Class forcing and topos theory

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Class forcing and topos theory
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It is well-known 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.
Area Axiom of choice Statistical hypothesis testing Functional (mathematics) Real number 1 (number) Materialization (paranormal) Algebraic structure Special unitary group Rule of inference Theory Element (mathematics) Morphismus Computer animation Many-sorted logic Lecture/Conference Natural number Universe (mathematics) Object (grammar) Series (mathematics) Set theory
Category of being Theory of relativity Lecture/Conference Multiplication sign Algebraic structure Morley's categoricity theorem Axiom Theory Set theory Element (mathematics)
Axiom of choice Ocean current Point (geometry) Pulse (signal processing) Functional (mathematics) Observational study Real number Equaliser (mathematics) Continuum hypothesis Materialization (paranormal) 1 (number) Water vapor Parameter (computer programming) Inequality (mathematics) Regular graph Theory Subset Power (physics) Lecture/Conference Meeting/Interview Different (Kate Ryan album) Well-formed formula Cuboid Series (mathematics) Set theory Condition number Social class Injektivität Pairwise comparison Theory of relativity Reflection (mathematics) Model theory Algebraic structure Line (geometry) Axiom Numerical analysis Category of being Order (biology) Universe (mathematics) Mathematical singularity
Axiom of choice Greatest element Group action Direction (geometry) Mereology Proper map Food energy Grothendieck topology Permutation Sign (mathematics) Mathematics Many-sorted logic Atomic number Analogy Partial derivative Social class Theory of relativity Automorphism Category of being Arithmetic mean Order (biology) Summierbarkeit Metric system Point (geometry) Existence Functional (mathematics) Spiral Regular graph Discrete element method Theory Power (physics) Hypothesis Frequency Lecture/Conference Analytic continuation Set theory Boolean algebra Topologische Gruppe Consistency Physical law Model theory Algebraic structure Variance Line (geometry) Limit (category theory) Equivalence relation Numerical analysis Logic Network topology Universe (mathematics) Object (grammar)
Geometry Computer programming Group action State of matter Multiplication sign Maxima and minima Student's t-test Grothendieck topology Vibration Morphismus Many-sorted logic Modulform Diagram Prisoner's dilemma Consistency Expression Basis <Mathematik> Limit (category theory) Cartesian coordinate system Rothe-Verfahren Category of being Universe (mathematics) Topostheorie Charge carrier Right angle Object (grammar)
Filter <Stochastik> Axiom of choice Functional (mathematics) Group action Presentation of a group Process (computing) State of matter Direction (geometry) Multiplication sign Limit (category theory) Regular graph Category of being Sign (mathematics) Many-sorted logic Lecture/Conference Object (grammar) Algebra Social class
Category of being Lecture/Conference Natural number Model theory Object (grammar)
Logical constant Functional (mathematics) Multiplication sign Direction (geometry) Insertion loss Cohen's kappa Inverse element Vibration Grothendieck topology Power (physics) Mathematics Many-sorted logic Lecture/Conference Well-formed formula Term (mathematics) Condition number Social class Chi-squared distribution Price index Limit (category theory) Sequence Category of being Curvature Funktor Oval Network topology Order (biology) Duality (mathematics) Object (grammar) Arithmetic progression
Computer programming Sheaf (mathematics) Infinity Limit (category theory) Power (physics) Product (business) Morphismus Category of being Length of stay Goodness of fit Computer animation Lecture/Conference Right angle Diagram Object (grammar) Uniform boundedness principle Set theory Local ring
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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
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