a part of something longer, it should have been but I couldn't derive it from the other axons. But by the way, some of these axons are quite plausible. You start with writing down the axons, plausible axons, and then you apply them. Then you need to have an idea how the interpretation of the arithmetic language into this language goes, because we don't have plus and times here. We can only see the sequences of objects which are similar. You can say they are similar. It's quite different from having plus and times, But, well, one is used to, the plus is just extending, adding sequences to each other. At times, it's building up sequences with separated symbols. So it's like when you use the beta function to build sequences in arithmetic. It's mimicked. And who has done it is Quine. There is a paper by Quine conformation in the base. So I'm taking more of that to Quine. and Quine does it for types. And this is quite different from the total contribution theory. And it doesn't give, it's not an interpretation of axiomatic theory into other axiomatic theory, but it is the standard model. So these are, I take the definitions from Quine, more or less, but it has to do quite other things here. Okay. Well, I take Quine's definitions write down some plausible axioms, and then I try to give the proof, and then it doesn't work, of course, and that's the way where you start inventing due axioms. But they should

42:30 be somewhat plausible. But that's how the axiomist eventually comes together. And probably some of these less interesting axioms are derivable or superfluous. For example, here The language with the concatenation predicate, we have the same length, predicate, we have the predicate, here is the stroke, we have the predicate, here is the stroke sequence, that is the strichfolge, we have the term, that is the stroke, But the job here is, for example, like T1 meter, and we have a project here, is a number, simulates a number. One can, some of these phrases can be defined using others, but as I said, I don't want you to go with the minimal data at this point, but some of the more interesting actions are dependent from these simplifications. And one can think about defining is a natural number by using the other stuff. That's the way I did it first. The idea was, being a natural number is kind of subtle, x is the stroke, or there is a W, W is the stroke sequence, and X is the consciousness of the stroke with W. That was the definition of X as a natural number. The problem with this definition was I ended up with an infinite, pretty long sequence of strokes. So the model didn't have only infinitely many objects, it wasn't infinite models, I didn't like this, so what I've done here, I've taken away the definition, and using this number as a provision. A number is not more than a stock sequence. Maybe I shouldn't call it a number, a quasi-number, but I don't know. And instead of having this definition, which shows that a number is a stock sequence,

45:00 So, what are the numbers? This was a number, and this was a number, and this, and this, and so on. So, it has to be something beginning with that number. And of course, one has other sequences here and here, and there are auxiliary objects to get some information about this number sequence here. But this was the number. Just one, one, one, two, one, three, and so on. Many other objects which are like the two and the three, but they aren't two and three. Why isn't the definition of M just, M of X is just X as a stroke seed? I don't understand what's going on in that. Oh, okay. Yeah, the definition as I've written it down, or? Yeah, the one written element. You allow infinite stroke seed. Yeah, and I don't want to have the reason why I got rid of the definition, but the theory allows it that there is not only this stroke sequence, but also here one, and here, and so on. So, I don't, the theory does not tell me that I have only one stroke sequence. And, the reason is, the definition of multiplication, when I define multiplication, I have to add auxiliary symbols, separator symbols, so at some other part in the universe, there has Then there is a stroke, then there is a square, and there are two other strokes, then there is a circle, and then again a triangle, something like this, and so on. And this thing is an auxiliary object which has to exist to get the embedding of the multiplication going. But, of course, this is not a stroke sequence. So there has to be other sequences of strokes and triangles and so on. That's why I cannot just work with one sequence of strokes. There has to be many. A stroke sequence is part of sequences of other things. But this definition of horizontally says that a number is something which begins with a stroke, with a stroke itself.

47:30 I could have many strokes, so to say. I could have the stroke 1, the stroke 2, and so on. And with stroke 1 I start the number 1, with stroke 2 I start the number 2. But I didn't want to have many strokes with number indices, so to say. Because that's too much arithmetic in the meta-series. But nevertheless, I've changed it because of the infinite sequences, written like this. And so, I have to replace the definition by some x. So, for example, the stroke is a number, the x is a number, then x is a stroke sequence. If x is a stroke sequence, if y is a stroke sequence, if c is x concatenated with y, then it is a stroke sequence. I can define a stroke sequence being a stroke sequence by using part of. I say x is a stroke sequence, that means for each y, which is a part of x, and is an atom, y is a stroke. Maybe we could look at this axiom, if x has the same length as a, and y the same length as b, then z and c have the same length. And if u is y concatenated with c, and x has the same length as u, then x can be decomposed into a y1 and a c1, which have the same length as y and c. OK, so a natural number need not be the stroke sequence, which starts just with the stroke. So it may be that this is the 1, and this is the 2, and this is the 3, but there is only one 1, and one 2, and one 3. So, we may have many stroke sequences, which have the same length as this here. So, this is the three strokes, but not all of them are the number three. So, if x is a number, and y is a number, and they have the same length, they have to be identical. That's just, there's just one, one, two, and two. Then the number sequences are arbitrarily long. And not infinity long, but if I have a long, strong sequence, there is a long sequence of the same length.

50:00 And if x is a number sequence, and y is a strong sequence, it could be that x concatenated y exists. but at least there is a y1, which has the same length as y1, such that x is contaminated with y1. So we have several existence axioms, especially postulated here, for the interpretation. Interpretation won't work if we don't have enough objects. We only have finitely many objects in the model. Now I have added some accent interrelating concatenation with part of, so that would be an extended language, and in this extended language, for example, I can get rid of such a thing by postulating new accents, which has an accent as a joke, and only if it's a joke sequence has the same length as a joke. If c is the concatenation of x and y, then x is the part of c and y, the part of c. If c is the concatenation of x and y, and u is the part of c, then either u is the part of x, or u over this y. The beginning are over the x, the end. If these are concatenations of A and B, and C is one of X and Y, and if they overlap, A B, and X Y, something else, but they overlap, and they overlap here, so to say, or here, or here. In this case, X, X is here over there, here too, and here, A, and Y. Okay, in this case here, A overlaps C. In this case, asymptomatic, and in this case, x-overlifts.

52:30 And the further axiom is type contendination theory. Now, what this expresses is somehow that there are no gaps. With a, the contendination of a and b, there is nothing in between, there is not enough in between, or not a gap, or so. And a similar idea is this here. So, here we have a stroke and a triangle and a circle. In type concatenation here we use the concatenation of the stroke with a triangle of this, of course. And of this job, with the circles too, because for each object A and B, but not so here. The triangle and the circles are different and are not part of each other. That's another type of material axiom which I haven't written down there. There are rogues, circles, and squares, and atoms, and they are not only different, but don't share parts. So maybe one could consider this one thing. I'm going to have a triangle here. A circle here, but that's not intended. it. So, they have to be on one line. In this case, it doesn't work, but generally, if you have x concatenated with y and x concatenated with y prime, then y and y prime have to be is identical, or one has to be an initial segment of the other one. There are no gaps between this and that, and it cannot break. Then we have an induction axiom, then we have some other type of axiom. It has to do with the definition of multiplication and interpretation, But since time is running out and these things are, again, a bit long, let me just give the idea.

55:00 Like, if you patient is defined, I followed my person here. By building up such sequences, you have separation of sides. And of course, this is not the same stroke as this one. You always have to talk about other jobs which are similar to the one you start with, which makes it quite complicated. But still, this counts how often you multiply and this is the result of one multiplication. And as a background, you have what you started. I started here with number two. Once multiplied, two. A similar multiplication, four. Three multiplications. And what I can postulate is, there is a star sequence, such a short one, and if I had one, I can extend it ironically, lengthening, at this time. And, well, I need to get arbitrary very long sequences of this time to have an alternative multiplication. And that it works for all the sequences, it's done by induction time. So, I'm going to postulate that sequences that exist directly, but I want to postulate that some sequences exist of this time, and that I can lengthen them, and there's a standard induction. So, it's Now, how do I get Zs? Now, let's interpret to Pa. How do I get Zs? Well, there is this famous result, which is sometimes called Banner's Lemma, or I think one should rather describe that you can interpret a recursively enumerable theory T in PA plus the consistency assertion here. This has to be recursively enumerable consistency, and this has to be a specific problem of tau, representing the formula of T. Now I interpret this into all this concatenation The interpretation of P-A in consternation theory I call I, and I have just I on set F. So this is a way to interpret P-A plus consternation for T into the consternation theory plus the further statement in the language of that theory.

57:30 Now I have to tell you that this is supposed to be a nominalistic theory, but after this is done, I specialize, oh, Q to ZF, ZF plus ZF is measurable, which I believe is consistent, and I get both the consistent, because of the innumerable mathematics, that I want, relatively interpreted in a nominalistic theory. But maybe this is not very satisfying, because the theory is highly non-finite theory. So maybe one should look for a finite theory phenomenon, more seriously. Now finite theory is something I would have liked to talk about if there has not been so much discussion on the nominees. So, yeah, what? I hope I have So, one interesting thing that I have is finding theories that include their own consistency. And, of course, there are examples where the consistency statement is formalized in an unusual way, but I talk about natural consistency statement, whatever that means. Don't work for theories like the algorithm. You have unusual consistency statements which are provable in the A, but not the natural ones. There could be quite different theories which do prove their own consistency even with natural consistency statements. So, one type of theories is, you stay with rather strong theories, that means extensions of PA or ICQA1 or whatever, but you lose the, you give up the restriction to acclimatizability. So, non-acclimatizable theories, which are quite strong, could prove their own consistency. And here is the, in a natural way, and here is an example, you take P-A, and I-sigma-1, that's P-A with the restricted induction exit, and you add all the true pi-01 sentences. This does not do the consistency, but the intersection does. This is one type of theory which naturally proves out. I have done this some years ago. Again, I don't have to blame that these theories are finite-istic. So this would be another

1:00:00 goal now, to have finite-istic theories, which proves out. To do the finite-istic reduction Maybe it could be interesting for this. And, well, Dan Willard has investigated these things. What he does is, he has theories around Isigma 1 or Obita, he's either the North or something, theories, and he somehow gives up the assumption that the operations are total. again. But it's quite complicated, I must say. But this gave me an idea. The idea was as follows. So giving up the totality of the operation was the main point. I write the on arithmetic in a relational language. We write it in a relational language. Then I have axioms of the type. For all x there is a y, such as y is the successor of x. If But now you have it as a separate axiom. Now, these relative existence axioms just take them away. What you have then is theories in arithmetic notation. Not sure if they are arithmetical theories, which are finite models. Models with the numbers for zero to end, and maybe other terms. And these are incompletes. They are sub-theories. But what you can add now is the sentence saying they are exactly an object. So this sub-theories, but they are exactly an object, is complete and decidable. These theories are not essentially undecidable like you, but they have decided extensions. If you have that exactly an object, they are categorical theories. And this is in a pinnate relational language. The idea would be to look at these models and they have Scott sentences, which determine their isomorphism time. So again, complete extensions of these theories. So, if we use Pa naught was dark theory without any relative extension claims, Pa at least n should be, at least n object added, Pa equal to n should be there, exactly n object added. Okay, these are complete and decisive theories. Now, take a consistent recursively enumerable field and consider the consistency plane.

1:02:30 It's a pi-0-1 thesis in the functional language. But it's not so easy that it is this simple in the relational language. But you can use some tricks, they are tied by Jacevich's theorem, to do it also for the relational theory. So write down the consistency assertion for an arbitrary, consistent recursively enumerable theory, as a pi-01 sentence in this language, the theory is consistent. It holds in the standard model, so it reflects down to an initial segment. But the models of these things here, ranging from 0 to n, are initial segments of the standard model. So it holds in the model of this theory, but this is the complete theory of the model. So a Pa is equal to n to prove the consistency of each consistent recursive theory. Also its own consistency, These are theories that prove their own consistency and are extremely strong in doing consistency proofs. They prove all the consistency we want, but consistency are in theories. So in some sense, this is a way to carry out human structure. These are really finite, finite, they are decidable, they are complete, and they prove all the consistency claims we want. But I think this shows only that not much is done by carrying out human structure. Thanks. I'll take questions. Jeff? You mentioned the, I don't know whether you want to come back to this being mentioned as well, but yesterday I mentioned the Shapiro business with the non-conservation. The idea is roughly that if a non-normalistic theory is sufficiently rich, then Goebbels incompleteness that applies to it, so it has a consistency statement, expressible in the nominalistic language. If your set theory is powerful enough to prove that the theory has a model, then you can prove that model theoretic claim is equivalent to the consistency statement, so therefore, by adding set theory to your nominalistic theory, you can prove the consistency, but by Goebbels first incompleteness that the nominalistic theory can't prove So it's not, there are some circumstances where I think set theory is conservative, but there are also other generic circumstances where, as we know anyway, going from 1st rule order to 2nd rule order to having a certain level of circumstances is non-conservative. So ACA0 is conservative, but it's non-conservative.

1:05:00 So it's a generic situation. There are certain kinds of set theoretic extensions which are conservative and certain kinds So I thought, I had the same idea too before I met Mr. Shapiro's data, and it's very natural to doing better mathematics, and my problem was I didn't find a mistake in his field document. He gives an informal, novel-theoretic document that his conservativity holds. I think he used his set-up downstairs in accessible, but I found him this. Where's the mistake? It's because of axiom schemes. A similar thing happens with adding truth axioms. If you add truth axioms, the task in truth axioms, you don't extend induction to the truth predicate, then it's conservative. a plot that has a nice proof of this, several nice proofs, one using recursively saturated moments and the other in the rest of our department. On the other hand, if you, if the new vocabulary is included in the induction scheme, you get more sets, as it were, to prove induction perspective, and you can now prove the reflection principle of the theory, so you can prove it's consistent. So if you, if you, if you extend any axiom schemes in the base theory new vocabulary, then you can get non-conservation. So if your normalistic theory has, say, in geometry you have a continuity scheme, a scheme of continuity, if you allow set-theoretic vocabulary into that, you might also have a comprehension scheme of some kind, asserting the existence of regions, kind of meriology. If you allow set-theoretic vocabulary, you can define more regions using set theory than you can define in meriology. So you get the more regions, potentially get new results as a result of extending the countenance of those atom schemes. So that's the detail of the atom schemes that makes the difference. What the philosophical significance of that is going to be. I think Volker at this point that people survive. So that's a small problem. One has to look very, very careful how he defines the extension of the So that's the lesson, so to say, but if you really look at what he has written, whether we take... Phil is right, yes, what Phil says is right. If the Exum schemes are treated as a list in the base language, then it is conservative extension.

1:07:30 If the induction principle is a list of sentences containing only arithmetic formulas, then the result of adding trim caps in the separate views of conservative. If you think of the Exum Scheme as a rule for a schematic letter, where the schematic letter can go into the extended language, they can be held in services. That's the way Burgess and Rosen will be performed. The theory doesn't make a technical mistake, it's just that there's an ambiguity in the two notions of adding a theory to another. Then the question is, what's the natural way to add two theories together? in the debate about truth, I think the natural way to add the notion of truth to Orypidic is to let the truth predicate be involved in the inductions, whereas Field, in fact, says, no, you don't do that because that's non-conservative and non-equational. So it looks as if Field is committed to the philosophical claim that when you add mathematics to non-physical physics, then you shouldn't allow the set threat in the cavalry to be instantiated in any schemes in the novelistic theory, which seems rather unnatural. OK. Anyway, that's the... No, I think that's it. We might discuss with it in the Burgess and Shapiro book, sorry, the Burgess and Rosen book, the subject is no object. I've got a kind of general, maybe philosophical, maybe strategic question about where all this is going, and I'm wondering, there's a problem that forms all kinds of discussions of this sort, which you might all be straining at anatomy and swallowing a camel problem. The question is, you're very scrupulous about what the logic language allows you to do, but you just give yourself carte blanche about the logic in which it's embedded. So, how about that? Maybe you could say, I'm an ordinary mathematician studying what these guys ought to believe, if they're not. But you're not giving anybody a recipe about the anomalous if you're, you know, I mean, for example, what do you mean by mod? What do you mean by functions? What do you mean by finite? What do you mean by...all the kinds of notions you need in order to

1:10:00 set up the syntax and the math of a person in your life. So it's as if you're living a schizophrenic life. You step inside this little world of the form of theory. But outside Okay, so, I didn't talk about models here, but one idea is, I claim that there is an interpretation of some kind of theory into this kind of theory. I didn't want to talk digitally, I didn't want to talk about models here, because I wanted to be a little bit nomadistic at least, and talking about these things. Then these things here, they are infinity metamorphosis, so again, on the meta level, I'm making infinity assumptions. to get a relative affinity assumption. It's much harder to be a finitist than being a nominalist. So I admit I'm an infinitist in talking about these. But then if I have carried out the interpretation of similar senses in a nominalistic theory, I can interpret also the theoretical definitions of X as finite, X as a model, and so on in the nominalistic theory. I can We can't retract them in the middle of it. Do you have, for example, tokens of concatenation, or arbitrary upon that thing? That's a little bit off. I mean, for all we know, certainly practically. There may be infinitely many planets which have infinitely many Swedens in them, Okay, these tokens needn't be strokes on the ball, so it may be three black tokens. Okay, but even so, you still get the problem now. That's the problem of infinity assumptions, or two large finite assumptions. But I think that's another problem. A strong cut between finiteism and nominalism, because most nominalists, I know, like good men, or skilled men, they don't care for finiteism. they don't view kinacism as a necessary part of nominalism. I know, that's exactly why the whole thing seems to me. You opt for a kinacistic version of nominalism, and it's okay. So that's what I started. It won't work if you stick to the usual mathematics, and you follow this reducibility claim idea.

1:12:30 It won't work if you want the Go-theories to have kinacom, just for her kinacomism. So, if this is demanded for me as a nominalist, I have to question. That's true. But let me add another point, which is... So, if I say there is an interpretation from Semino Senkel set theory to this theory, I seem to be talking about a function, about this interpretation function. So, how do I know this? And that's a further problem of legal nominalism. Now, this interpretation is given purely syntactically, So I have good hopes that I could prove, in a theory like PR knowledge, that there is such a proof of interpretation. I haven't done this much work, I've known to never do this, but I still have some experience with transforming existence claims for interpretation into proofability claims for the existence of these interpretations in PR. Now, if I'm convinced that PR is interpretable in concatenation theory, theory. I have reason to believe that if I can formalize the existing claims for the interpretation, it will work by interpreting in KT. But this is somewhat gappy, because I should have done the whole thing in a concatenation theory, or in a normalistic theory. Now, in the end, I should prove the reducibility claim in KT, or in some other concatenation theory. I would prefer a theory which is not only about strokes and triangles and things, but it talks directly about letters, about vocals. But I have done this, and I probably will never do this. It's like when you compute 2 to the 10, it's 1024. It should be brutal with the authorism. I'm sure it is. I will never do it. It's just too much work, and it's a little bit... Thank you. So, you continued to strain it and now you're trying to convince us that the camel is in fact a man. Sure. But still, I think this is the way a nominalist should answer. If you ask him for being a meta-nominalist at this point. He has to show, or to derive a nominalistic theory, this interpretation. And I think chances are good, because one has experience in formalizing what you think in PR, and one can use the experience to try to formalize it. I do not know if it's worth the effort.

1:15:00 Because there are so many other questions, like what is a normalistic theory and what is a good reduction theory, where you can attack the soul of the world. And I think it's more interesting for me to do, to work more this way, to try to prove There is an obvious point, but we've already discussed that, I mean, under the concatenation interpretation of arithmetic, every mathematical sentence is now an empirical claim, in particular the consistency of ZF is now an empirical claim, and that's perhaps somewhat off, you know, but I would rather like to skip that and ask you a little about your concatenation of relation in order to understand it better. For example, did you exclude that there is a concatenation of x with x? You don't need that, I guess, for what you're aiming at, but it's part of the intended interpretation, I suppose. I would say I'm rebuilding the calculus all the time, and as far as I remember, I do not exclude it. Yeah. Just do not postulate it. But in the intended model, if I may refer to the model, you would say it's true, right? concatenation of axiom. You said this axiom on the right-hand side, you interpret it in the way that between two objects that can be concatenated, there is no object in between. Is that right? And by that you do not mean, for example, if I've got that object and that one and that one, then it's not really excluded that one is physically in between, but in the sense of the concatenation relation, there's nothing in between, right? So if you concatenate that they are close to each other. There might be one kilometer between them. Okay, yeah. So in a sense, concatenation relations does not have a very specific physical realization. I mean, it could be more or less everything, but in the sense of the concatenation relation, there's nothing interesting. But if that is the case, why not simply state that there is a kind of successor relation? So you start with that overhead trajectory, and then you've got a successor in that sheet of paper, and then there's the successor, the whiteboard, And just postulate that there is this chain in a useful way, and then I guess you can again get a result that you're aiming at, but it's much simpler then, right? I don't know if it's simpler, but I have nothing against this in principle. Of course, Volker mentioned using substitution instead of concordination, but I think there

1:17:30 are other simple operations on expressions or things, on concrete things, which could be used in a standard, nothing against it. I just thought concatenation has been put forward in this area of research, not only by different clients, but still they view concatenation of purpose as something nominalistic, and I want you to speak to that, to be accepted as being nominalistic, and have some intuitions, like talking with Tarski axion or so, for For some axioms, I know if I want them or not, even with this year, why it's a concatenation of A and B, and there's a long gap between A and B. The objects I start with performing the intuition are such that there is nothing between them. After the axioms leave room for non-intensive rephrases. So that's the reason, I think, because, I mean, what I said would just amount to, you that there is an omega sequence in the world. That's it. That's trivial and uninteresting. But empirically, it's less strong as an assumption than your assumption, because you say there is a bunch of things, and there is a longer bunch of things, and there is another bunch of things. But you have to explain plus and times. That's the problem. This long sequence is trivial in my case, too. The problem is you have a simple operation, and you still I want to strengthen Hans' remarks, so he said, what you need is that you have an omega sequence, right, in some sense, but I think that there is a, the problem is even worse, because this omega sequence need not be of concrete objects in any clear sense, right, so we have light advertisements of moving levers, for example. Jeddikin's thoughts. Jeddikin's famous thoughts from his theorem 66. The thought, and the thought about the thought, and the thought about the thought is an only sequence, and that's how he reads the job as well. So then the point is that the concrete tokens are much more, are not really physical objects. They're occurrences, and occurrences are not. Yeah, and of course the action system leaves this open. It is formed by starting with having

1:20:00 a look at things together and things like that. Alright, so for example, a simple example, right? I said we have a number of words here on each letter of the alphabet and some of the words. And now I produce a text by holding up one letter at a time, right? And so in some sense of token, in two occurrences of Okay, it was the same object, so how do you know, so what is the notion of object here? What is the notion of concrete particular here? It should be occurrence somehow, so... Yeah. You had the events in your example, thank you. The telcons are events. Of course, the excellence leaves it open. Yeah, so there seems to be a kind of intuition, so the nominalist wants to have it, so everything should be concrete. Yes, but I don't think he can force this by giving axioms. No, no, sure. But the system only announces the same nominalism if it comes along with an area of human interpretation, which the nominal is in the back of time. I think the interpretation is that I have typical examples of concatenation, where I have concrete objects and I put their line together or I put them together, and these are taken as the basis for choosing the axioms. So if the axioms are getting paraphrased by talking about concrete objects lying near each other, they should turn out to be plausible or correct. The only thing I think that remains, I don't want to talk about models at this point, being a nominal. It doesn't work with concrete objects. Or water drops concrete objects. Whatever, you just stick them together. You don't get two, you just get one big one. that you breathe on it, and it gets back to the side of the original. So, and it's not the physical relation, so complexionation is not the physical relation, and similar for the authority support. Maybe it's a relationship, but it's not only when you look at the size, or the form, more complex. So it's at least a relation, or could be a relation between concrete objects, no abstract objects. So, do you agree that concrete objects cannot move faster than life? Okay, but the letters on a light advertisement can, in principle, do faster than light, so they are not good. Oh, really? Just like wave tool sessions, right?

1:22:30 Oh, okay. What does this show? Well, it shows that... It moves the concrete up to that. I agree with it. Sure, sure. I think that what we intuitively draw letters are by no means easily recognizable as concrete objects. Yes, I agree with this. It's hard to say what concrete objects are, but again, the accents should be plausible under specific reading of what concrete objects are, not what concrete objects are in general, but if you have enough trees and trees are standing near each other and things, and you concatenate in this sense, follow concatenation, paraphrase it in this sense, the axiom should turn out to be plausible. But that does not mean that there are other paraphrases or other reasons or other models of these axioms where it's very questionable whether they are concrete objects. But I think I have another goal there. I don't want to have all models for this theory to consist only of concrete objects. I do not see how to support them. But if you're a sincere novelist, just the problem of what a concrete object is, it would be easy, because everything is a concrete object. Everything that is anything is a concrete object. Yeah, yeah. But still, I can't... That doesn't help, actually. Right? I can't state it on the board. That's what I mean with a concrete object. But the latter is a concrete object. I just want to push a little bit further. I mean, the reason the appeal doesn't seem like syntactic tokens seem nice is that it's supposed to give us something that seems somehow to be a concrete or nominal, and at the same time seems to give us this infinite stock. But, I mean, one thing that I think Robert said a minute ago was that if you really think part about what the notion of syntactic tokens is to do that is not so clear whether it's really whether it's all nicely we have their common object whether it's nicely anomalistically acceptable this is a problematic notion if we drop that and go back to trees we don't really have enough trees to to build from a model so it's not really clear whether

1:25:00 So it seems that we are free to postulate infinitely many abstract objects, that's no problem. But not infinitely many concrete objects, because if it might happen, there are only finite objects. That's true, and that's the reason why this theory may turn out to be false. That's what Hannes already at the beginning said. I have to live with this. Sure. If you think normalism has nothing in favor of it from an intuitive point of view, you're happy with sex or numbers or so. The whole thing is nonsense. I'm starting with the intuition that I don't know what abstract objects are. I have no reason to believe that there are any of them. But I don't want to treat number theory or so as a pure casting. I want to talk about this without talking about abstract objects. It also emerges you don't know what a concrete object is either. So there's the kind of, we're really up between a rock and a hard place here. I think this is a concrete object. I have examples of concrete objects. I don't have a definition or a definition of concrete objects. Yeah, but then my point was to simply say this is a concrete object. But isn't this description of the token to the concrete object always going to hinge on consciousness to make that description on the cell phones about it rather well, when he was saying that if you're making a Turing machine out of cats and cheese and mice, which you can, according to some viewer principle, do, is that machine is not a Turing machine. That is only a Turing machine once you have ascribed what things act as what tokens. So you've always got a blank question. I don't know how you see that, in terms of introducing the abstract in that concept. I don't know, but it's hard for me to say anything about philosophy of mindset today, because it goes into the direction of philosophy of mindset. I'll just put you correctly. There's always an intention that something is about that. Well, if you're talking about a token, surely that's only made sense in the lives of understanding that to be a token, is what I'm trying to say. And so when you're talking about concrete objects and stuff, you're kind of talking about these sort of... It seems to me you're talking about and you're totally devoid of any rational addition to that, any kind of interaction with them.

1:27:30 So you say, this is a chair, then it's an object. And you don't need to have a constant entity around to see if this is a chair. But if this is a standard or something in a system in which you're trying to do mathematics, there's not a relationship. But for me, these tokens, so to say, maybe I should have two different tokens, are just like chairs. So what I do is, I have a theory like piano arithmetic, it's a calculus, and I have this theory, and this theory should be plausible when I read it as being about chairs or trees or so, concrete objects which stand near each other, under the assumption that there could be infinity, many of them, and that's the problem. This is a totally irrational assumption. I don't know. I don't know about that, but... Well, okay. But I mean, it's quite plausible that it isn't. And to assume that mathematics falls or holds according to whether the universe falls... But again, the problem is that pietistic demands are not fulfilled, and usually knowing this doesn't imply a pietistic demand. is another project, and this provides no answer to the finances. Absolutely. But that's why I started with talking about finances. Reducibility to financial theory, in the sense of having finite models, and usability, and you know, it just doesn't work. It just doesn't work. So you have to grasp these infinity assumptions at this point, just for the sake And so, the theory has to be plausible under a certain paraphrase, and then what I do is interpret one theory in the other, and whether, but this sign here, this expression, of course has to refer to something, but if you say, when talking about the reference, you have have to have an intention of our fellow. Or perhaps I could... got you wrong? Yes, I think so. You're saying this acts as these chairs are used to do mathematics, or this constitutes a physical incarnation of a Turing machine, for instance. You do have to have... I mean, again, it's not necessarily binary, you can make the assumption somehow that there are an infinite number of caps, but it's only with the intention that it comes... that the token only takes effect when it has an intention on it.

1:30:00 There are some type of arguments why nominismus couldn't work approximately as follows. The nominismus formulates sentences, so sentences of a predicate term structure, and you understand these sentences. In order to understand these sentences, the predicates have to have an extension or to express a property or something like this. And if you assume this sort of semantics, there has to be some entity given to the expressions, or perhaps even some relation in the mind between the expressions and the entities. Nominalisms don't work, I think. But I think one is not forced to this position. say, in the quine area, whether, say, meaningful predicates have to be extensions or semantical values objects or not. I cannot say more there than quine, stick to quine. You better get 24-1. We're going to have lunch, and we're back at 2 o'clock. And what have you to say about the University restaurant, or what do you propose, Philippe? More or less, as yesterday, if you want a more substantial sit-down plan for me, you can go to the University restaurant or...