Very, almost, and so on, ...

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Very, almost, and so on, ...
(when fragments of the language find their way into Topos Theory)
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Very, almost, and so on, ... (when fragments of the language find their way into Topos Theory)
Mathematics Elementary arithmetic Natural number Multiplication sign Infinity Variable (mathematics) Numerical analysis Set theory Element (mathematics)
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Axiom of choice Group action Finitismus Multiplication Process (computing) Matching (graph theory) Eigenvalues and eigenvectors Multiplication sign 1 (number) Maxima and minima Algebraic structure Mortality rate Axiomatic system Subset Orbit Goodness of fit Lecture/Conference Energy level Object (grammar) Physical system Directed graph Finite element method Stability theory Chi-squared distribution
Point (geometry) Finitismus Process (computing) Closed set Multiplication sign Decision theory Algebraic structure Equivalence relation Theory Numerical analysis Subset Category of being Frequency Arithmetic mean Lecture/Conference Finite set Natural number Theorem Object (grammar) Cycle (graph theory) Resultant Set theory
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Point (geometry) Functional (mathematics) Existence Nuclear space Incidence algebra Numerical analysis Element (mathematics) Orbit Connected space Morphismus Category of being Angle Lecture/Conference Natural number Uniformer Raum Network topology Energy level Right angle Identical particles Sakoku Resultant Freezing
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shh the 2 of them and on the other of case the think I will skip almost just because of lack of time here is what I'm supposed to say and as you see it's a very long and I won't have time probably to do everything so I prefer to sorry you could make the whole world will see so the 1st paragraph it is the behavior of various if you say for example big very big very very big and you continue there will be a with later called the saturation phenomena adding 1 more very won changes in the value of what you say end the language is more clever then all of us together Nicolas it can even indicated this saturated and should point if you want to say very very it's up to you say he's really very big and that means he's very very at no need to go you cannot see he's very relieved variables that very really very big indicates the saturation of there at the end of this I can explain to us 6 years all he will understand that but adding 1 more very serves nothing so I want to see 2 linguistic phenomena it's not linguistic because it's I think it belongs to all languages In French shores I want to model what costs the immediate idea is to say OK let's call this said all adjectives to reach very applies L soldiers X is equipped with an and Morpheus and which I call you because I'm French and entities initial today and saturation phenomenon will b given by for all X there exists and end number and such that the committee again Bruce 1 of X equals to Tulelake genes yet united as 1 of day and sorry the at end of next OK that's very pleasant but has nothing to do with the language why because if you ride this thing it if there is an end to end being well although there would be a smallest and nobody will agree on the smallest and I didn't put this in the data just save In this week from his days with the disease .period OK so suddenly this wasn't working insects so we might try To see because sets the nature and numbers are well although OK so we might try to say instead offsets let's suppose that X is a big X is an element of the Groton Iqbal policy in that case and is not when all and this might work 1 the in that case it doesn't work why because as I say I can explain this 2 of 5 years old boy he knows nothing about the sorry sorry sorry I'm speaking English I will had the rather give this spoke in French so you will excuse me if my English sometimes so as I said I can explain this to me the 5 or 6 years persons and he would understand if he or she but he has no idea at all about nature and numbers he doesn't know what's in the current number so he doesn't know infinity this can be saying and without any user infinity therefore it cannot be done in Groton it doubles because in working the top posts there is the nature numbers OK so all this is elementary language show what can we hope for after that maybe nothing maybe this is fail anyway but we can try to work In an elementary doubles without nature numbers object getting ideas of gold sorry I have decided that this was where that was my problem to prepare the talk because I need I should show how we use that I need elementary but nothing has been stayed here about elementary Koppel says and maybe some people I don't know what they are or how to use them so I I almost decided to cancel my book because I don't like to
speak I just follow my pleasure and we've all very few persons understanding note that they say persons OK so I do as a challenge and when I finish you will tell me is the challenge has been met so you don't know about elementary proposals and I don't think differently sorry my shopping list yes but I mean you didn't prove anything but without knowing some call during the children of a language in and saying that what used to located at but you didn't say so the OK that people here just on the basis of all of your pork which I'd like very much to do in order to be able to twofold what I say a provided don't know how to use the language so don't panic in the language will be the language of set theory didn't have the level and sorry the major yes OK those who don't like giving names to things if in the discussion we have the possibility I that I'll tell you why Mitchell has nothing to to do with I have moved when the sulkily it's it's the naive language also said the difference with the languages that Olivia presented to us is that it is not a 1st order language it's Ohio older language you can quantify not only on elements but on subsets and for example the most striking example is 1 of pianos actions which are certainly used mainly what characterizes and many actions but the more the strongest 1 it is that for every subset who with quantified on subsets of N if 0 using and the X or when an if this is stable by the Canadian s implies the best teams usually a cold successor then I have all em that's the most powerful action of then was but you seem to requires the possibility to quantify over substance OK there are and they should talk about them all the versions all over nature members fried or a little over the year and I should see how fits with what they do OK no supposed text is equipped with an angled Norfolk X now is an object of these elementary top posts which you don't know what to How can if X in but let me the 1st S please contained in X I shall see that as the stable if deal West is contained in no need to and now if X is an X I couldn't take the intersections although all stable so well objects which contained X and this show called the orbit collects OK what is this alarming it's X T and T TAX etc OK so we have the notion of sector think the all the act with I didn't want to write was said so all we the growth in the top bosses or etc. or what not it is for all X the arbiter of X has fixed nor need to say anything about nature numbers OK so the Ex-Im fall vary it is and the morphism such that each or be has a fixable if you have objections and just barely and confused finally I am confused it is a lesson that you in the know as I see if I want to express the phenomenon of vary as it is in the language I can express it by saying for all the eggs the albeit appeared to fall next year because you have 1 answers as well as I say I a very has the property that
very very very big is at the very very very very rich stops very very that's the thing the phenomena of his also confusion just sorry that your capital it's kind of our moves on or little excesses in the face of the whole world staying in more than a month after all the only thing that you have to write everything I will never forget but this piece of news about have use of the group a lot having this priority Andrew LO alarms have a fixed point that's very hands this ii exemplifies vary by saying it is an object the various yes it is that and then morphism of and object which has the property that every albeit as a fixed this is the hinterland of course that's what I need this can be saved in the internal language of the bottles since I have a specialist here he might then be what it X doesn't have any element but this is the sorry this you know I see nothing if X has no elements and I 6 in the orbit of every element stops that I said nothing because initially and this will be true for no it could even if it's not empty it can have no global elements I was so it is in the know it can be a empty their followers million Canadian you can be empty I don't exclude end whether the internal language permits even if X has no global element to write this formula but if you don't like these things quantifying what all X where X it is something an element of the capital X Witchazel elements let me say it in another matter because it would tell you the power all elementary top bosses sure right for the time being I share right I shall define a radiation which I write like this next smaller the new life it is why all that and the game what would that that mean when there are no amendments but they can define now something which perhaps might satisfy this is a binary relation OK as a binary relation it is characterized by it is the smallest what the intersection Of all 3 alders on X such that X this can be proven OK at that this would can take intersections Central etc I could Hughes and what does it have to to do with this this although relations because I want to make a remark on the power of stop losses this is a binary relation on the next a binary relation on X can be viewed as a matter From banks To the power the central OK and what is this map it's camapign Texas the all of X's and now I don't care this is a binary relation well-defined sold now With the X has elements will doesn't have I don't care OK so I will the right as an abbreviation if you want that's why "quotation mark X pullbacks is an abbreviation also thinks it's more than have for this really OK now I think 1 more remark just side remarks to show the power all of elementary Koppel it has to do with family there are 2 old notions of femininity families offsets few for families of anything the family also said index by I it is usually
like they send this notion of family you can do for any category say with full backs because you want soups the truth there is another notion families which is the 1 which is used by this said so I quote I should call this an implicit families but there is another notional name the family of sets indexed by i is a map from allies In 2 sets or better it into sunsets offsets that's the a comprehensive scheme which saves well we find have a map from our into 2 sets yet in cold weather the power of stop losses is that implicit families can always be made explicit why if you have this intuitively for each you know the stakes are high for the year inverse the mind of the Holy it's what I take to Clemens 1 of us and this is a subset all of big X so this doesn't mean this implicit family I can write it as an explicit families and this is 1 of them most fundamental to of stop losses implicit families can be always made into explicit love OK on or beats and etc. I don't have to be quickly each state orbits and etc. bits of exactly the notion of etc. all very simple this central now I'm also fits dictated the Itsekiris than this 1 but they won't have the time to talk about the so the idea is to study in the doubles and end morphism by looking at its Long Beach 1 can understand properties of and the Wolfensohn was
1st look at or wise properties things which are and then also the releasing see how these these old bits fit together this would be done has indeed gone as far as the eye can see and many of the things which I went say and not be done systematically many of
the things which I will say I think our New England attending offsets OK so all the show calls and Bobby an object X together where an animal x 0 and and then the such that please there is such that the Baltics is the albeit of excessive 0 that's what I call on all parties to the conflict all of 0 fault that all of the 0 is based on the law and the army of the 0 is physics the orbit of AIX 0 is 6 hours ,comma during the course of the sun will be the start of an abstract all notes that x 0 0 I need not be unique or anything that has a real well it's a generator but maybe there are many in the OK so you know the requirement that Israel is what properties have such albeit that is formed in the early hours of the morning before no no no and talking about in general I said I want to study and morphism by looking an arbitrary and morphism by looking at it's all bets and then seeing call little bits fit together so the 1st step is what property albeit have 1st of all and can be seen as the week nature numbers objects specialist explained in what's a week In old as I said before there are many notions equivalents of any and there are the Slovenia definition they're all on the Friday definition which some people may not know but I will recall it and there is the ban would be finished let's see let's compare with low definition if this is bitch endit if I have any object twice take quipped With 1 0 and the translations I would always calls to broker this is a little bit there is at most 1 ever making this diagram commuted so in involve area if is 1 and 1 s exceeds wink and oak replaced there exists 1 and 1 A S is there exists and most 1 so this shows how it is a weakening in the sense all In that sense of relief In the sense of In this sense of panic it is stated that the nature numbers set the slide the most important action will be the 50 of recursion end you had 0 is not a successor let's call it bears the image of him as for 0 0 is not a successor and the math he sees is monarch and the banks With current actions when an orbit and also that everything is either 0 or a successor In all when do we have from the UN we had a big atoms recursion action but we also have that x 0 unions S by it is all of X's another way of saying is that in an albeit all the elements are either it's 0 or successes there more of a elements so it's a game away the various media manager who said he saw a median age of the overnight at the United yes I I S called the image of the S meaning the successor :colon OK sold again we see that the we have weakened a little bit when a big deal but nevertheless we have weakened the definition of nature numbers of Donald yes he was instruments used in response to the vehicle also relied on last year the concentration of the demons and send it to Honolulu and October mentioned pride I mentioned the I'm and I mentioned over here and not talking about more otherwise I would be here tomorrow and the series of tests you have Mendes's utilization as I am making an albeit 1 0 can be in In the no-bid 0 can be in OK well since you are mentioning this I anticipate on something which I was going to say it anyway namely suppose we don't even have
to talk about all suppose that I have office I should say that an element x is the psychic at this until next it is in the orbital what does that mean start with X right TEXT TEXT Teti Teti acts sometimes come back to where alone I think that is known as the exigencies the and he takes a sickening agency this that is sorry who thinks that I was involved with the thinks so that means I want to understand that that means stop with X right TEXT TEXT TO at that that's about that and you find 6 that's correct for cyclic OK and that's 1 more thing I shall face I but to Is there a sidekick it not the doesn't have any cycle the syndication doesn't work at all they need is a psychic if and only if that's my definition every cycle set psychic commitment In the is fixed at is a fixed .period if I suppose I start with banks and then it's it's just at the very 1st step I think this is a psychic there was with what about what they're because of this 1 well there then it's the X's notes if it has a fixed point that every fixed-point decide but as a site click means that the only click elements of fixable the that that's my definition is OK it's a cool let's call sealed makes them through the object of cycling tenements let's call fixed X the object of fixed elements this 1 is they're easy to describe need of higher-order it's just fixed X is the equalizer of the identity and but psychic elements you cannot define it in that category unless it is a proposal or bottles like we have always this inclusion and to say the tape is a psychic Is this OK and we just I know everything that I might forget the future government no 1st the fears that and Albany is a cycling if and only if the PRI although relations on the orbit is another relations we're going to have to be proved it's not difficult climb up with difficult questions but later for example the major numbers if they exist we are a society OK the because the order relations induced by successor it's an order well the which show I said all are weak nature numbers let me give another example if at isn't all bad then it is anomalies With 1 generator and such Montoya's or commuted to so 1 electoral that's the nature of numbers isn't a commuter 2 1 lead for edition it's a crime has generated that's true for any so many things 1 problem formation members control for all this sort order for example for a society called albeit then you have but OK they were the Florida thing is killing the cycle I think it has How do
you know that if you have a pre-order although relations you can't take the Associated although so let's hope let's right X equivalent to wine an egg smaller than and why some of them and let's write X doubly equivalent to y if X and Y belonged to the same cycle these things are almost at the same when it is X equivalent y if and only if if X it all or the X and Y all causality this is this relation is symmetric transitive and not necessarily reflexive the because X and X need not be perceived go cycling if it yet they are OK so we can take the but cautioned that all banks bye yeah this equivalence relation this commotion it is now at the respects this equivalence rendition so factors as an open and morphism as all this however this it is and other relations with the pretty although becomes more the relations 1 X N the old now fish albeit of X gives an albeit here but no the obits are on therefore this is a psychic so what we have done it is just by identifying 2 points which are In the same cycle we have replaced the whole cycle by a fixed and this is sees what let's call by 12 notations deal aimed at the category of objects equipped with an angled morphism let's call a Old the subcategory of over this formed only with a cycle I think what they have just proved his contract and joint but just guided by just killing the cycle killing means reducing them to move to a single atom and in particular if every albeit of when X as a cycle then for the associated a psychic element the object every albeit would have a fixed point and we will be in the situation of vary no there are many more things to say but I have to take only a tiny bit of each paragraph of from 1 to respect the time for what comes next I there is a difficult clearance which sees the following this 1 is read the other ones are middle exercises each 1 takes maybe 10 lines had at most provided you give them in the correct order but but here is a difficult theorems and all of it it is they cited the source is finite if and only if it has a cyclic element that's that's a fact why when using the company I'm sorry I don't want to the volume of just "quotation mark proposals the fear is that the the Hollywood from time to time finite To Rutkowski final there is a notional finite but again you feel don't know elementary top bosses Finkel finite that's fine I and I want and I want to speak of to things that people can understand the soaring fuel
system services that the money was I'm not saying that anything about that some of choice of the other 1 was national phone on a nite that things the on being in an elementary top 1 defines correct Orlovsky finite elements some which in the case of offsets are just the finite elements in the case of growth in the top bosses they are a bit more complicated but think of them as finite well that's all I can do the livelihood of the levels of his authority it would be good for the services you find that there is someone the emotions of the people of the world view of eluded him year he said I hope and I don't care provided she doesn't take on the my time of talk I think we have a well it is in a sense In a sense 1st it is difficult but it is it has some philosophical connotations that means that this abstract definition L finite mess or even insect definitional finite this has something to do with orbits With etc. finite has something to do with that says all bets are axiomatization of etc the annual just thinking of you and I are in 100 years this is isn't the way you want him to describe using their home you on the questions of common sources of food it's because when you have have a single 2 of the missing matches is new and higher finances are generated value from union of single so you that the smallest to what some of the views those on the union I was told in a single single and as you do that you get all the good of the service chief finance officer because of the finances it is something which is common in the smallest legal and policy and this will be the 1st P 5 new faces some discomfort in August the majority of of OK well let me say I don't like your definition should give an equivalent to definition I don't know why you that it's not secure left by the way it's not your definition it's the definition we might get focused solely on local I showed him another definition which would perhaps make it more intuitive and why there is something between all rates and find evidence instead of saying yeah this is joint sublet this means you can suffice to say that it's stable by the falling cooperation if Esq koizumi follows that of end text In 6 then it's stable by not taking the union of 2 things taking the union of 1 thing and the sink you you obtain all of X by it directly in the the process also taking the Union of X and single what does it have to do With etc well I didn't want to talk about this but the etc. I mentioned it is a very simple ones there are more complicated it's across for example I suppose we have X equipped with a family although and amorphous not just 1 the etc. which consist hitting they think something the kinks or something the orbits are obtained by picking something and the TI all that something not just the thing this isn't the kind of multiple etc well the positive effects has this structure namely the expert said it will follow said similar I it has a destructive bye taking all subset with cyanide equaled the subset you know what Coleman toward the west and the north and I'm happy to say anything that isn't just for publicity would you just did to use soul what's the final object or more generally we look at the old mates in the sense which I will not defiance of the steel the obits call all the
finite Seibel objects of life and seeing that close the why is finalized is that OK OK here we have a lot of decisions so that he can literally giving you know that theorem this and all the today that is finite if and only if it has a psychic and and that's a difficulty How can you see these because I want you to see to have a vague idea of what's going on but it's a it's not trivial suppose you have a computer and the repetitive process starts with 1 thing take that the England and teaching it's a dried CA and it's supposed you'll Dave the company I know nothing about computer science by the way sold on and suppose you give him this is the only instruction start with some headaches dollar repeat the computation and stop when you find something you have already the results you have already met and it stops which remains TAX it's it's it's up has a cycle but it was stopped certainly In a finite number of unifying time conversely if it stopped In a finite time that means that has meant the city can but this very very but that's how it is except that in that case it's fine I think the usual sense of set theory where find is as has been defined by you'll now or by other persons but but those were the point is that this finite mess for example doesn't have some properties 1 might expect of finite for example the subject of a finite object is isn't the need not be fine if we all believe that the subset of vitamins satisfying Of course because we leave when in the 2nd set theory into bullion gold I'm not supposing it's bully here is another very strange and To me it was surprising result here too at the back suppose text is smaller than mine OK with contended that his wife and this is a little bit of it we can define at the close at intervals it's why as being all the zeal which will between X and Y no the following are equivalent full all these intervals are finite is equivalent to all the x x this can be pulled but this equivalent this equivalence is not drawing general it is true if and only if the top bosses not only do wouldn't get in their final near the structural walls structural or football pools what is that determined by this and of course if team is a psychic this 2 equivalent conditions but if they are not assigned the top post has to be born that says also something a sidetrack means maybe there is no fixed point if there is a fixed .period OK
it's as society but maybe there isn't for example for the nature numbers there is no fixed .period however every interval Of the nature of numbers clause interval is fine we know that for the
nature of numbers but we know we can for every ace or not only the thing which goes all the way the albeit which goes all the way to the fixed-point if there is a fixed point is 49 but only intervals of 5 and the surprising thing is that morning this is needed and that would assume the Olmert's the thing is to recycling water and is needed in general for this equivalent Seoul the game some properties of etc on private properties will the whole topples no 7 is connected components this days I shall give an ideal of approved because I like it and because it doesn't take a long time we have this order relations so it's a category OK so we can talk about the connected components OK when 2 object to elements X and Y action and other equivalence thing in the same connected companies the answer is there is simple namely that are in the same connected component give the orbit of X the orbital wife has a common enemy I shall give Liverpool it's not so
obvious analysts that I'd missed something but this relations is always reflexive and symmetric it remains to show that it is transit usually to get that connect company you have to Meigs exotics like this but here that cities just 1 zigzag but I suppose that's why the act and there is a new indictment you in the old of X and there is an element this here in the orbit of said intersect Gilbert but these Q and both In the overdone wife and we have seen that in albeit the 3 although it is not real so we have either you In the orbital vehicle all V in the orbit of you but then that's trivial to check the it this transit OK so by 0 users you hear is 1 more thing for more application of suppose we have X equipped with an underwater isn't I want to talk about 2 composed with the composer who whose etc well
you look at the X at the X and it is equipped with an angle morphism let's call to blacks for each S take "quotation mark "quotation mark boldly so it can take the end of the day albeit of the identity and the orbit of the identity it is exactly what we want so we can have not only at 2 level of elements but all functions for example suppose you want to suppose you say OK with native members against the rights of all text the existence of any such that the handprints wonder I don't think it was but suppose I want to say something more namely not for all X there exist and hand but their existence and on X which is for Inc but so you the writing formulas as if there were 2 numbers but there isn't How do you say that that means the orbit of the has a fixed point and this will give you a uniformity of the Gold so far he has for him what you he added is a mail that for all this holds that want to be able to see that this fixed point there is a uniformity at the uniform bound so you see that this notion of etc covers many many many things even when there is no nature numbers object and you want to do as if there was 1 so this albeit that albeit of the identities along the likes calling the medium albeit and when property does it have the for every the leaving albeit meps subjectively on the albeit backs take the evaluation if you have this was if you have been there in the orbit of the a role in the leading more but he called on acts the leading albeit then for each X evaluate the attacks and this is a mapping of this In true it will be prominently and we know that such mappings all unique and we know they also rejected so every albeit of any
element is a caution also this CD albeit for example if this indeed albeit has a fixed point then albeit no so we finish with just 1 more remark important what they have said that the very has the property that each wall because a fix .period OK I claim that this is a very important notion must have criminals what does it mean that team is such call for Texas I always thought it was a but I call such an annexed and 2 I rooted Y H connected component is a result of tree which means all the orbits of finite a fixed point that separates that's the correct definition of a tree so it's important to this since that very is rooted Florida and that at the some so you he was mentioned in the files if you would please "quotation mark although I don't and How can you believe it you mention that this of National Endowment of the the most of the names of those fears about it is because the yeah the solution to the nation's precisely I don't know what use of In the mantra proposed well I suppose you and love so much the reality is that there will be little from the general doesn't of course cities to the piano good to want could make it but how the value of the yuan some of his music of give him incidence is the proposal to these of these onto Madeleine yacht the I said that the motion we then little freezing they don't anymore the jittery investors point that out but at 8 40 albeit seclusion nuclear sick Yupon it is the sponsor me up there should be University it would avoid discussing this
action the piano this is what the president of the see the XIV only by successor is the successor it said it took a tumble No 8 2 8 was was successor to pursue the Treasury among dozens or hundreds of good there were gone the successes for inclusion in the Australian was all sitting summer somebody could say on it by successive prestige is somebody good to be the successor is summer you seek which successor on 1 of us could embolden films that come with said the 2 8 disk as the press the best effect could be accessible by the console good create good next so too should be on diesel good quality but mobile this the quake see 6 for exact India good company could the clinics to build a spaceship could anymore but could make the electric when the parliament the disk in the western Gulf boosted the sources for successive hired shady movie Bengals renewed vigor in Milwaukee and in almost all these people up and the usually young off with Goosen boss most admitted making could be angle at this crucial a short while please for the Langley they'd waited for a longer it's a lot fibula Sacre du John Swan Bradley appalling the big time if the