Class forcing and topos theory
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Formal Metadata
Title 
Class forcing and topos theory

Title of Series  
Part Number 
12

Number of Parts 
28

Author 

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. 
Identifiers 

Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

Content Metadata
Subject Area  
Abstract 
It is wellknown that forcing over a model of material set theory co rresponds to taking sheaves over a small site (a poset, a complete Boolean algebra, and so on). One phenomenon that occurs is that given a small site, all new subsets created are smaller than a fixed bound depending on the size of the site. There is a more general notion of forcing invented by Easton to create new subsets of arbitrarily large sets, namely class forcing, where one starts with a partially ordered class. The existing theory of class forcing is entirely classical, with no corresponding intuitionist theory as in ordinary forcing. Our understanding of its relation to topos theory is in its infancy, but it is clear that class forcing is about taking small sheaves on a large site. That these do not automatically form a topos means that the theory has interesting twists and turns. This talk will outline the theory of class forcing from a category/topos point of view, give examples and constructions, and fin ally a list of open questions – not least being whether an intuitionistic version of Easton’s theorem on the continuum function holds.

00:00
Area
Axiom of choice
Series (mathematics)
Statistical hypothesis testing
Real number
Element (mathematics)
Materialization (paranormal)
1 (number)
Algebraic structure
Set (mathematics)
Functional (mathematics)
Rule of inference
Theory
Morphismus
Computer animation
Manysorted logic
Lecture/Conference
Natural number
Universe (mathematics)
Object (grammar)
03:01
Category of being
Theory of relativity
Lecture/Conference
Multiplication sign
Element (mathematics)
Algebraic structure
Morley's categoricity theorem
Set (mathematics)
Axiom
Theory
Set theory
03:58
Axiom of choice
Ocean current
Point (geometry)
Pulse (signal processing)
Observational study
Real number
Equaliser (mathematics)
Continuum hypothesis
Materialization (paranormal)
1 (number)
Water vapor
Parameter (computer programming)
Regular graph
Inequality (mathematics)
Theory
Number
Subset
Power (physics)
Lecture/Conference
Meeting/Interview
Wellformed formula
Cuboid
Subtraction
Set theory
Social class
Condition number
Injektivität
Pairwise comparison
Series (mathematics)
Theory of relativity
Reflection (mathematics)
Algebraic structure
Set (mathematics)
Mathematical model
Line (geometry)
Axiom
Functional (mathematics)
Category of being
Universe (mathematics)
Order (biology)
Mathematical singularity
10:08
Axiom of choice
Greatest element
Group action
Direction (geometry)
Wohlgeordnete Menge
Mereology
Proper map
Food energy
Grothendieck topology
Permutation
Sign (mathematics)
Mathematics
Manysorted logic
Atomic number
Analogy
Partial derivative
Social class
Theory of relativity
Functional (mathematics)
Automorphism
Metric tensor
Category of being
Arithmetic mean
Order (biology)
Point (geometry)
Existence
Spiral
Regular graph
Discrete element method
Theory
Power (physics)
Number
Hypothesis
Frequency
Lecture/Conference
Analytic continuation
Boolean algebra
Topologische Gruppe
Consistency
Physical law
Algebraic structure
Variance
Mathematical model
Set (mathematics)
Line (geometry)
Limit (category theory)
Local Group
Equivalence relation
Summation
Logic
Network topology
Universe (mathematics)
Object (grammar)
19:15
Geometry
Computer programming
Group action
State of matter
Multiplication sign
Student's ttest
Grothendieck topology
Vibration
Morphismus
Manysorted logic
Diagram
Method of lines
Prisoner's dilemma
Consistency
Expression
Basis (linear algebra)
Limit (category theory)
Cartesian coordinate system
Bilinear form
Maxima and minima
Category of being
Universe (mathematics)
Topostheorie
Charge carrier
Right angle
Object (grammar)
24:29
Axiom of choice
Filter <Stochastik>
Group action
Process (computing)
Presentation of a group
State of matter
Direction (geometry)
Multiplication sign
Limit (category theory)
Regular graph
Functional (mathematics)
Category of being
Sign (mathematics)
Manysorted logic
Lecture/Conference
Object (grammar)
Social class
28:12
Category of being
Lecture/Conference
Natural number
Object (grammar)
Mathematical model
29:00
Logical constant
Curvature
Multiplication sign
Direction (geometry)
Insertion loss
Cohen's kappa
Inverse element
Vibration
Grothendieck topology
Power (physics)
Mathematics
Manysorted logic
Lecture/Conference
Wellformed formula
Term (mathematics)
Condition number
Social class
Chisquared distribution
Price index
Limit (category theory)
Functional (mathematics)
Sequence
Category of being
Funktor
Oval
Network topology
Order (biology)
Duality (mathematics)
Object (grammar)
Arithmetic progression
33:26
Computer programming
Product (category theory)
Sheaf (mathematics)
Infinity
Set (mathematics)
Limit (category theory)
Power (physics)
Morphismus
Category of being
Length of stay
Goodness of fit
Computer animation
Lecture/Conference
Right angle
Diagram
Object (grammar)
Uniform boundedness principle
Local ring
00:01
he so tall and on to the end of the year
00:15
very honored to be out of Tokyo the birthplace of accomplices and in light of all the excellent talks To my offering may be slightly more humble but in the lively and the conjunction of some ideas existing and a a couple of new ones will be an interesting some people might feel like that looking into the further the case so you have to terminology and disclaimer and there was references to universes and so forth earlier in the week and if you like take 2 universes inside V things in you'll be small things and they will be moderate anything not a newbie large anything 90 we very large however refer to this again because it's in the background and sometimes you can dispense with this technique if it makes you feel more comfortable taking out also unnecessary otherwise watercourses I'm dealing with elementary troubles with a lot so it wouldn't that be an auto bosses after completes with natural number objects but I want donors say take currently letter troubles management object again test disclaimers terminology cited his terminology which is sort of not yeah maybe 4 or 5 years old not widely circulated so it's itself to keep in mind especially when talking about the relation between of forcing techniques which are coming from set theory in particular that sees theories and set theory not using that material sets His where elements of sets have independent material exist 3 elements of a set In Z sees a sea is optional or have abounded Zoe set theory choice optional again from those who knows the history of of besides this area this this 1 turns all allotment Serie these materials that spammers assets to which objects to the theory structural sets areas In where everything is as my good technical definition that roughly where Everything is invariant on a small morphism you don't worry we say take the rules and then have to think joining the palace said the natural numbers the functions from the nationalist natural numbers you know Sandoval a out such the Bubble blood you it's just a real numbers it is what it is that lost
03:03
Norfolk and I have to worry about the government's accidently ,comma coinciding with elements of the structural sets cost is elementary theory of the category sets and the primordial example maybe some variants
03:18
on this and 1 that I like to advertise cold set elements and relations "quotation mark CIA and due Marchal whose name refers to a couple of times already but this is not published but it's only unless you will have yes thank you but it's is a structural set theory there is not a categorical set theory of pneumonia he people talking categorical sent whatever's NFC it's a it's a structural set theory with axioms a purely on sets how they behave elements how they behave relations how they
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behave 3 sorted out for axioms shorter than the idea that he TCS list and In an afterlife 2 and a half vaccines you got itself most of its costs it's really quick say materials instruction sets crisis so that senators in history of this he he he he he has slept in her desk it's quick it's as saying strength has rebounded generally with choices box and despite the threat of injury mentioned in his talk I stuck semantics and structural and comparison material structural set theories as gives a formulation immune this extended in Internet
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extended internal language which is which recovers schools in city that the property called tautology and you're winning you can shop that formula is here are actually classified by some objections various so I can say structural set theory and recover full ZFC strength has issued and other people to McLarty and others who have come up with replacement axiom analogs the agency s protects so that history but In story was set to consider the type of sets and consider subcategory injections and then preorder the partial or reflection of this is the category of cardinals and the usual relations strict water relation he said what's his name they don't couples easy half of your was life in 1 line at the symposium tonight is that the powers that restricts 2 sets injections is positive said of a substance is a subset of the current set of chips that handle Universal property here this restricts recall was on the side so everything here classical and get some more general to post later and and this is a continuum function because continuum of officers was what value does take when you plug in an actual numbers what is that service but some and a strict inequality here so it was real nice general construct a model of presented city where there is no such that any such that's in between here please potentially equality it's the nationalists or spouses and then current came along and give us full and he said Well I can construct to set a wellconstructed Museum of a new model of set theory and I build its explicitly with the set India these 2 different foreigners right after this in Monica Maclean's book I'm from Canada to the series point of view but I have some nice you start comments so forth and I say you can actually see the Sharia the turtles theory argument even Cohen's paper so it because this is more general question this is a partial however set aside as a Class I special order has been in a function look this in a function do given that we you can find different categories of sets where are you it makes clear reminders values taken more so there's to cases so sorry to say behavior to cases a regular case say it is a 10 year conditions singular case studies a conditions on Cardinals regulars nice ones like the numbers a regular assuming choice things like the real numbers have a regular cardinality singular things is like an and analysts on if that means anything to you and senior cardinals on 2 wheels this hard and this is solved the eastern we know exactly what the behavior of the continued function is operated In sense that we know we we can characterize precisely what behavior can have when it comes to Exeter and so on sir yes and people like share last right literally hundreds of paid Marichal I'm so putrid brought hundreds of papers on this but of here hundreds of papers generally now trying to attack this problem for those the singular columns of pulses and so so why wise piece of important you know this is a set threenation what we care about us pose serious so is a definition of an elementary tackles that I believe we've had 1 word I think 1 given in another
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direction he wasn't happy with it and is 1 I'm sure and some people may like and on limits however objects hands articles as incredibly incredibly rich internal logic and do little things marriage :colon limits limits stand to certain things limits his is pretty weak everything else come from the power objects and in fact you can relax this even further back the service and gets a bit so this is this is this is saying I've fraud "quotation mark Jason X I have analyst object P X relations makes the PX such that any of object why relation critics to this a unique map In such that this group this just all the magic happens from this it's just incredible soap look at the behavior of how object from counsel said from said theory case In somehow he has incredible freedom as receivers in theory but in fact gives us lots of what's structures as good to sort of understand what's going on here when this this this is solved purely In classical modular choice you throw a choice said theorists run screaming from the room and say I don't know what Kurt finality is doesn't make sense what you wanted to it the committee and so this part so forcing him to hold 3rd forcing its she's is .period this is nothing mysterious about it now details perhaps there is and his accounting alike to draw uh traditionally start was in at the start of a small partial order as a sign the delegation apology coach she's it continuing what they had 2 options here intake internal logic the Indonesia's reasoning that perfectly good and then you performed construction you variously to call his foreman and Hideyoshi various facets of the and then you get set year's call Lillian valued model of Zenith C and the bullying is the the thing is is the topology on this partial take the woman out for Europeans and then is Boolean algebra is where truth values like something as another the way to do it is further taking an antsy in his book that was partial the top of of the I said and it's Melosa generality since the company that's the based there is the hold was always is that some this is the site is a small side domination quality and are downward closed this sets sorry there's an analog of this apology for auditory categories the danced only across most the delegation topology when you pass to the preseason category in the sort of of its unique 20 cents and then so get partial partial finger is a category where but this top element and you think of as things going down and the further down the go it tells you more and more partial information about the things you get in here and I say that wanting to do with this law you take a new category of sets the existence of objects of the of the general objects can you take as a filter uses this is a bit like a point but not quite you quite different things on this list 5 before what's called the fruit
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aggression construction and details about employment this year is now now well appointed the internal logic in the external logic now agree in this category and so you can reason exactly as if you're in the intelligent but you just go well this is how we object to this it was works and then he performed this construction again very various people and then you get In a forced model the citizens sort back between these 2 reports as convenient and now thing is this the start of lines is a rather delicate and complicated construction evolving transfinite wellordered Richard trees set the stock market if you say that set theory is a summary of world founded .period injuries but that is not really but that's what we hope is studying so you have be so this is a step that we like and props just attack on the internal logic that's where we're as well action happens energy paper you can see he's describing exactly these constructions and sums of sensors moderate McClain mentioned on there and also is a variance you can consider topnotch group acting by automorphism partial order and then considered equivalent she's because metrics of models and you just take a topological group .period and consider continuous actions of this but his work on permutation morals don't give you model gives with atoms so this predated :colon by about 40 years and it's basically forcing the centers like a spiral at 3 Qassams sorry but it's it's fits into the country your picture perfectly well the case is important .period taking a person and when the effect said many continuing near the values of continuing function if we start with the model of set theory the collecting of function do forcing the continuing function changes but this procedure would price 2nd effect said many and this particular down about facing the commonality of the no the problem is Eastern solution involved adjusting proper class of regular the continuing function proper class regular Cardinals so you can't sort of going yeah it literally you take all the regular cargoes of which there are because in most of the universe that just keeps going up he was able to shift them the subject of the hypotheses of his theory this is something we will can all sentences and military means as I 6 years assets and that is the cardinality of a number of the values of continue function which you have altered so you can't affect arbitrarily high values of the continuing function by taking she's on hold partial take and what he did he did could hear generally didn't he adjusted a proper class of values of continuing function 1 at a
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time this might make you pause I had discussions with the rest of the way no 1 will miss the it's like a spate of and arrest so as a nation existing emissions from this jury which found which very beautiful there is a nation of 4 officers and this is true for all recently converted here so on this is because they're thinking greatly purposes and so for us this isn't unbounded sex traffickers solicitor posted geometric morphism set but he's not underestimate In particular so the example gave is to take a large program many consider so can universes as needed to take a limit of his program and then consider continues actions of this and this is up to some small carriers to you mean this is a perfectly good locally small category which is Interpol's would Dumitru was intercepted and there it is that's a beautiful and moreover if right down maybe a sort of definition for this it is indeed enlarged the application skin later in studio logical coming this year and then actually gonna spend uses this unbounded qualifies internal logic and back this is too restrictive is not worrisome used to to consider a it's state utility and well founded category initial objection you we can consider diagram of sites like to advertise their morphism which of vibrations of sites student 1 Dick Cheney's costs mounted more prisons and this is a good example that gives rise example of the final trip also will send his cheerful and they consider the case where our small when you can take the territory possibly likes of workers from what was usually limit of this and they get a troublesome people apparently applied and regulatory the FAA as
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large you can you just can't do it if you forget to the inventor of the world the book is the story bounded this aggression it troubles this shit so this say this time of reasons of sites said this is probably what you call a morphism sites in the growth style but in fact this is just the right enjoined to vibration and this is especially heating of sites as In like a basis for a top post in this we had today of vibration sites is always exactly analogous to Jordan form you coming under the door much true pixie take a basis for 1 year courses and then you pick internal I've definition further the trouble sitting over there she's on this and you can pull is down and you get various expression we have is a joint founders would cover preserving properties you can characterize extending this site characterization to characterizations of geometrical designs why these particular vibrations right covers about 2 examples of these I just think it should be advertised widely given this sort viewpoint of using the basis of I icon back to him was the the continuous but I like this is this is now very large category site I'm
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thinking of surpluses have their sights and she's overcome sittin on here Lecce's sort of signs creep up just 1 more year go through what was a very good news is this is a so this time is that they said that if you're talking about them continues the quality of just as driven courses I could continue contemplation and taking in charges here take the 2 hours which make the Convention's work we have seen different conventions you pick on then so what is the world is finally limits clear limits what is what you might call presentable category but minus the actual presented the and because because these are artifacts through presentable I'm sort of direct mission following go to Linux because what happens next is I take this could be done differently and on at start the opposite so far until next CC R is a filtered wellfounded category with think of it as the Orioles the With that sorry initial object initial object known as 0 object vision of this you have a lot of these whose sizes locations as well yeah and then you can take top bosses and then you forget it's taking those images In this image but this is like the algebraic viewpoint on top officers of forgetting the director Mitchell because the directive which functions will no longer exist in the next step take a limit on 12 he said from the now this is not likely small category in general but in you have some investors already take the 1st on categories and this is what still does he thinks appropriate are in fact is just a class of regular cardinals and To find appropriate science which in effect 1 regulate the 1 value the continuing function at a time and then takes account of the 15th hole the regular condos for sitting in mind died and 3 others were could not just snap pursued sorry that article is the Eastern and it's not topless opery Lori said the state of here the company has very specific choices yet so quickly he is it's always such a process could complete hiding creator costs finer Terry financer
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Wtypes types Fiji natural number object to this is the end the
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internal here firstquarter and intuition is ecologically hiding hundreds as your subject notices and 30 types means it has is for the best in charges for polish your front is associated to match which a models in a slice category so very literally fall upon the new National Missouri example of this and if if so notation as above said
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fairy indexing objects every was every object in a safe house said of is eventually constant anything formula based in eastern straightforward technical sense means is you keep going by you invest in a trunk is eventually the power Objects stabilizes and then as a small business all the way up it doesn't change the the various phenomena what followed if this is true then he is the type of this is exactly the condition that Eastern finds and characterizes the special case of vibration of sites in that he's using have that makes this work and by using locomotor troubles so he needs is not for every object in indexing category just in the base category of every satellitebased category if it's power said eventually stabilizes after passing by the inverse image functions after this class long dogged by then it it's a loss so yeah about adding are objects so who you miss our this this will be the weather the qualities of the that the reason for the lack of progress it's exactly and an affected only all of them you just take some kind foreknowledge that sort of collection inside the inside the order of cardinals missing for large enough you need to call of news and unbounded sequence of condoms and so he we have internal here have internal problems pertaining to pass it you get over the top churches but I think over time so I was going to give em explained Eastern's
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example Harry characterizes this eventually constant in terms of properties of Presidents but in Sydney here interested in Cuba the little a lot of reasons cited by U.S. yes let me just say that the use of vibration of categories the United cutting it's on and it is method joint too fully faithful functor going in the other direction and the presidents Thomas the knowledge of the please just Is it sites so I didn't write it apologies here but this is the underlying categories and that their topologies union and all this stuff I'm I know but it without loss of generality if you have complete freedom you can actually choose see how handy to have followed limits you like In fact there are important cases when that I haven't working more generalities because most of the problems with those who hold dual front like flatness or something related the what yeah the image of a currency generates even ever since so it's in moderates as in the ocean and not expect to come from I should say what used to uh side such as a surge in 1986 so Xever
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newest nation and it's only newspaper and in the elephant and doesn't operate on so I think should be pushed more Kansas counties if you can't remember how policy nonetheless the blockages I I really think we have in the world know know what you want to know what to say that the United Methodist we don't want you there so small so small hiding if you like you have and it's not even while palette once becomes at a cost that has power objects and hands because its lack of local smallest that you don't get the joint funk fear to get all that directing the chances that you might think from the the commitment of the diagram this is like a limited exception it's really only in algebraic morphism of triple since he can't find the right into so we don't want to be bounded morphism almost all of you view your 100 that was only a few minutes 6 minutes like purple /slash set in Krasnodar purposes over set then automatically morphism between rebounded have to forget set anyway because I no longer have the and the right to during the global sections 106 so I have to forget them sliced a set policy of goods produced the U.S. are probably had something like top here instead of like talked with set as the president of horses have Lex categories on the and so I have constant she's in his big Kelly that I have "quotation mark sections I would say it's I'm used to hold the city regardless of program value you yes sorry dear on it's an infinite inventory critical sorry the :colon infinity pretopos possible but we have invented proposes that get credit local pocketbooks smoker products the including the organizers Yugoslavia because of an injury could insulation a thank few
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