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Logic and topology
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Title  Logic and topology 
Title of Series  Topos à l'IHES 
Part Number  17 
Number of Parts  28 
Author 
Coquand, Thierry

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/20758 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  The logic of topos is naturally described using intuitionistic higherorder logic, an intuitionistic version of a simple theory of types, a formal system designed by A. Church (1940). Two important axioms of this formal system are the axiom of extensionality and the axiom of description. Recently, Voevodsky formulated the axiom of univalence, which can be seen as a natural generalization of the axiom of extensionality, and showed that this axiom is valid in a model where a type is interpreted as a Kan simplicial set. This model uses classical logic in an essential way. We present a variation of this model which is carried out in an intuitionistic metatheory and explain how the axiom of description is validated in this model. 
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00:03
and what plans on that was it I don't want 1 1 1 1 man thanks so fast that we like to think so the organiser of very nice tutorials interesting colorful and for this indication so far only hours Williams has
00:27
already connected to the universe sexual and he has explained that the Solorio Ski Club the problem with that of the sexual In the emotional space well was places the completion simplicial complex dance initial set and so so as I will explain so they're all so this when we look at this argument that it's not intuitionistic securely and so cannot be done in the money title push so the main goal of the season had this talk is to present to introduce valued her emotional 1st page submission so there it was a plan of the talk is so 1st I will explain why the Sumerian section is very important in logic complement of those with the intention and then I will explain the home so so simplicial set down simplicial set of intrinsically classical music in implicitly where it is political classical logic and and then I would present this the mentality of notion of "quotation mark that he died so far so good because definition that on the Quinault of the calculation of deposed so it opposes upon presentable locally and was continuing with the subject has failed and online services started to explain to which is set to be this corresponds to so I will do this by using instead of title school yeah on the Net no so it's a good indicator because there is presented don't this characterization so and what they don't know is what kind of form set teleological respond to this this notion so look at the captors and was so in particular also it's got enclosed so it does expansion and so is there is actually something the fault is so new additions will also change the design of a very simple very betting on and former system because the simple type which corresponded to the logic the sexually that on his so this user habits or with his nostalgic so you have a type of thing opposition so I mean in just was walking in a classical from walks of the sky was he that this is a type was only 1 but for top posts this will be the subject of the present failed to propose another type of individuals and the only information that we have a lot of the building like is a function that so I'm in this sector leaders in natural said it examined whether the problem is that there is a time for you to do so the fix so typical the idea is a type of food that along with 6 of this type and so fortunate just was not really using it was using a type of thing you know enough yet election despite was included in the document on this is what he was reading the type of federalism this a ball set will be idea I will when the 2 committee at site by Powell were so as a bible for positions and you assault just minutes after the public you can add confusion .period this if you want to correspond with captors and was continuing in some cases said that is similar to that have to say something to be said that about the policy "quotation mark this year so there is an attractive presentation when you don't have a function type that you have only got product and and this is usually and sold in American Scott for instance the book came categorical logic may sound like an Scots you use this formulation of the dated but actually I want to use this formulation because I want to stress that the In this formations resulting official functions so when social function is Newman plays this creative and there is another muscle function which is usually emotional functioning versatility only half in mathematics so it's a function which is given by functional so in shops system there is I questions to evoke predates the social function given by functional loss and this notion of primitive functions and this is done by Soviets with 1 here and what is the connection between distortions so so if we have a I mean fall having a good connection between this promotion we need to introduce an operator which is known as the description of it as so all next and this is needed to connect social function and the notion of functional and this wonderful fibrinogen fundamentally mathematics a top presents us with shrimp and this week said that if you are uniquely forms then you have an object which of his players so it correspond to In mathematics to new name will object if you you know that was the subject of Mendes of your property which is you satisfied by only 1 element that you can give the name to this subject His meanwhile the user the stronger operation that there will there too 2 operations so which is a former stronghold of actual injuries but so the Sobel Marquis White told that usually strategic it's good and in that computer until operation which uses platform of actual choices so if your existence then you can use name to this list of but the charts and so if you have this attorneys this implies classical the church and which includes the ritual of operations if you existence then you can give a name and there there's this operation so as I said it's a little bit and I think souls so that's why the stateowned logic and pilgrimages to this operation is fundamentally logic but I think it's only the walk of the district Walk of widescale please stop to other mathematical analysis of what's going on in this region I have so as I said so long as its operations with is used unique existence so uniquely responses which approves a motion of liquidity so only quantities Lopez opened in this former system so it's so good it can be calculated by the time you defined during the system but it can be also characterized by actually for the lost 1 of its reflexive and the 2nd 1 is that the new 62 due to poor sleep as and any product you visually always satisfying they 1 to and then you have the answer lay conceptually so fast actually conceptually easily actual effects such amenities like conceptually and you have an extension injection and after the fall of this extension and adhere to the 1 that you expect which is that if you have 2 functions on May to meet the pot .period weighs a quarter of them like Florida and the 2nd form so it's the purpose of analysts expected it said that if you have 2 propositions I quoted then this is it that often educate grounds then there was this propositions equal it so on and so the union announced so that actually In this former systems designed the opponent but this was the cancelation as going to see is a generalization of 1 look at this form of extension into and if so In the setting but I would result no this generalization 1 implies to so it's a great idea still full of extensions so as a whole
09:38
said the set TO the corresponding to topple still what phone is weak can he denied the ball ball asset so we can for portable tied to that of course and so on but we cannot talk intelligently about the families of such slips .period so we that the man in the world ,comma conceptually so this is a natural fit investments that I was too busy to talk about this venue some of the yeah simple fact allegiance to so long refinement of painkillers at the moment In the final defiant we we present you can do it and more importantly so if you want to present to the mathematics in this sister and you cannot talk trolling produced an arbitrary structure so you can takes a structure on the type I am on the Type II but you cannot include you let's X new type and that will consider all structural Knicks game and this is a very strong the limitation if you want to talk about public transportation a majority of tolerable and good things about books some people of designed for an extension of which is also where do we find a lot of time with the types so what is the personality basic so I would not give details many of this system the manager is that what is really taking his policies system is emotional family of pipes can already sowing this notation so you analog of so when you have vibration after and all types it's not you ought to look at this and there is a point of crooked the fiber of students saying so high relative to the sanitation which is used in the state clearly so it's not unexpected that this former system will have connection with emotional language and actually this former system is kind of meaningless former system in which you can talk about the notion of anguish and in this former to primitive operations the dollar which sees the obligation conceptually what all this primitive operations museums and I to do some locations as the 1 used in the wiki institutionally so this is a set of section in the formerly of this is a type of section of this familiar can this is a type of buyer paid you will there it is an area under these new so settling in usually this is considering this case funding and time and evangelism song and would and have to set of section positions good if all my generalizations unknown 2 types instead of Sept and so what is being noticed is that afternoon you can explain the politically operations on generate use logical progression told operation on vibes later following dictionary so you would define the Cartesian product which correspond to the conjunctions 2 in the number days and the functional types is among the prominent function and then you will so if you see a proposition so the idea is to support another type of its pool so if you have this new and you can expect you can look at forward as being equivalent basis the poll .period and very strong form of existence as being present at Wednesday's servicing
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and so and this was done soaring the entire goes away
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mathematician the wine with this attitude is this format is demonstrated by official another medic who on the computer what is this so nothing to do with the vibrations at this point and this is a very nice between description like most of what going on so and what I find interesting is that they're so this is the logic which condition of religion all you do is you need a lot of the time and you looked up to explain logical so know job she had to list all surgical on there you go paperless political the following the so and so what is the source of little diets and holding you put it in the context of sources universities and both types of so usually it's going I don't have the time the since we want to see the connection with topology and we put them above so was our unit that's so you know that's so it does mean also universe is being produced by the wine and the not in love so you need don't see that guy which is of equivalent types so it's a tightly amongst all types of this type of pipes it's screwed by the following so in this kind of formal events also sparred looks does not apply directly but usually Geelong was able actually to walk results ability for ponder has had said the disability is all the nose of all all went from the day following geologist I kind of went from the type of work from the donation of with political issues and so Sugiyama was able to show all the results despondency issue of a type of food so it's similar to settle legal attendance at the Ford said so why not the law selected in following goaltending explicitly image Angleton explicitly include using steadily hindrances all want to go there and the extent of the taste and it should SUN disclosed by those operations but UN contends winds and so where a man from the rest of the general have not known the mothers but it's also so if you're the type of is any type in news you always also typing you away so it's very natural to me by allergies 2 OK so resists you can knowing you cannot present the notion of structure mathematical structure because you can when you have style cues you'll see you can introduce an army probably X you and the structure exposed to this type for instance between what with all the collection of pipes In news also by reservation only operations will experiments in X and the cost of the discussion of on the discussion of the 1st and was earlier the agenda she wanted very candid finally alone but also point of course is a college and then you have this type of you another have type of thing and this is the standard for presentation which is used by the offer the to so that if you want to introduce a of facility you will have qualities in this binary operation so you need a notion of liquidity and this was a very important continuation of some nothing Love who introduced so the new rulers have fallen short of liquidity but no agreed to the best of the thought of as a long so this will be the type of meetings between you and me so that amount of this type of result inequality between the so it's natural to think of this as a boss connecting if we think of as the if we look at spaces for these guys the greatest just said civilian population yet there in the other global initial frequently the company was 1 of the what prosecutors yeah because somehow you have to yell and there's this adequate heating and if you look no exit the goal of liquidity it's very similar to what you're full of select for Bossi on distant past we can and which correspond to the approval by exceeding and that each of the past connecting I mean I said that to defend the and the popularity of the to would be no you can look at this as an issue of a path between now and then you have a way to passports in nearly all of this to the amount of its so this is exactly so if you look at this picture I mean this is exactly the past lifting poker we the nominations the buy or if any of the see convinced he's the man so grueling that this is true there was also a product of former years although this the has had a hand in high over the bar as yet so still faces no immediate Monteagle excellent but did you know it can be a structure mathematical structures which values which we believe is possible between the Socialists and social and so I know also that someone in was the binoculars the hallmark of a continuation of the His only where they want to use this form of systems that exist already and he was about to form relentless on various environmental conditions which corresponds to the important fact into procedures performance also is defined the .period phase applied so there isn't in some junior events you defined type from which means that there is constructively so when does mean it means that you have a point in sewage at any of them .period it areas according to of meeting in 6 came with the repression of this factories it is the weakness of the space there is conflicting the new defined the fiber so we have not wanted to and a needle pointing the fight but also all reunifying fund so it's only if you are interested in the space travel lot that all cash and the solicitor out of this there a violation yes yes so unity so the types of John complexes and the phone was born to a confederation but can't find it the fastest battle this year by the 1st thing I can do this in light of wasn't they are so you so if you this you have the witness stand so you have a witnessed this with its talented visual proof that there is confronted with such a view have to look convey festival not home is going to add to that which saw the report that there is conflicting with the new other than and that simply cost projection so and then you define when a lot of this is because of you wanted to have rules that will have a house Confederation yes so that's it so there this week ,comma Teresa at I have a Christian .period and this the government so that it is a conflagration well above and in this former exist and that they will come and then you go when asked so if is a lot to you can define when efforts of immigrants and this is simply to fall in the fiber is conducted and then you define the space of the criminals between June and has the system of types In the cruelties of bad so it's the functions 102 in pool that is in many different types of opposition sought by political opposition if the any once on court even if I knew that when the Tigers is the set the
23:09
died space softball usable position and then you can define when title bullied and recently formed the Society for the History notes so that not so that's and soul to reassess the nation's solely on a nutritionist explain what it meant that you utmost above between 2 and you have also the notion of decided what time this will be so you have to have resisted calls for Texas in the process of so negation Of course it said so if you have a pool for this following a meeting of excellent in text so and this is defined as beings implications of 12th so is in itself a space which is a space so the respect that well this is the antidote so this will be X is the definable and 1 of the festival and of this formalism is that if the type decided then it is a set and it is welldocumented and because it connects logical motion and on what appeared to be the kind of society and this will face there a lot of this area because government bonds as suggested by the year of the sorry Of the 1st of these was the result of this contacted if so elected is so far and the content is 1 of is something victories this type is a proposition so that's this is so for instance you can pull the toll any day so this type is aware the proposition and following the end of the study is published villagers here so it is also his local position at the end of his contract would die and then there's and that's a lot of the school and so whenever you want something like that it is in this type of and so the actually formulated positions that and so forth just give an example and so and so and topology so whenever you 2 vibrational fight was based so sorry that you have to look at you don't have and then you hear a lot between its rivals continues in India then you can define corresponding up on the part of space then it's a deal and that the army this not home but especially crude family for which a is also something of a firearm yes yes and this is not the answer which you find in in under white topology and Villaescusa was able to put it in this former assistant like Johnny it's impractical manipulation Yukon components and so when you have also so this notion of equivalence it's a very strong emotional you can sure of course what you expect is that the up between 10 days and equipment than it doesn't end there so in particular talents and logical any grounds is a logical equivalents and not a critical analysis only unions the German mark in both directions but if you have 2 types of all opposition won the notion of logical agreements coincided with initial agreements so here you have to do the types of propositions then as this logical equivalence is actually in criminals with you friend we're going to war in the is this in the formation of the year the year of so the flock to both the quality time that's so resistant notation so often enough land for Kenny Kelly logical that With it could use this law soon which means that for any type so you have a new rule remains essentially ideological robotic it said that following the end of this time this contracted with OK and what is it starts this expressive is explicit about the space of evaluation defined by suspense of Boston in the religion is contracted through this fact was mentioned to care character you have to do something to do with this hassle this liquid is a little under the command of the facts that should always positive qualities it's not something I'm not actually want to do that other diligent antiquity it was yet and displays the change this fact and that is really an important holding on for instance in the pool would use the stall and you have to use suspect and this fact was introduced for John ideological reasons why not enough and what the market at its this factor to topologically this is exactly the starting point of space middleaged and white as it was we have of the this is my home that he was a little over right half of the time I was implacable exactly so exists is formulated using the locations including when we want to do from some sense said he he that was the of it have the the question of the students in people riding in middle of the back the report to show that he had L no I can come to the house so the statement a few months so fast a form which is close to the 1 thing which is 1 of 4 Boesky and so it's so that they and this time the universe you so you have which you which is simply an ex so points of you know the type so you have all unit of Texas to the police said the marketing you this is a total aspects of this on this speculation and there are so I'm still working on states that this at this site which antiunion that only this quite soon and the formulation of the weather so if you have a path so inequality between you and me then so you can pull that in ICU won't simply because of a sequel to an and the liquidity is a list of extinguishing the disease and the formulation of this canonical among the and we use a lot of competition which is equivalent which is the His noticed lost scalpel which is sort of like will statement is this flexibility if you're OK so we know that if we look at how is this house is set by the rules for nothing of the looks of this type and scope to do when the system is going to do so what happens if you replace spot by agreeing to give them a little point at which is this 1 the entrance the underwent a similar way to follow relates to section of the New York area also showed that this type is completed and although few look at this so this and then you have the resisted types so you have situations that you have to family of fights between for need so today if you have not when path to equal and an inopportune which is the general like soldiers in uniform and any Knopf we increments but still and on the crew put space because you have got to move effectively so any amount multiple between 2 conflicting is an acquittal so if you have this actually this formulation is actually a criminal In a danger to
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you fix a time numerous and look at the things as what kind of person would not have lost their ability to it's been stopped it has stopped using the technology however I so what's the connection with have actual description so many there you can see this time but which is a proposition the former insults at any point in the inquiry effective uniform generalization humidity I mean when you said that the state is complicated but you can define this as being and you know as I said so if you're in school hoped to use the new this consortium based in my 1st projection so you have in direct while you have a description of the nation's because this type incident in places where and so if you have a point volcanic stance then you get a proof of this signal and sway Frost from you get to the position is unique exist also implies that effectively so this is to say that you have operations and you look at the fibers which are concerned that this operation will have a section of no no position Over 1st of all over the place Wisconsin is complicated this area of England only because new so we can use it in the hands have left the on this beach but specific look at plants in any of case the basis of the it is as good as is from that none of the dozens of mourners said something about that but those spaces of language that's so you to say that the post those regrettable as it is unique in the form of a job yes but you don't use it you will use a violation of the health for what had happened at him but then it is a little distress is that so this notion of sequences unsuccessfully generalizes it's small and uniquely Wisconsin set so it will be a unique exist also plays his own unique existence in a categorical level to categorically 1 unique it is ought to criminal intent with which have so we have a we can say that we have an explanation for this fictional but somewhat what is it what matters is sold in the 5th inning Sosa was developed development I'm going to tell you a little things so for instance title then American paper is going to cut identification of all of the people so fast music can because certain that these intrusions and won the the 1st 2 surprises in this notion of space it was noticed by more innocent enough In 55 in that the space sector is contrary to the usual to produce goods the former interested so but as I close my expressing more you just don't want have to go out there that the yes so the tenant is yet so that yes so it was that they had computers this kind so this has announced his it in jail it contends all of 1 must use was the attitude this spaces formal model of type with you the factories so that was a fast that I wanted to understand and also in particular losses this is if you look at the foot of this in this couple singing and suddenly all the couple talk about these guys and the resulted there is the proof proof was in the hands of the value of X will launch of new mollifying some but that during this crisis .period President Asif technique and they need and if you look at the pool me it's quiet it's technically and war and if you want this to be a bit because ultimately Jake and you don't you want to of the simple justification of and then you don't need fall within effectively unilaterally returns this place ,comma look any conclusion so which correspond to people .period and if should look at this things With the names so you seem so if you look at this and say producers to the facts that the previous article violations of stable buxom confederation and this fact so which is considered as basic people reach Schwartz who is done complexes and the truth is actually quite complex and that and I think that approved using admissions think that that during .period complex and so we would like to have that long it's a good argument for that so anyone can say is think that maybe so will not clearly enough maybe we have not found a simple model for this but actually it was negative negative negative but joins the causes as was the source of the intrinsically nonaffected so it's not a question of what that tune in and find what proof is architecture is not political intentions to do so even if you use no strong intuitionistic system picture digressions only wanted you can prove that the following the basic facts of what were so this deal and that too dance complex on close by expectation this is not what it would have been and if you have vibrations so something only completely missing so if you vibration from whilst connecting points in the days then you expect to find us to be a lot of people sort reconcilation cancelation feature on the basis of the facts of yes or yes exactly so you can think that the problem is because they're when you the Kunkel blacks to serve you that existed the effective feeling so you can think of is not that duties as well as that to was among the few victories so that's not way the authorized the faces include 1 it's not enough to say that this is only the last fall the negative ones and whether or when you look at this sort of artistic intuitively formerly from working at the pool really that was as only wanted to do that he was resigning by charities without neglecting their genitals so that's and this year's you would be so this is a lot of Salomon years apart well or those with owning a is dubious didn't like disease if it's not indigenous to the world I have visual yet because you don't you will fall said it wasn't you at the time and you ask your results and practice the distance and this is not decided when India and as soon as you explore so also a mathematical reasons scoff at the fact that the argument or not uniforms no not in public so I think we present is "quotation mark election as argument uniform with students have so we know little to pursue an effective political and which will produce use where all the the arguments of no element on the this and and so what I want is emotional space which is really as simple as possible and that what they want to do but if it's not I don't think this is what the people want to do that and you want to have yeah well motivated and that each of these simple fallout for getting over that of the 2 many not for their opinions about the produces the following so let's start will walk with students because it's more
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convenient for With effectively and I will have so what I want to work with and whether we want I want to continental presentations all of this collection of tubes so if I define it said want a formal presentation of history and what I will use his game so is headed wide publicity of the old 1 so it's not too much us under 1 thing that it's over and it's something which is good Donald magic on but it's not really very close to distributing legislative manager wise among distributed late this year but unit intelligent 1 so I want to focus on this but in a community with so I wanted and marginally what they want is if I a fine except I want to Continental conditions so the category of children will be open communities and what they saw as you want as a structural model and was of no longer marginalize along the disputed that there was no longer on dilution so what is it with the moment invitation itself and we will widen got to it swaps and and so it's 1 reminds you always want 1 minus 1 that's what the 1 the and the use and 1 a suit 1 1 that is the Nunavut is just at the end of this year and was effective that's what isn't the wellknown and also in the sense that you have the more he said suggested that this this pollutant in which eventually did so no 1 can 1 landscape with whom over you that the company failed sections executive sorry don't come into solid and hazardous the window closes in general and so what is a prime example of a little knowledge of what it's like to you unit intelligible 1 who is the locks and many Corp positions among the definitive at this and this operation I'm satisfied with affection for that what I will use of force "quotation mark we think 0 1 to look around for many what is this half the capital of Jews so I have the time this is a purely what nations like you and your white kilotons so I went to Northwest is freedom alone and baldness and as fallen distributed he fastest to democracy out of the water fire at this time however this week come from and I won't be as good as was Chile ,comma Paul Hogan antigens ,comma when you will the following is the various that so if you have 10 elements than you were when you exist they became known world number Free distributed letters from finite set and so amounted to a visit that the canonical Flintridge it so have to live with this development free if you only 1 elements however if you have 1 elements what is the free capital knowledgeable as you know I don't mind this time periods in conjunction with the zeal of his decision to commute everyone so you in this case you have 1 2 3 4 5 6 7 1 which is dedicated of do that's a number of that's normal for the month of and finished utilities onto it so it's attributed to his is history to have created and of what it's like to buildings of previously utilities it is good to so this is defined among other on the getting finance sets and so the statue of Cuba will be the opposite of the place because you think you do what I would do anything but that's only and what is not only June 2 In see it susceptibility platform 5 to the freedom of the world on To thinking so we will continue the size of the year to think of writing seems to me that it's essential that is at the heart of the matter is selling green and very carefully yet it's so and bettereducated what it amounts to is that I will this cargo operations the top in Ireland while Parliament position was in no upon what politicians what I can do is to use likes men and women 1 that's what I want to capture former consumer chemical sets when the question and this this category so completely innocent of Amelio With transition functions from excited to exchange at all the political opposition last year was more but it's certainly isn't usually that's so which correspond to the meeting quickly what you want to do any of your Parliament files using maximum and then you problems so I mean as you and I think that this was 1 speaker of Italy's is so I will identify 1 object of the base continually resulting wood from Delhi differently so sold the finds of finance I will widen slides open and they were from friends so what so I'm walking was is based at the scene and I'm walking in the office of Christopher particularly of standards which is commitments From because lawyer this formal presentation it should find said I had a look at this whole accountable .period has a phone line open edition of the tube on time and so as not for instance an example of what not that they have so his his face not since I consent as I choose you'll and would wants to themselves so I get to it's not like there's a Windows into script and I get so usually composition of animal feeds on his decompose uniquely face mask with 2 statement With this factor fueling Monteleone interesting thing but not as much as a single consents so since it the only 1 hazard along and it was so I can define the functor homesick talk which to ISS it is you want the wood and so any "quotation mark space X they find a single document could set this effects was based pixel exercises a set of Clinton's on July 1 .period works as this would be a it's so so that's the a good thing it's the 1st time that can result allegiance from doing so because so if we use can usually definition among many of the city's funerals this will not water it will want to have lost which owned a solid fundamentals chemicals at all so I would like for them they say 0 1 star objects is that Honourable 0 1 s serves only on its own the old unit until 1 was a are and where there real and say Where are all eyes of finances who will lie the wording of world the unit is to specify no I send items into its objectives of this category I associate exist but would you consider all the elements of the region were not going to say what progress has not been alone in the final minute defined this winter will be put with we get prospect because of Pennsylvania was much more the there is also the place right oranges and locate their societies in yet those on the way time in the morning you have the president of the
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European Central time I don't know how do so so let all the license so it you could go on for 2 years said so if I have not hold so this is how we want to confuse me not qualified to judge is acetylated dropped from January to them on freedom of religion 1 please so by "quotation mark following the verdicts AT and AT and not when resistance to change .period and then because this is a little more and you get a lot of people knowledgeable "quotation mark items you want it enough to know not because I was so muffled as 1 needs a lot of 1 morning Eizenstat its finances almost from the moment I silence Johnny assembly so that I get a lot in this this this and this is a lot means Divine Secrets so Noonan both boats can be concept it is all about Jerry could no longer of subject that's what this is all so this defines the pressure and this is all presentations in Dublin and what is when I get this final is the so is convenient for them to do that by all those imported petroleum pinchhitting Lopez a based fast please OK so what is next to the line exterior so I have this interview rates should correspond to have 2 children of dimension and I should overture will file the motion which connects them and this is exactly what you get with this so excited Jr is essentially given by the Cuban Ireland it's had so this is really means it is that it's much better than much of myself a lot of things to do than simply set because distribution said that part of the explanation is that the 1 is even more complex to disclose them OK and also what they get for free is this from among the fact that the Zeus of contact you go get it on delivery of because set was overcome condition because what is it what it is it's a lot of people coastal then the laboratory type a condition of his square positions while users cancel any past 20 parts like an associate of public opinion coastal bathing so I yet for free this is the wrong place at the moment it was fiction OK I have so this is the 1st of the that this suitable fund and then sold knowledgeable than another and so there will be true objects that the importance of the alleged amendments acquaintances and so this is so that too only after 3 objects of who plays 1 of the reasons intolerance and then there is and as always there is a subject of the and then he and I have to nobody in 2 0 1 1 even then so I have a lot of valuable not which is Clinton intelligence defined as they put value of 5 . 1 on Friday and I can look at the marriage of this month In that defines this opposite Madison and I can define and look at the nature of this month and this will be the subject of some Christopher Flamengo and that will visit with icons App let's face like this because it's it's actually the lettuce which is did not so it's a set of seam and there are exactly the same result all denied them by faces union offices a consolidated Nunavut faces which will play a fundamental role but several fire is a set so intuitively so I kind of just so I opened alone and this willing to the facilities so gratifying to that come so this will be postponed because of like all of a sudden bordered on the procurement of the children Maryland and that's what they needed in order to stay because it can conditions that conditions saw how this and this and that this can be defined likely as if we equivalent to utilities United with just 4 months in wardens this condition and this is where local vibrations we've come so fast reflexes cycle which interviewed so ending up which is classified as we do which that's and constructively so we cannot hope you can conditions set in the classical I'm not on 1 local but this is to this weekend so if I have a lower standard so I will simply for medical conditions for this and for this source for that if this motion so the whenever level up .period prior to Sciences is defined as some objects seem to have concluded so withdrew from the study that the subject so interested so the so this is a presentation of the cumin they want to buy and this is a local politician of some of the faces of what "quotation mark so what I would say we standards is not musical language but not to the recusal say so given use the example of what it means to be compatible and stuck with this and so what is usually in the form of opportunity usually Marshall party meeting that the issue of a standard Texas conception that you want to say that the Texas conflicted with what you say usually is that too if you have not only suburb where you belong the June genentech collect on this locked 2 2 at a time when you can fill it in so when you can focus on what I ask is that only final of this which is uniformly so what they say is that the final objective in jail and a lot of good category has continually to buy so I can I can put it like this so it it can change shape and so I will have a so why composition executive it's not that we have an extension of you and what I ask is that I don't give only 1 extension .period earlier that they give a uniform collection extension so I asked fourpitch hired you the that's of definition so we haven't coordination which is uniform which gave this filling uniformly given that sweetness defined emotional and similarly I can define what is a confrontation but no it's a uniform national conference which and that so that's what they wanted the distance of 1 extension so it's natural learning to get something constructively put the exhostages on its that and this is the condition so I can show them to the space of closed below expansion in an effective way and if you want anyway and they don't need this or just an example of perhaps this the creation of uniformity it's actually quite surprised some lessons
1:00:40
opensource you can show that this effort is contracting resisting the conditions only got conflicted with this uniform definitions I it's a very interesting word because it's so full each item you can find you can find an extension but it's impossible to find it in a uniform way that's very nice but submission Crystal City and then also eventually define this and then 1 of the 1st result that you can check is that if you have a topological space next to to the single persons property single because that is firebombed was a strong social uniformity and this is for this and you take the usual all you do is this I need put this usually pools at the box is a part of the field in books and if you look Texas must not toward proof this means is proof is actually uniform and book can I think it should stop so yeah so that's all you did effectively model of society which satisfies the union concessions in an an effective way each of which which I I would like to get the usual you last year the preamble that slowly the interesting people walking this will be means to almost 1 and crystal subtle and equaled only now and I think so I'm not sure what to do about the status of this wasn't thinking this was a call walking on getting a hundred like which with the working poor and then on to the meeting of the governments of animals that were there the that's about all I can do all the end of the day of the city that is that of his life a entire thing that yes also salt crystals that :colon economy in all 22 was pieces when went through of Korea but in of the soul minnows of aggression where so you do this but you know what is instead of tried denunciation when that you take your will decided 1 of so of which "quotation mark positions upon this and they would and you try to use only models at all this and I think so similar results at the Concorde expulsion but I'm not sure about the actual human being section and for the Union section which is you use the crucial away is the have attitude on this and under the new tend just after the hour and you have a kind of elimination of quantified for which she was thinking is that if you look at the zoo meticulous screening the full of potential exists and that is used in the "quotation mark for Fourchon and I'm not sure if he gets the timing this is also looking for that's the last thing they hope to get things you should know about this should enable Mitchell where the His stock funds would explain we also mentioned that for had there want which is the area where the division called for more than a quarter of its virtual also a supporter of the death lined up and incidents in this battleground the general manager of worldwide this year has also been discontinued the largely from his motion for human use hyperextended right disintegration and the back marble however ,comma this it's not that people groups like the 1 of the assets of the force of law and the emotional side we need to hold him the area making contact with his son with as the result of marriage the meeting with the new please contact me and get on with expect the next moment in the sun can just want to know you the main thing some of wanted the money I have to say yes or injured men application for a similar applications is designed by system whatever you do don't is good and where everything conscience you can call it illustrates the government please we have Virginia the tied with the implementation of this the meeting you the president this was a around
00:00
Axiom of choice
Group action
Building
Musical ensemble
Graph (mathematics)
Multiplication sign
Correspondence (mathematics)
Sheaf (mathematics)
Insertion loss
Propositional formula
Axiom
Parameter (computer programming)
Complete metric space
Grothendieck topology
Variance
Mathematics
Topostheorie
Military operation
Logic
Position operator
Amenable group
Descriptive statistics
Physical system
Injektivität
Product (category theory)
Spacetime
Propositional formula
Thermal expansion
Price index
Term (mathematics)
Functional (mathematics)
Bilinear form
Connected space
Category of being
Physical system
Existence
Presentation of a group
Set (mathematics)
Modulform
Frequency
Natural number
Simplicial complex
Operator (mathematics)
Liquid
Antiderivative
Addition
Element (mathematics)
Mathematical analysis
Reflexive space
Set (mathematics)
Field extension
Calculation
Computer animation
Logic
Function (mathematics)
Element (mathematics)
Dependent and independent variables
Object (grammar)
Distortion (mathematics)
09:33
Existence
Transportation theory (mathematics)
State of matter
Ring (mathematics)
Set (mathematics)
Multiplication sign
Modulform
Sheaf (mathematics)
Propositional formula
Student's ttest
Cartesian product
Number
Frequency
Goodness of fit
Mathematics
Positional notation
Meeting/Interview
Natural number
Operator (mathematics)
Analogy
Logic
Antiderivative
Position operator
Fiber (mathematics)
Curve fitting
Physical system
Area
Sine
Moment (mathematics)
Basis (linear algebra)
Propositional formula
Algebraic structure
Set (mathematics)
Limit (category theory)
Bilinear form
Functional (mathematics)
Connected space
Field extension
Computer animation
Logic
Theorem
Element (mathematics)
Game theory
Family
Arithmetic progression
13:45
Complex (psychology)
Euler angles
State of matter
Logarithm
Multiplication sign
1 (number)
Sheaf (mathematics)
Kontraktion <Mathematik>
Propositional formula
Axiom
Mereology
Total S.A.
Mathematical structure
Vibration
Grothendieck topology
Explosion
Summation
Duality (mathematics)
Mathematics
Positional notation
Military operation
Logic
Universe (mathematics)
Fiber (mathematics)
Descriptive statistics
Position operator
Physical system
Area
Logical constant
Product (category theory)
Spacetime
Process (computing)
Point (geometry)
Propositional formula
Nominal number
Functional (mathematics)
Bilinear form
Higherorder logic
Connected space
Proof theory
Category of being
Lattice (order)
Phase transition
Theorem
Uniform space
Mathematician
Resultant
Directed graph
Point (geometry)
Suspension (chemistry)
Expression
Flock (web browser)
Presentation of a group
Divisor
Observational study
Sequel
Paradox
Student's ttest
Inequality (mathematics)
Event horizon
Rule of inference
Frequency
Binary tree
Formal power series
Operator (mathematics)
Liquid
Analytic continuation
Subtraction
Units of measurement
Condition number
Standard deviation
Multiplication
Sine
Physical law
Projective plane
Mathematical analysis
Content (media)
Algebraic structure
Set (mathematics)
Density of states
Equivalence relation
Inclusion map
Field extension
Formal power series
Computer animation
Logic
Function (mathematics)
Factory (trading post)
Network topology
Universe (mathematics)
Element (mathematics)
Family
32:30
Complex (psychology)
Group action
Logarithm
Water vapor
Solid geometry
Vibration
Expected value
Maxima and minima
Insertion loss
5 (number)
Descriptive statistics
Physical system
Spacetime
Basis (linear algebra)
Uniform convergence
Mass
Sequence
Category of being
Intuitionistische Logik
Lattice (order)
IRIST
Mathematical singularity
Theorem
Arithmetic progression
Directed graph
Point (geometry)
Free group
Student's ttest
Infinity
Open set
Number
Frequency
Latent heat
Goodness of fit
Helmholtz decomposition
Energy level
Directed set
Units of measurement
Standard deviation
Model theory
Operator (mathematics)
Set (mathematics)
Line (geometry)
Formal power series
Equivalence relation
Game theory
Musical ensemble
Building
Euler angles
State of matter
Multiplication sign
Decision theory
Sheaf (mathematics)
Propositional formula
Insertion loss
Parameter (computer programming)
Meeting/Interview
Position operator
Fiber (mathematics)
Stability theory
Area
Process (computing)
Closed set
Moment (mathematics)
Distributive property
Bilinear form
Functional (mathematics)
Hypothesis
Connected space
Proof theory
Funktor
Normal (geometry)
Uniform space
Fundamental theorem of algebra
Existence
Divisor
Presentation of a group
Set (mathematics)
Modulform
Tube (container)
Distance
Discrete element method
Formal power series
Causality
Operator (mathematics)
Euler's formula
Free group
Condition number
Sine
Forcing (mathematics)
Element (mathematics)
Projective plane
Disintegration
Incidence algebra
Commutator
Factory (trading post)
Finite set
Object (grammar)
Marginal distribution
51:16
Euler angles
Multiplication sign
Mathematical singularity
Sheaf (mathematics)
Mereology
Vibration
Military operation
Square number
Cuboid
Position operator
Physical system
Area
Logical constant
Spacetime
Topologischer Raum
Closed set
Point (geometry)
Moment (mathematics)
Distributive property
Uniform convergence
Coordinate system
Ext functor
Thermal expansion
Bilinear form
Maxima and minima
Flow separation
Concordance (publishing)
Category of being
Proof theory
Arithmetic mean
Lattice (order)
Hausdorff dimension
Order (biology)
Relation <Mathematik>
Uniform space
Cycle (graph theory)
Fundamental theorem of algebra
Resultant
Presentation of a group
Observational study
Set (mathematics)
Distribution (mathematics)
Distance
Frequency
Gaussian elimination
Goodness of fit
Centralizer and normalizer
Field extension
Lecture/Conference
Natural number
Free group
Associative property
Field (mathematics)
Condition number
Standard deviation
Forcing (mathematics)
Model theory
Physical law
Division (mathematics)
Set (mathematics)
Mortality rate
Line (geometry)
Incidence algebra
Cartesian coordinate system
Local Group
Field extension
Element (mathematics)
Object (grammar)
Pressure
1:07:08
Computer animation