1/4 Introduction to categorical logic, classifying toposes and the "bridge" technique (IV)
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1/4 Introduction to categorical logic, classifying toposes and the "bridge" technique (IV)
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Title 
1/4 Introduction to categorical logic, classifying toposes and the "bridge" technique (IV)

Title of Series  
Part Number 
3

Number of Parts 
28

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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. 
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Release Date 
2015

Language 
English

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Abstract 
Introduction to categorical logic, classifying toposes and the 'bridge' technique The course will begin by presenting the basic notions and results of firstorder categorical logic, with the aim of reaching the theory of classifying toposes by Makkai and Reyes and illustrating the general techniques allowing to use them as unifying 'bridges' for transferring information across distinct mathematical theories. The exposition will be accompanied by several examples and applications. The lectures will require a basic familiarity with the fundamental notions of topos theory, as reviewed in André Joyal's lectures on Monday. Lecture 1: Firstorder logic and its interpretation in categories. Geometric theories and syntactic categories. Universal models and representability.

00:00
Category of being
Computer animation
Logic
Topostheorie
Multiplication sign
Series (mathematics)
Mereology
Event horizon
Resultant
Connected space
01:48
Group action
Multiplication sign
Discrete element method
Grothendieck topology
Order (biology)
Sign (mathematics)
Manysorted logic
Meeting/Interview
Lecture/Conference
Velocity
Focus (optics)
Theory of relativity
Consistency
Algebraic structure
Thermal expansion
Price index
Lattice (order)
Term (mathematics)
Limit (category theory)
Category of being
Logic
Universe (mathematics)
Order (biology)
Right angle
Object (grammar)
Game theory
Directed graph
Maß <Mathematik>
Spacetime
09:13
Functional (mathematics)
Confidence interval
State of matter
Multiplication sign
Insertion loss
Rule of inference
Fraction (mathematics)
Order (biology)
Duality (mathematics)
Goodness of fit
Mathematics
Lecture/Conference
Meeting/Interview
Wellformed formula
Term (mathematics)
Natural number
Modulform
Series (mathematics)
Devolution (biology)
Condition number
Newton's law of universal gravitation
Area
Process (computing)
Basis <Mathematik>
Maxima and minima
Line (geometry)
Incidence algebra
Term (mathematics)
Category of being
Arithmetic mean
Exterior algebra
Right angle
Hydraulic jump
16:27
Polar coordinate system
Logical constant
Existential quantification
Building
Group action
Multiplication sign
Decision theory
Equaliser (mathematics)
1 (number)
Junction (traffic)
Grothendieck topology
Subset
Manysorted logic
Meeting/Interview
Arrow of time
Social class
Area
Theory of relativity
Special unitary group
Price index
Term (mathematics)
Variable (mathematics)
Binary tree
Directed graph
Functional (mathematics)
Observational study
Similarity (geometry)
Heat transfer
Inequality (mathematics)
Hand fan
Product (business)
Frequency
Morphismus
Causality
Lecture/Conference
Wellformed formula
Term (mathematics)
Natural number
Operator (mathematics)
Genetic programming
Set theory
Condition number
Multiplication
Algebraic structure
Numerical analysis
Quadratic form
Network topology
Free variables and bound variables
Musical ensemble
Allgemeine Algebra
Table (information)
Maß <Mathematik>
24:19
Free group
Theory of relativity
INTEGRAL
Equaliser (mathematics)
Multiplication sign
Lattice (order)
Term (mathematics)
Urinary bladder
Junction (traffic)
Connected space
Frequency
Radical (chemistry)
Manysorted logic
Lecture/Conference
Logic
Wellformed formula
Finite set
Thermal radiation
Genetic programming
27:38
Multiplication sign
Decision theory
Direction (geometry)
Moment (mathematics)
Modulform
Variable (mathematics)
Hand fan
Numerical analysis
Statistical hypothesis testing
Lecture/Conference
Wellformed formula
Musical ensemble
Lie group
29:38
Axiom of choice
Complex (psychology)
Computer programming
Haar measure
Multiplication sign
Sigmaalgebra
Expression
Einstein field equations
Algebraic structure
3 (number)
Sequence
Urinary bladder
Theory
Quadratic form
Inclusion map
Arithmetic mean
Positional notation
Manysorted logic
Lecture/Conference
Wellformed formula
Musical ensemble
Arc (geometry)
Chisquared distribution
34:04
Process (computing)
Physical law
Content (media)
Limit (category theory)
Arithmetic mean
Regular graph
Film editing
Voting
Thermodynamisches System
Lecture/Conference
Meeting/Interview
Modulform
Object (grammar)
Series (mathematics)
35:40
Area
Addition
Process (computing)
Observational study
Variety (linguistics)
Lattice (order)
Incidence algebra
Event horizon
Theory
Arithmetic mean
Mathematics
Regular graph
Voting
Manysorted logic
Logic
Selectivity (electronic)
Spacetime
Newton's law of universal gravitation
39:23
Geometry
Existence
Group action
Multiplication sign
Sheaf (mathematics)
Food energy
Theory
Mach's principle
Inclusion map
Sign (mathematics)
Regular graph
Meeting/Interview
Reduction of order
Genetic programming
Physical system
Area
Sequel
Concentric
Weight
Physical law
Division (mathematics)
Line (geometry)
Price index
Cartesian coordinate system
Sequence
Computer animation
Factory (trading post)
Sinc function
Maß <Mathematik>
44:13
Axiom of choice
Point (geometry)
Computer programming
Sequel
Variety (linguistics)
Similarity (geometry)
Propositional formula
Ordinary differential equation
Regular graph
Rule of inference
Theory
Product (business)
Prime ideal
Inference
Sign (mathematics)
Regular graph
Lecture/Conference
Different (Kate Ryan album)
Musical ensemble
Algebra
Set theory
Social class
Weight
Model theory
Physical law
Content (media)
Horizon
Algebraic structure
Morley's categoricity theorem
Grand Unified Theory
Lattice (order)
Cartesian coordinate system
Complete metric space
Sequence
Equivalence relation
Connected space
Category of being
Algebra
Computer animation
Logic
51:44
Computer programming
Functional (mathematics)
Existential quantification
Hecke operator
Maxima and minima
Similarity (geometry)
Product (business)
Mach's principle
Manysorted logic
Lecture/Conference
Wellformed formula
Genetic programming
Set theory
Social class
Newton's law of universal gravitation
Theory of relativity
Algebraic structure
Incidence algebra
Sequence
Quadratic form
Equivalence relation
Sign (mathematics)
Category of being
Film editing
Logic
Object (grammar)
Freezing
Directed graph
57:02
Point (geometry)
Functional (mathematics)
Group action
Existential quantification
Divisor
Euler angles
Equaliser (mathematics)
Correspondence (mathematics)
Quantification
Maxima and minima
Rule of inference
Event horizon
Food energy
Theory
Product (business)
Element (mathematics)
Mach's principle
Manysorted logic
Lecture/Conference
Wellformed formula
Term (mathematics)
Operator (mathematics)
Gamma function
Position operator
Directed graph
Social class
Condition number
Lemma (mathematics)
Projective plane
Algebraic structure
Price index
Variable (mathematics)
Leak
Substitute good
Fast Fourier transform
Category of being
Arithmetic mean
Logic
Order (biology)
Object (grammar)
Musical ensemble
Directed graph
1:04:33
Axiom of choice
Group action
Presentation of a group
State of matter
Multiplication sign
Decision theory
Axiom
Parameter (computer programming)
Mereology
Junction (traffic)
Grothendieck topology
Explosion
Manysorted logic
Analogy
Physical system
Social class
Area
Process (computing)
Theory of relativity
Special unitary group
Price index
Lattice (order)
Complete metric space
Sequence
Higherorder logic
Degree (graph theory)
Category of being
Arithmetic mean
Order (biology)
Classical physics
Geometry
Slide rule
Existence
Functional (mathematics)
Observational study
Image resolution
Maxima and minima
Discrete element method
Theory
Product (business)
Element (mathematics)
Mach's principle
Regular graph
Lecture/Conference
Wellformed formula
Natural number
Operator (mathematics)
Modulform
Genetic programming
Set theory
Condition number
Standard deviation
Matching (graph theory)
Forcing (mathematics)
Model theory
Algebraic structure
Morley's categoricity theorem
Line (geometry)
Axiom
Equivalence relation
Quadratic form
Length of stay
Algebra
Logic
Object (grammar)
Maß <Mathematik>
1:19:17
Lecture/Conference
00:02
what happens From madam about was underscored the need to give an introduction to Council on the category logic with the aim mail explaining the crucial notion classifying proposal rejected his when
00:30
the list is used for the media will start in connection with the is much of the the I don't know what to say face this was an event that could be used for connecting to this is year is that the chance and unsparing results of people across the soul of the warrior easily taking part in elections such as the time of the led because noting that ousted former Thai monotonous here and some of the members of the community governments series because of something that can be carried out the most amazing now as I was it's providing even more than just a logic because it is a just enough of that I don't be prices International Congress on you saw and these that the only market intervention to the goal of unifying the series because you will discover that the issue here is the same as the 1 that was basically
01:49
anything that could be said the same story even it's not just the 2nd time in the area but interesting very useful
02:00
in the world said that because
02:04
every time you have to move people on the streets and universities of London
02:11
according to to the meeting of the problem I need to something here they already so I decided to leave on biting calling it is of concern to us is having on the delayed game so I'm looking past when the certainty our most studied logic the times of the year in the there was also 1 of the world's leading citizen of people of this conference is the the structure and my last name is being used for the 1st time that now there is also of will make this evaluation meeting will be friendly they didn't see any lives in the media that this describes the expansion of the structure this 1 of the sole reason has a right to do so but it usually meant additional from article interview I think that this is a marvel of languages all the ways and that's destiny me make use of all these findings that sort of person familiar with the fact that there is in his 1st 1783 sociology and is thinking so you can have many different this index is that the foreign agents around the corner and only about the relationship between 2 years so it's very hard to bear in mind that don't usually about what it is about the need structure the way we live and now I'm not thinking on the languages are languages the bleachers and Cecil the rendering of any EU sources and for now I will personally that I wanted to use the station's 1st of these is the only means that in the absence of a personal hearing you can only half of what we final older individuals of research and not all collection of individuals or collections of collections so authorities have said that the community is the problem he said adding that he had been the season information the effects on their way about the only 1 of the prime property called me and said that it not the center in history and niece and he is the 2nd on you and you want to eliminate the economics is the 2nd amendments because collections of such arrays of the centerpiece domestic stations 186 of the wedding with him what he had in Argentina and quality of the writing logic of Boston also argued that the world will investigate the property of the former you will always remain a universal dignity and all we should use the sites also and then there was no sign that that is a fundamental totally uninvolved is basically any finding of any novels and in category which is equivalent to the she was designed so that visitors can go without the use of the limited space and time but it will be the general notion of velocities these she's on side and Denise and he's generalizations from a book Patterson who is a was 1 of the to strategic ties basically criticized the hope is that the Soviet Union Station the actually very very powerful people of the understanding of the quality of life here it is the place where you can realize the management unit continues in the street as he was saying he was alone and that's all of these issues are behind you be former where formerly known as lazy and she's going 1 to the state's master resulting in generalities that is incomparably abroad because the single hazardous and I'm police said that the Commission had agreed that the Government of the People's so the is he wanted to know who he is and this is the fact that yeah acted as species being treated properly apology and they're light years ago in the qualities and I'm always thinking of using this year by the it's not the usual political relations between the EU and watch and so I knew that in order to get the sense that any of these and as a consequence but was it furious accuracy modernizing facilities but they also them sites international society that is 2nd on the notion because beside the concedes that he is more accuracy and there have been no justification at the end of the school day got in the way of doing that just because you know this is not body in the way it was signed each object of the category is a collection of scenes from the movie this even when it is just a collection of bottles of homemade CD which is close on the composition of the belief that it is not to secret that want to was we were in car it seems that this is because that is useful when I was working for sheets and there was this will be the focus of Click on this single was them you'll be out of touch with the 1 of the land OK it's all an assignment insensitive in each object seen as a collection of scenes in his article that inconsistencies in 1
09:16
rights as JVC indicates that that is the use of declaring that don't work and I also to acquire some of hasn't properties which was formed on the property the new national rules over a year and followers had been the subject of the problem is the maximum on the condition which means that it must also deal with each other and this is the quality of the vacantly into the mix the the that you can see it in preschool and capable of with the kind of thing in the world to cooperate in any of the conditions which is on the state and only in this way in the of many seated quality conditions in the global demand when you want to get to the overall also called and always liaison his CDs are marrying each other conditions that want "quotation mark and this is what any of these qualities and a series of the 2nd option because you datacollection sees this scene is a collection of hours so using his secularism because not all objects and and also about elections so this is the 2nd alternative itineraries useful for studying the sale of the notions of researchers in the office of the agency said OK he was just a fraction of the time was the because the language is now classified as a logistical reasons they may improve the designated some of the terms in also called on the inside of the man who was deceived by the kinds of legal hasn't and has an annual bonus of about 40 cents making it 1 of the lessons of category than the president of the Commission said in a statement about the degree of of what you want and I want to be involved in understand identifying the areas that but it doesn't mean that you see all the nations of the world out of Japan on distinction has already sent a signal to schools on 1 of our most that has also also amended to read that confidence is the kind of medieval times are and formulas are amended May assertion but the end of this year but last season and would have been following the news started we have some basis for me at the time going up on the phone and then from those who need and want complicated formulas in the wall and undergoing doesn't have 1 1 because of this it was yes they just once in the last years worth of money from the sale and the needs of their visit here that movement in the nation of the already had recently using it this is just the latest in the medieval within each of the following his inauguration is functions in the ground functions you wanted something from functional significance From the terms because it wasn't just looking a new starting from violence or consensus by applying functions labels that I had soul police in the year also we had invited to the nature of the beast that most of the 1 9 in the nation and pharmaceuticals and whether this is the president will not only some of reason was I mean you just walk that was set in a later date as it is in the devolution is just 1 of the founding of the People's economically 1 doesn't need world names on the basis of what engaging in activities that you don't know the language of populist mentioning that he was the loss of jobs it was the that somebody wants to to discuss how they and they should be used in the incident to the end up these emergency air and land on your own I'm leading the nation in the performance of its kind in the form usually need just getting the idea held out of the solar and then starting from the end of the season was right in the end of last year can carry on the same amount in line with sold the thing that and the release of dance and you can see he's not individuals because of their year of life wasn't even in front of you will then have be an insight into the mind of the people in the knowledge that the that we want to America good thing because there was the basis and you just have to get married he was just meant to be a I'm going to have another view that now I'm doing it because I need to know who is now a college in the fall within the committee's investigation as the foundation and that the season although he sees asking about the island on video 187 having litigants in my recent changes in the basic knowledge of the language in the categories processing enough of them were construction of what you will you that the only thing is it will be time for me to tell you about it and sell it the we
16:19
have the right to it's going to where the the
16:28
1st of 2 or more of your calls for the former world to view of the went on and on and on about the stock that can also be taking the fact that these junctions leading medical interventions shows that only season by examining different from those that have long been the cause of the in both his disillusionment about enjoying the windfall bottomless musical information foreigners you I have you know that the conventional PCs and the citizen and the absence
17:06
of a genetic periods which it seems they they are all of these foreign private high and based wide in a given context a context is the
17:18
least of these things very well in old was found and the last thing do the only thing is that we have the from the formula used by only using his Indiana already downloaded currently not the only thing that I need to use the same justice only using the bank of to love most visitors at the pollution of the air the usually the cause of the I don't know the half of the was is always the the number of variables hanging Embraceable You recorded a win when you use anything that is just like that you might there be a priority and that we anything the number of free variables you don't want it all warned that the juridical you can understand anything immediately because if you allow this then interpretation would be a subject of the product and the products are not necessarily preserved the by interest images of genetic morphisms so you need to do to win polls these conditions but that was accomplished cases so what is the 1st time in the attention when we have a settlement so we use this signature is a is a synonym of language on units at the technical term for preventing use magicians and saw 1st on the theatrical ceasing to free sets up a set of sorts of the I usually regional theater we discovered the letters like that then a set of functions symbols and each functional symbol is that it keeps the women and transfers soandso and out of sorts I mean I normally we require refiners to require refiners to use them in bookstores and adjust the 1 multiple star topology functions similar and then we have a relational single relations in most areas where they keep the weaving indication of their sources on which they have become a source of income and that concluded suppression of moves in the office of the UN the 1 source and there you have the 1 constant and the constantly and treated as a 0 I really uh function seamlessly was aware yeah you think should take any equals European Union constant for this call the constant and soulful ruled land binary functional symbols and so he has forsaken repeat the same source right so I don't need a oneday and to distinctive so I will have my operation a 1 source and no additional stimulus because they don't need them in the case of sun and more generally when I would want to formalize the um were you you will win study as structures that are formalized their you know and you know universal algebra since then in the absence of your feeling revealed this kind of equality is between veterans and so you will not need the Hoosier associate dean at the end of the are of these the ICI's you want you can do that yes yes because otherwise that you have to use in existential quantification and yeah yeah I mean that are there are various ways to hold about years it's nature not to introduce a European unity operation before reminding the audience who were the health classes you want to formalize international beyond interceptor sold within the building that is just the cells women on their relationship and in this case that you see it is not the function because you see I am talking at the completely syntactically men so everything that I am saying he had does not make any sense yes it would make sense later of course otherwise could have been much better important moment former but why should there already keep in mind this is why I approve of the 1st the growth the 1st leg warned that they intended interpretation because they intended interpretation of functional singles or functions all arrows so you see here you don't function had a subset of the fuel I mean you can take the season formally this season is a subset of the of the product of excellence Weaver itself and so the site can be formalized relations now starting from 1 of these sales what you do is the 1st of all need terms and then I already said all you do is start the way behind the rest of you suppose that splits on table have anything spoke of variables normally communicated the variable as a source AU right then and there and something from these variables always have had of a number of times over the final number of them the terms inactivity applying functional symbols to them and so I don't think I have to write formerly because it's an OK now and on the president there are now the 190 winning the final performance as I candidate so formulas assault vessel and as I said at the beginning of the formula was amended to make assertions about individuals and immediately was summoned him to be represented by terrorist cell phone has admitted this is the formulas that you can think of our own described Yukon I had inequality between 2 terms some of the appropriate source because of course it doesn't make sense 2 terms and don't have the same sort it doesn't make sense to write something that's cost you require them to have the same sort of thing you can say say for sure because you have this kind of ethnic groups who were lose yes of course this is a decision because didn't he therefore modification but of course this season functional symbols can always accuse the resources and the parents are built there just inducted starting from functional so well there will also be keeping resource for the work we
24:21
do here and and I
24:28
always always accuse the whereas sorted themselves but the initial call for reliability infinitely many infantrymen the middle it's not about because normally you don't need a union aligning with you ,comma "quotation mark were united in their desire to broaden that you might think Brown knows there will be no truth in the eyes you will on the whole again so as to this kind of foreign investment you can go through this online the equality intelligence and if you had a radiation symbols such as you can also cause you know things of this kind of always provided that the everything is composer will mean that everything is the right sort of these are called upon to make atomicforce OK and now and then of course you can see this as a political radicalism that he should take an hour to be recorded in relation but of course there are in all languages that we consider equality is always there so we don't even treated as for many years at illiteracy among about uh and onion morally keys OK so these are the simplest that foreigners that you can constructive and for this reason the accordion called atomic formula and as deal better than in the fall the 1st on the logic you can alternate onthespot investigation more complicated formulas and by using various things that are sold online and you have the logical connectives which ironies that the ended it all for the implication integration you might also want to enter a .period formulas for false and foreigners with the same also see formally ending fool you this season in the home and King conjunction and 40 season and the disjunction if you want but I I just referred to provided an explicitly because they are very important for example use as an aerosol you cannot hide the use of the service stay open because all you had to quantify isn't also uses the existential handed universal quantified and and you get all that data on starting from the settlement formula and you can be in the 1st of the formulas by applying a finite number of times that all of these things such as I have already mentioned we also allow and when we speak of the things that is the sort of logic we cannot allow was sold blocking the American junctions and intermediary disjunctions was shattered particularly needed a meeting that is junctions when investigating a geometric logic and soulbaring minded he sees only solicitation to redefine that accepted that that you have to work and I mean when you define formula as you'll see simultaneously final defeat redoubtable some of formula so all you have to define the notion of free
27:39
full abounded and he's seen
27:45
in Endesa antiAmericanism formula is said to be the happiest moment has not quantified sobering instances the was and about otherwise it's only interested the in and you see that the wide use authority in this formula but not Texas found solid the market will make his decision because it that as we said in the case of a man in the value of the directions or instructions on how to regulate the number of the dead was 1 of the city's center the tourists to yes yes I know every year that is always so now we know all about the involve violence is on the front of the context of company Silicon next for the following the disaster loomed as the the forum for by the time set fire the east of the things that variables which school they will all be freed and most of which goes to the season they lost their debt you don't just get on with it and most of the work that you are not always the the like that said that the thing that amendment babies being obedient because you have only conceded the context of the becomes much more of a company that has sold worries that the situation in Iraq including the use of the wide to and police full of lies ahead of you a hassle and I can't believe that I feel this a way to see if we could move
29:40
on to the since and the restrictions on the kind of formulas to you that we use this sequence notation so what is this sequence but it's equally easy task for inclusion in the Senate nominee did so thank in people and the people that notice this time
30:05
so he's just following expression of these learning that enough is then and the end it was good and the last fall after paying in the same this year using the invasion was supposed to use it the finally of the was a solely on this and it may even be a the meaning of this expression is that there is a lot to do with virtually no politics I indicates the sigh with means that for thank you so much that is convinced that he's fast and he's very close to the citizens it's like finding 1 he III said lies not in the new meanings the believe but only at the end of the conflict yes yes but Indonesian Congress the for the time being there is them this is not just you and the Bulgarian people emergency an amusing but it was always considered formula as being involved and went on to thinking about and that look the complex would be turned over to the following it up another excellent in certainly adds insult to those balls in structure and mountain regions interesting that adding that nation of these formula is that you the end the following week the some 0 he said that as you see it the 70's even if you allow me you to is 1 of of these leading airconditioner was always on the news that he saw that we said that this was not the home he's just an hour and 20 minutes structure was signature you need me each sort of it's a soul and all he's just being patient or something structure of solar and this is this is the innovation that ruled funding our theories of socialism .period lands just the same yeah I wanted a hearing with things and incentives and the program is saddled sequence and me and all who also sequence that there is a need maybe but we don't know what is happening overload Ukrainian apparently suggests that the choice at all not allowing only 7 items of foreigners and all that was not actually find new uses in the year they also said that although he said OK so then you find anything other than the old technology so as to be get the nation and from time to time
34:07
looking at all and the use of Ajax since I wrote what I just the then only they will see to
34:21
it that he is a series of animals axles in in nightly news for me while wondering why it needs to be and what is the meaning of the needs of data from coming from another vote on it he was being fired from your job and it's essential that we had calling you remove the tradition and not in Saturday's Daily they the 1990 anything that contradicts the limitations in their it has become a problem he that in an almost got hold of it I saw the whole season for use that seems to be finding suitable legislation not only in the form of social and cut and all that is left is that they bring their use of it in the priorities the president each 2 of them all this environment is because the content of the letter of the law is the law of the land preventing the object was of and not the founders
35:43
religiously based on the findings
35:45
of the study said that mothers of hearing is a veteran was nearing the end of the year and how the
35:54
absence of your theory was being gations and used in automobiles reasons given by the end of the unique of thing and then you
36:03
have to that you have ruled that using the scientists here believe that the of and that's the thing that the images and events in the sense that they all want to find the means and I was ending wind up into the single and so you have to exclude the situation of the body's going around because you European you can make history you cannot you just aren't interested for instance in the that baseball disappearing with within East you make history instead of India's so you know I can kind of like many many interviews and then you can major in college stadium and all the songs and the men who on a career ourselves on July the absence of any changes in the seasonally and called and your fault in the slaying of a monopoly situation here and this is a process that includes a school of morning essential all his remaining years anything that has come from the house to stand in front of store the during last the proceeds from the center authorities have been monitoring the vote was different because the area on the eastern end of the month golden and widened his whole life among the facilities in general because they wanted to do but it also says that this is all you have to think of that the In addition to the when you include dozens of committees so I'm not getting involved incident yeah I'm just suggesting that what you can do this on the delegation or a sort of litigation due to people that the Forest Service who will not limitations because of high lessened the changes in the sense that intervals in the world will see that the only thing that forces us to the meeting with leaders the advantages that from now on because of the possible reasons for the soul of the necessary we will support you less frequently in the UK Hoosier politicallysensitive logic and found meeting of the season something you can do and that there are very few of them this is not the least being on the manufacturing and you will see that there is a wide selection movies and that really comes from the from the yacht the Eagles 18 although the beleaguered spaces or requires more than variety and he's the rise of the Internet more easily the associated tho the conciliation and the United here and in the innocence of the things that the optional Mortensen the mission of the lottery handling of the media is the sole owner of the nation's who fear it
39:25
could work insofar as as it did in
39:30
almost none of yet I'm getting a little this is a good kid who only came to me and I will remember him as he was seeing I'm moving through the United States because you know that have this is something you might as well not being used solely for the IRA to realize that and like the falsity of the region was used in the island the Caesar Faller said while that also that there was no unique on the final day of the men had already that I certainly gives you this is the only thing left is no woman to the disease there is 1 indication of how we discuss this on of the idea of concentrated on the hands deal the he went on to already "quotation mark the area following the law said the deferment of the coherence of the same the how the absence of the sword that have a union wanted to know what has to be clear and we need to get a lot of these units of his finding that these journalists had visited the factory during the fault of the hopeful that he and his about these show because he is an entity in the of use of solar the symbol of our review not the CCA that finding that the inclusion of all of them including the of the name line is not mandated when you visit clues last fall to make it into the net in the consideration the energies reason of the ensuing hearings using them he said the inquiry's inquiry so long as the geometry of the soul the dissenter from adopting prominently on using federal judges pulsating being very generous health and the existence of the Division of the Soviets had been named fragments on the Net might want to consider and thought we had to hold now that we have defined international theory and we had only as the latest sign that did so much should that only about who he is and what does he mean the company is not the subject of interviewing members of the union was enough to make you feel you have these sequences that we have pulled out of news called the axis of the beginning of the end of the year and most of all we need is a starting from the US you also be used the sequences from the and then when you do these things go around my neck and head of the School of reductions sister Cecilia lending actions now you have a very high and weighing on the yen after I think it is the 1st time since before introducing elections the that's because of cost each fragment isn't so detection system and this section the sense of the same even tho there is a big problem in the beginning of year by the fact that the idea now is the time to arrive at the base of his election season the images in 1920 that of just you and me and and but the season is closed you got on and on and on and on
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and on contentions he's
44:24
trying to get to the laziest guest houses and they that all had the same variety thanks also to the cities that have the beginnings of England and you would not want to use it in a little over who will be added and all that so you have to have connections the reason for each of these prime Minister of State didn't the analyst said that article but did you know that it will cost you may have yeah you for instance in 1 of the community and there was a lot of media and political and as for the regular season and someone in the eye and say yes the nothing is the same he said and it's it's easy to see we shared the history it's a single application of this year another instance of the and was that is of mutually school an ailing so we have a lot of believing his side of the experience so we can probably sequence so we think there is a sequel to to the rule of law he has didn't hear the head of the delegation education is just what the hell is going on in your life is even worse than hostilities and and what you get the idea and conclusion you can call it fearing inside the European OK so and now we don't about that these which could move from the American public he has exactly exactly yes I mean when you when you walk into thinking that logic saying only don't give out their regular sequencer which means the sequence of the former when you walk in a dramatic logic will only consider or you will yes yes yes yes yes yes I mean the old then was that you couldn't continue to come up with a product that is an important exception that there will be excluded in don't How on "quotation mark we will have to be the United States has not yet presented in a slightly different way sees we have presented any inference of sequence so they might not be recognizable easily or hiding in the linear sequence that means said you needed India India presented in this way so I didn't answers to their in the program so you can continue to work on this and deploy a strong today with so this is completed this year among is something that the affairs of this Saturday's Bazemore there's always something here now and it acquires the actual choice if we have a great advantage over their dead which is the fact that we we can consider models of our fear is not just a sets up but did not any categories of provided that these categories processing enough categorical structure for us to be in the interpret all connectors and quantifies the horizon you know you know a fragment of this to to prove it is still the completeness of the which is even stronger than that of the affordable strapping therefore through the notion of fine back category that I'm going to introduce an hour because the I'm on the notion saying that the category well now maybe it is indeed the premature to talk about this because I haven't even introduced a categorical semantics matzo balls that I am then at you could cut the number because of the developments from the question and that there is a positive sign that the refugees in a certain categories with an appropriate structure that the notion of saying that because I think it is very nice because it allows you to lead them into son all your feeling inside and sign that the category which would end an old like that so we will have a universal model was headlining a universal it is universally in many senses that 1st of all it is universal because as we shall see in winning a little thing all the other models of the fury by pulling back this as long as certain funk but back that follow from the point of the completeness and is important because in this modest satisfies the property that would his money that he is probably the fury has already a the and ordered that bikes said fertilizers completeness and this is constructed in a completely insane told a logical way like a among Pascal Djurovic proposition fearing his see when you have a proposition hearing you while you can take the algebra consisting of equivalence classes of all the proposition formulas of its language and so on and this conveys a model of this hearing and in which what is valued at these are precisely what is probably the future and so these analyzes completeness and use the difference between what you had in this framework with a single differences apparently wanted to take all all this Saturday's models of the future and more or less last letter to suppose the choice here we have always said that the 2 articles in the modesty of Atlantic Avenue in in this isn't that the category where we find that this model and then this is modern English syntax and semantics of the fury meets it the thanks to the recent meeting with Italy's unification you can get them back and people living in here and this is 1 of the main advantages over from Tonia semantics it is about administration of the fact that they're looking ahead to what the evidence suggesting the Senate's is married and has very limited because you don't get the faithfully image of the fury as the main man in the fight I mean you see the forefront of any beneficiaries need national choice and when I make a wish that some of which you can do it but there is a major issue having initiated a theory that the natural choice will not suffer as I have from my to political structure the fury from more you will have had her and then you are you are lost if you want to stand in Philadelphia is revered a classical technique similar funerary lost the weight yet even days is such as you might if you had these things and there is only of this is called a stronger completeness because it is not just because it this is something more because everything is concentrated in a single mothers OK so I don't know how to find money also saw what is what we saw the
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1st of the signature signal what we can do is to consider the notion of a sequence structure the accounting only 25 whether
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or not you need a justifiable because the only thing you need is there to be able to take the certifying products here and if you want to interpret computer logic that we should know when I can't find it in because we have rebound from the absolutely justified infiltrated this would be useful if not today but don't worry confusion similar conclusion hearing so I can inferior here you require a decent formula as currently there are about only to this hearing so he have 1 has a really believe that with the notion of probability inside the fury as it means that the government so far formulates confusion if it is needed the from atomic formula my only using the final 30 conjunctions and existential quantification switch provably unique in the fury this is not logical McCarthyism categories In this sense that I should take a a few of the infuriated that meets the confusion saying that the category will be confusion and conversely any character generic category can be seen as a disinfectant take a category of such a few this is a new findings from today and yesterday this is the community that you lose yourself in your office otherwise you constrained finally completed the CIA but I Kincaid will use of confusion and misunderstanding always so suppose that we had a cut including had grown scene and we want to the define interpretation for this back the conventions that we have introduced so we have to interpret the sorts of a functional singles and the relations singles now sources we interpreted that as a objects all other categories function seamlessly with interpreted that as a proposed that supposedly had a function similar to that this is going to be commencement of this is going to be interpreted as a I'm not only going to from this kind of their products this is no appeared on these generalizations with evidence 90 classical dance dancing 169 because you don't understand semantics assaults on interpreters as a sexy and the replaces sets up a we've objects because we are going to do so in the custody case CD is equal to set by the category of sex and functions between them religious symbols such as will interpreted as a assembled just equivalence classes a oneoff incidents you know what I think we are seeing in the 2 no adjustment freezing so so this will be among more feasible under the cost of this there will be interpreted as a 0 I functional symbols of sodium and will have enabled the um product that and I believe that can a depending of also on the category so and equal 0 also stunts so this will be a thing OK so we know how to interpret all of these UCI everything that goes on there is no free from um from whom it is a very small visionaries solution from a weather Benson sets of tools to continuing and economic adequately funded programs but yet another is that we have just interpreted very very busy competing survivors we just interpreted on function was and relations health questions and you want to go along with an interpreter complicated formula as we needed to require some categorical structure
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present in order to be able to do this because all the places that you want to interpret that as a conjunction of
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formulas while he had to find that the products in Japan and will not accept this anymore because I would like to take at the intersection of the interpretation of this we the position of the and such intersections formerly next year category and a source of finance for events we know general subsidies and we have to acquire at least that uh and the affected communities can and of course there are the more complicated formulas are pleased you can and do you might want to interpret things like the use of it .period and although quantifiers and it is also possible to particularly in the doubles to interpret these things but you you need to use that particular structure in your category to be able to do this you cannot avoid it in every category and there is a solution see each of these deferments as a corresponding class of categories in which it can be interpreted but their lives lost a wheel on the OK assault was that we have defined decent in the preparation of this basic ingredients and then we will describe how Altintop before talking about foreigners will have to think about the terms although interpreted those Wendy's is completely stifled 1 because you see functions have interpreted in this way those are just the things that trying to slightly the variables functions soon applied that be amended so it is isn't completely clear how you interpret ending will be as ROC will really be interpreted as idols of the terms and he said you don't need anything more than a fine its broadest Lucas and the whole an informed instead if you want to interpret that the leak formula was a they're not this only if it is a constant it is a valuable including its projection the projection or just didn't think of it depends on the context is larger than previous projection otherwise it just didn't think salt and now In terms something really now of course the authority says in these formulas of this former imagine for instance that you'll want to interpret the formula was only signed at Formula One year union likely won't winter formula kind when you necessarily need them more than capabilities as this formula then you will have a better the interpretation of the school terms these are this would be true of Arizona a certain things and the interpretation of the Senate will be the equalizer on display so you again if you need them that your category also sees this equalizer and yet will need the backside of the that world will make a point where you'll considering the Provisional IRA which is a civil jet and an adhesive the stones will give you an hour and then reform for the pullback and this will be the interpretation of your formula now as and after you have clarified all to interpret the meaning of the as all the other acts that are formed the foundation by using introduced a logical next season In Florida the quantified and the local police later and this is stopped so I did not feel it is you don't know anything more than financially in itself so that you can also use laser Sony's means that the pages well actually just already final that of his because of this fall the accidents are all this former Soviet agents in the categories refined products he had abundant you want to interpret the ball in the final that a conjunctions and existential indications of source for sure find that the conjunction also you need the full Republican pullbacks such enormous and that the real objective so it means that you need a contingent category but it doesn't stop this because in condition category you might not have the possibility of interpreting music is essential qualifications in in a meaningful way and now you have right there when you come and when you will wonder about halting the reportedly so you just try to to find out that the categorically devotees of interpretation that situated in the provision of the old existential quantification and US authorities as with Museveni's there when you have a lot or just the function and when you consider its image this underrated as a total about admitting that he is the cost of energy so this operation is very important that this operation although the effect of rising the manager and hazardous objection afforded by you a more feasible because it is the key to the whole Alcantara but these exists there shall be quantified because they can look at all this is defined you see that there is a sort of existing quantification him so that he is an element of such a thing it was OK so much of it might be latest the relevance of such an attitude that exist something which is assigned by fluently and that's all basically yeah you need them the notion of ignorant of the ways of a cabinet I categories which are currently and English tomorrow that any harm as a factorization is an image of a viewing the managing general is characterized as least some objective for which you are factors and require moral that these images to be stable and the pullback a disobedient Quebec is related to to the fact that you knew fragment of logic you wanted a day the substitution rule to be sounded and therefore you need also the search for america's Community College now form a coherent theory is unique on because you see you this being here and there so unique that what is called the poignant and cutting I'll
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call you indicate agrees on the categories in which some or all of less and you have to find that the although some objects and this spine unions are against the war and Quebec wife because you want to that but it is Junction although I do what it's supposed to hang on hoping that day and Agencia like this defined his job just because the clearer and logic and I know that the interpretation longer feature formulas we need a subject all of my my final product line that all of these would be subject to using and also in doubt these junction of these well you to take a the unit of the soldiers nothing and this is something that you can always meeting said that by the end 1993 "quotation mark categories so you need not to impose on the existence of the state of the union also understand that unions of some of GE's annual moreover have required them to be stable and Quebec and businessmen emotional coherent and now the shape that notion of that there is an awful lot of geometric and what is a particularly important for us is that they don't think that was his argument that the catalyst and so on we should be able to interpret any kind of Jamaica fury unit and a study of 10 years before OK thanks the whole case sold no but I think you have to to great think a wedding if you want that have not yet you don't get any you just under false and whose you know what can you do you think we can turn on you know the definition of matter and in what you can get going here is the same the following me you can write things like it's something you can take a step forward this you can find that this is the only kind of allegation that you invite you can look like there was something completed in more complete all on no the the yet the sequencing the sidewalks said the only problem is being treated the staple classical nature your system because the usual stands for the start of the logic of the classical so they had little excluded me and this is the precisely and this is the only thing agency that there and it doesn't want the snowy investment market in another critical positively United 93 category because the sun in order for this to be told whether this means that the identification diesel is a complementary to the some object and in general and you don't have the uniting the category which is what you would need to interpret the oldest litigations and implications in any meaningful way in uniting category on it categories William and Neukom under Houllier categories about 30 unaltered frequent anybody today are the majority of couples is Arnold Williams saw and if you want to interpret this anatomy that it opposes so you'll you'll you don't want to know excluded those who is to lose according to the report it I used history to think that this is not of things for suggested this season has sort of what I suggested to Bolton earlier when and personalized the media our what I what again I believe I want to introduce and just the 1 particular obligation for instance I might want to introduce an indication of the quality so if I to introduce this I can do it that way remained inside there are and logical and the way I do always assumed that I call on diesel let's according to any foreign additional the quality of the Sweeney it by adding relational similar and what I need able to the of a book that axial which ensure that the center would be delegation but litigation in the sense of a compliment all the younger 1 of which is that in the provision of the yellow cards this season adjust them to death stock and the other 1 is that when you take the intersection of toward usually against the forces of using these completely coherent Axelsson because he and I have a seat at this juncture he and I have the the fullest and this is going on so we have inquired entomology and this is an operation that there willing going to talk about the later thousands will do that to you as the need to understand that anticipate because tomorrow the simulated with threeyearold because it is a natural thing to do when I when you have a fury to and through on injected 5 8 in the sense that there you want the category of modestly the same except for the fact that you want the job rose to become a werewolf seasons and this is the way to do this you will add to the signature of your fury this thing that automatically and you will and you will get there and in times of Saturday's Mandela said the models will be the same as balcony areas will become the injected and a lower fees and so forth for your theory and of course you agree to work with the recovery of cars without a hearing through hours of course you can always so what they said to me What concerning the in what is said earlier concerning the modernization means there is a general procedure and that doesn't look at the fury as specific adding faced an invoice for interfering in the driveway and solve fraudulent mortgage in most situations such as you do this way I mean you just take a specific formula and the conceded his knowledge you try to make it through metaphorically reinventions of the whole mess and of course you have seen an illustration of these indicators of litigation which can seat unhindered implication is more united than that because litigation can be seen as implications of forcing and you cannot make a decision 3 years went 4 for the implications so embodied in situations where you just do the same relative to the genetic purity don't to need the tool to win floppy life disability hate the machinery of modernization OK sold all match and all Californians know what to do what I can say OK so what
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should I say now whether it will also called his 1st on the way talking about the soundness and completeness labeling already alluded to completeness earlier laughter I OK here's said analog Kaneda told little something missing in virtually all OK and hour OK we had said the what the what he user and a structural is seeking a structure in indicating that he will find it proponents of course we are now ready to say what it is a model of the fearing when I intend to define their own refuse sense of structure this is completely stifle warmer than in a structural fees and is there something that has the nation and behavior with respectable for functions Engleson relations symbols so you can write to System for 1 who commanded the condition yourself and he doesn't present any surprises and so I guess the police and the standard I'm sure there's something in the diffusion of mother of hearing so you fury achieve all that is certain a signature and the I supposed that the In a fragment of the refusing a fragment that is interpreter in the last year that the utilization of categorical structure for 1 to interpret these things so and suppose that the series in fragmented is intolerable in a certain category C and then once a and that this is the a structure and is a model of the he will always the axioms all key trends on the actions site of the original sequence of these former soldiers would things of this kind I'm satisfied mean and what does it mean that the slide in the Udinese cemented their interpretation of the unprecedented ease contained has of objected until the preparation of the conclusion this is a clear because any man and entered the intended meaning of a sequence was that for all the excess and fighting and various this so this is the same thing as saying that there would be another element state accusing his interpretation is also increasing show so now we know what they're in modeled fury and thereby conceding today the category was objects are he has said this with the layout of a widening actually not you are there to make the cinematic about that and if the substance to check the Axelsson because we had the soundness and I am about to think about the soundness so solid tender the only business year that lucrative we were in government is just a part of the logic in which you make the choice of restricting your attention to certain particular classes of forms the Socialists tightening security tools will consider just some the not is just the way you define infuriate you say if you use a set sequence this sequence of called the actions of a few major image might have initially along that anyone can nonaligned in Kenya within a few days and presentation sold within the whalers to degrees and then just east of the sequence electrical accident on the of the unknown not at all not at all also on otherwise it could even be appropriate soul analyze it is a set of sequence in and there and then and there is this very important for now just the location location is the 5th of Jean Monnet the This is an occasion for the category was the objects are the mothers of the inside of a Category C and the gyros are being on a structure or more seasons so the national movies and he had a little more than some of her were reasonably search is exactly the same Seoul you will hear the finer than than ordered the category of structures in the season .period got the OK this is just the former patients now soundness and completeness season precisely what the answer is that there question has something me is that there any see plenty which she will and in seeing his body can embody dissatisfied insisting on that in all modern some of them came in categories the product the categories all the appropriate kind of a relatively fragments of course although the appropriate time and knowledge of the managing the sees a feeling that it is very easy to do because the only thing that it's quite a sight toward the only thing you have to check user will eventually need for resolutely the brain uses a satisfied and also because it because the whole body the Committee said that the interpretation of the understand and he's conveying the influence the seasonal notional value the Indiana OK and completeness of his adjustment in the economy yourself I mean the expressed sequence is valid in all the knowledge of the fury in in the categories of profound and then it is probably and I have already given up on the director usually don't adjust and completeness but we had strong the completeness of thanks that exist themselves and that the cataclysmic this it would be for after that tax time mean