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Alternative Set Theories Yurii Khomskii Introduction Alternative Set Theories NGB MK KP Yurii Khomskii NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories Naive Set Theory Alternative Set Theories Yurii Khomskii Introduction NGB MK Naive Set Theory KP For every ', the set fx j '(x)g exists. NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories Russell's Paradox Alternative Set Theories Yurii Khomskii Russell's Paradox Introduction Let K := fx j x 2= xg NGB MK Then K 2 K $ K 2= K KP NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories ZFC Alternative Set Theories Yurii Khomskii Introduction NGB The commonly accepted ax- MK iomatization of set theory is KP ZFC. All results in mainstream NF mathematics can be formalized AFA in it. IZF / CZF Other Yurii Khomskii Alternative Set Theories Philosophy of ZFC Alternative Set Theories Yurii Khomskii Introduction Philosophy of ZFC NGB Everything is a set. MK Sets are constructed out of other sets, bottom up. KP NF Comprehension can only select a subset out of an AFA existing set (avoid paradoxes). IZF / CZF Certain definable collections fx j '(x)g are \too large" to Other be sets. Yurii Khomskii Alternative Set Theories Logic of ZFC Alternative Set Theories Yurii Khomskii Introduction Logic of ZFC NGB MK Classical predicate logic. KP One-sorted. NF One binary non-logical relation symbol 2. AFA In this language, ZFC is an infinite (but recursive) IZF / CZF Other collection of axioms. Yurii Khomskii Alternative Set Theories Other set theories Alternative Set Theories Yurii Khomskii All these factors are liable to change! Several alternative set Introduction theories have been proposed, for a variety of reasons: NGB Philosophical (more intuitive conception) MK The need to have proper classes as formal objects (e.g., KP \class forcing") NF AFA Capturing a fragment of mathematics (e.g., predicative IZF / CZF fragment, intuitionistic fragment etc.) Other Application to other fields (e.g., computer science) Simply out of curiosity... Yurii Khomskii Alternative Set Theories Alternative systems Alternative Set Theories Yurii Khomskii The alternative systems we intend to study are the following: Introduction 1 NGB (von Neumann-G¨odel-Bernays) NGB 2 MK MK (Morse-Kelley) KP 3 NF (New Foundations) NF 4 KP (Kripke-Platek) AFA 5 IZF / CZF IZF/CZF (Intuitionistic and constructive set theory) − Other 6 ZF + AFA (Antifoundation) 7 Modal or other set theory? Yurii Khomskii Alternative Set Theories Alternative Set Theories Yurii Khomskii Introduction NGB MK KP von Neumann-G¨odel-Bernays (NBG) NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories NGB Alternative Set Theories Yurii Khomskii NGB (von Neumann-G¨odel-Bernays) Introduction NGB We have sets and classes; some classes are sets, others MK are not. KP NF It can be formalized either in a two-sorted language or AFA using a predicate M(X ) stating \X is a set". IZF / CZF You still need set existence axioms, along with class Other existence axioms. NGB can be finitely axiomatized. Yurii Khomskii Alternative Set Theories Axiomatization of NGB Alternative Set Theories Axiomatization of NGB Yurii Khomskii Set axioms: Pairing Introduction Infinity NGB Union MK Power set KP Replacement NF Class axioms: AFA IZF / CZF Extensionality Foundation Other Class comprehension schema for ' which quantify only over sets: C := fx j '(x)g is a class Global Choice. Yurii Khomskii Alternative Set Theories Class comprehension vs. finite axiomatization Alternative Set Theories Yurii Khomskii Introduction NGB As stated above, class comprehension is a schema. However, it MK can be replaced by finitely many instances thereof (roughly KP speaking: one axiom for each application of a logical NF connective/quantifier). AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories Alternative Set Theories Yurii Khomskii Introduction NGB MK KP Morse-Kelley (MK) NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories Morse-Kelley MK Alternative Set Theories Yurii Khomskii Introduction Morse-Kelley (MK) NGB MK Morse-Kelley set theory is a variant of NGB with the class KP comprehension schema allowing arbitrary formulas (also those NF that quantify over classes). AFA IZF / CZF MK is not finitely axiomatizable. Other Yurii Khomskii Alternative Set Theories Consistency strength Alternative Set Theories Yurii Khomskii NGB is a conservative extension of ZFC: for any Introduction theorem ' involving only sets, if NGB ` ' then ZFC ` '. NGB In particular, if ZFC is consistent then NGB is consistent. MK KP MK ` Con(ZFC), and therefore the consistency of MK NF does not follow from the consistency of ZFC. AFA If κ is inaccessible, then (Vκ; Def(Vκ)) j= NGB while IZF / CZF (Vκ; P(Vκ)) j= MK. Other The consistency strength of MK is strictly between ZFC and ZFC + Inaccessible. Yurii Khomskii Alternative Set Theories Alternative Set Theories Yurii Khomskii Introduction NGB MK KP Kripke-Platek (KP) NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories KP Alternative Set Theories Yurii Khomskii Kripke-Platek set theory (KP) Introduction NGB Captures a small part of mathematics | stronger than 2nd MK order arithmetic but noticeably weaker than ZF. KP NF Idea: get rid of \impredicative" axioms of ZFC: in particular AFA Power Set, (full) Separation and (full) Replacement. IZF / CZF Other Instead, have Separation and Replacement for ∆0-formulas only. Yurii Khomskii Alternative Set Theories Applications of KP Alternative Set Theories Yurii Khomskii Introduction KP has applications in many standard areas of set theory as NGB well as recursion theory and constructibility. MK KP One example: KP is sufficient to develop the theory of G¨odel's NF Constructible Universe L. AFA IZF / CZF L is not only the minimal model of ZFC, but also the minimal Other model of KP (this is because \V = L" is absolute for KP). Yurii Khomskii Alternative Set Theories Alternative Set Theories Yurii Khomskii Introduction NGB MK KP Quine's New Foundations (NF) NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories NF Alternative Set Theories Yurii Khomskii Introduction Quine's New Foundations. NGB MK The idea is: avoid Russell's paradox by a syntactical KP limitation on ' in the comprehension scheme fx j '(x)g. NF AFA NF has its roots in type theory, as it was first developed in IZF / CZF Principia Mathematica. Other Yurii Khomskii Alternative Set Theories Stratified sentences Alternative Set Theories Yurii A sentence φ in the language of set theory (only = and 2 Khomskii symbols) is stratified if it is possible to assign a non-negative Introduction integer to each variable x occurring in φ, called the type of x, NGB in such a way that: MK 1 Each variable has the same type whenever it appears, KP 2 NF In each occurrence of \x = y" in φ, the types of x and y AFA are the same, and IZF / CZF 3 In each occurrence of \x 2 y" in φ, the type of y is one Other higher than the type of x. Example: x = x is stratified. x 2 y is stratified. x 2= x is not stratified. Yurii Khomskii Alternative Set Theories Axiomatization of NF Alternative Set Theories Yurii Khomskii Introduction Axiomatization of NF NGB Extensionality, MK KP Stratified comprehension scheme: NF AFA fx j '(x)g exists IZF / CZF Other for every stratified formula '. Yurii Khomskii Alternative Set Theories Finite axiomatization of NF Alternative Set Theories Alternatively, we may replace the stratified comprehension Yurii scheme by finitely many instances thereof, each having Khomskii intuitive motivation: Introduction The empty set exists: fx j ?g NGB MK The singleton set exists: fx j x = yg KP The union of a set a exists: fx j 9y 2 a (x 2 y)g NF ... AFA as well as other \non-ZFC-ish" axioms, e.g.: IZF / CZF Other The universe exists: fx j >g The compelement of A exists: fx j x 2= Ag ... The full stratified comprehension scheme is a consequence of these instances. Yurii Khomskii Alternative Set Theories Properties of NF Alternative Set Theories Yurii Khomskii NF is weird! Introduction NGB V := fx j >g, the universe of all sets, exists, and V 2 V . MK Therefore: KP · · · 2 V 2 V 2 V : NF AFA NF ` :AC. IZF / CZF Therefore: NF ` Infinity. Other The consistency of NF was an open problem since 1937 till (about) 2010. Yurii Khomskii Alternative Set Theories NFU Alternative Set Theories Yurii Khomskii Introduction Ronald Jensen considered a weakening of NF called NFU (New NGB Foundations with Urelements), weakening Extensionality to MK KP 8x8y (x 6= ? ^ y 6= ? ^ 8z (z 2 x $ z 2 y) ! x = y) NF AFA IZF / CZF NFU was known to be consistent for a long time, NFU 6` :AC Other and NFU 6` Infinity. So NFU+ Infinity +AC is consistent. Yurii Khomskii Alternative Set Theories Alternative Set Theories Yurii Khomskii Introduction NGB MK KP Non-well-founded set theory NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories Non-well-founded set theory Alternative Set Theories Yurii Khomskii Aczel's non-well-founded set theory Introduction NGB We already saw that V 2 V holds in NF. MK KP NF Peter Aczel considered the question: \how non-well-founded AFA can set theory be"? In other words, is it consistent that any IZF / CZF kind of non-well-founded set that you can think of, exists? Other This leads to the Anti-Foundation Axiom AFA. Yurii Khomskii Alternative Set Theories Pictures of sets Alternative Set Theories Yurii Khomskii Idea: sets can be pictured using graphs, with \x ! y" Introduction representing \y 2 x", e.g.: NGB MK KP ◦ NF AFA } ! IZF / CZF ◦ ◦ Other ◦ Yurii Khomskii Alternative Set Theories Pictures of sets Alternative Set Theories Yurii Khomskii
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