Alternative Set Theories Yurii Khomskii Alternative Set Theories Introduction NGB MK Yurii Khomskii KP 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 ϕ , the set { x | ϕ ( x ) } exists. NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories
Russell’s Paradox Alternative Set Theories Yurii Russell’s Paradox Khomskii Introduction Let K := { x | x / ∈ x } NGB Then K ∈ K ↔ K / ∈ K MK � KP NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories
ZFC Alternative Set Theories Yurii Khomskii Introduction The commonly accepted ax- NGB iomatization of set theory is MK ZFC. All results in mainstream KP NF mathematics can be formalized AFA in it. IZF / CZF Other Yurii Khomskii Alternative Set Theories
Philosophy of ZFC Alternative Set Theories Yurii Khomskii Philosophy of ZFC Introduction Everything is a set. NGB MK Sets are constructed out of other sets, bottom up. KP Comprehension can only select a subset out of an NF existing set (avoid paradoxes). AFA Certain definable collections { x | ϕ ( x ) } are “too large” to IZF / CZF Other be sets. Yurii Khomskii Alternative Set Theories
Logic of ZFC Alternative Set Theories Yurii Khomskii Introduction Logic of ZFC NGB Classical predicate logic. MK One-sorted. KP NF One binary non-logical relation symbol ∈ . AFA In this language, ZFC is an infinite (but recursive) IZF / CZF collection of axioms. Other Yurii Khomskii Alternative Set Theories
Other set theories Alternative Set Theories Yurii All these factors are liable to change! Several alternative set Khomskii theories have been proposed, for a variety of reasons: Introduction NGB Philosophical (more intuitive conception) MK The need to have proper classes as formal objects (e.g., KP “class forcing”) NF Capturing a fragment of mathematics (e.g., predicative AFA fragment, intuitionistic fragment etc.) IZF / CZF 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 (Morse-Kelley) MK 3 NF (New Foundations) KP NF 4 KP (Kripke-Platek) AFA 5 IZF/CZF (Intuitionistic and constructive set theory) IZF / CZF 6 ZF − + AFA (Antifoundation) Other 7 Modal or other set theory? Yurii Khomskii Alternative Set Theories
Alternative Set Theories Yurii Khomskii Introduction NGB MK von Neumann-G¨ odel-Bernays (NBG) KP NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories
NGB Alternative Set Theories Yurii NGB (von Neumann-G¨ odel-Bernays) Khomskii Introduction NGB We have sets and classes ; some classes are sets, others MK are not. KP It can be formalized either in a two-sorted language or NF using a predicate M ( X ) stating “ X is a set”. AFA You still need set existence axioms, along with class IZF / CZF existence axioms. Other NGB can be finitely axiomatized . Yurii Khomskii Alternative Set Theories
Axiomatization of NGB Alternative Axiomatization of NGB Set Theories Yurii Set axioms: Khomskii Pairing Introduction Infinity NGB Union MK Power set KP Replacement NF Class axioms: AFA Extensionality IZF / CZF Foundation Other Class comprehension schema for ϕ which quantify only over sets: C := { x | ϕ ( x ) } is a class Global Choice. Yurii Khomskii Alternative Set Theories
Class comprehension vs. finite axiomatization Alternative Set Theories Yurii Khomskii Introduction As stated above, class comprehension is a schema. However, it NGB 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 Morse-Kelley (MK) KP 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 ϕ involving only sets , if NGB ⊢ ϕ then ZFC ⊢ ϕ . NGB In particular, if ZFC is consistent then NGB is consistent. MK MK ⊢ Con(ZFC), and therefore the consistency of MK KP does not follow from the consistency of ZFC. NF If κ is inaccessible, then ( V κ , Def( V κ )) | = NGB while AFA ( V κ , P ( V κ )) | = MK. IZF / CZF 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 Kripke-Platek (KP) 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 Idea: get rid of “impredicative” axioms of ZFC: in particular NF Power Set, (full) Separation and (full) Replacement. AFA IZF / CZF Instead, have Separation and Replacement for ∆ 0 -formulas Other only. Yurii Khomskii Alternative Set Theories
Applications of KP Alternative Set Theories Yurii Khomskii KP has applications in many standard areas of set theory as Introduction well as recursion theory and constructibility. NGB MK One example: KP is sufficient to develop the theory of G¨ odel’s KP Constructible Universe L . NF AFA L is not only the minimal model of ZFC, but also the minimal IZF / CZF model of KP (this is because “ V = L ” is absolute for KP). Other Yurii Khomskii Alternative Set Theories
Alternative Set Theories Yurii Khomskii Introduction NGB MK Quine’s New Foundations (NF) KP 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 ϕ in the comprehension scheme { x | ϕ ( x ) } . 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 A sentence φ in the language of set theory (only = and ∈ Yurii Khomskii symbols) is stratified if it is possible to assign a non-negative integer to each variable x occurring in φ , called the type of x , Introduction in such a way that: NGB MK 1 Each variable has the same type whenever it appears, KP 2 In each occurrence of “ x = y ” in φ , the types of x and y NF are the same, and AFA 3 In each occurrence of “ x ∈ y ” in φ , the type of y is one IZF / CZF higher than the type of x . Other Example: x = x is stratified. x ∈ y is stratified. x / ∈ x is not stratified. Yurii Khomskii Alternative Set Theories
Axiomatization of NF Alternative Set Theories Yurii Khomskii Introduction Axiomatization of NF NGB Extensionality, MK Stratified comprehension scheme: KP NF { x | ϕ ( x ) } exists AFA IZF / CZF for every stratified formula ϕ . Other Yurii Khomskii Alternative Set Theories
Finite axiomatization of NF Alternative Alternatively, we may replace the stratified comprehension Set Theories scheme by finitely many instances thereof, each having Yurii Khomskii intuitive motivation: Introduction The empty set exists: { x | ⊥} NGB The singleton set exists: { x | x = y } MK The union of a set a exists: { x | ∃ y ∈ a ( x ∈ y ) } KP NF . . . AFA as well as other “non-ZFC-ish” axioms, e.g.: IZF / CZF The universe exists: { x | ⊤} Other The compelement of A exists: { x | x / ∈ A } . . . The full stratified comprehension scheme is a consequence of these instances. Yurii Khomskii Alternative Set Theories
Properties of NF Alternative Set Theories Yurii NF is weird! Khomskii Introduction V := { x | ⊤} , the universe of all sets, exists, and V ∈ V . NGB MK Therefore: KP · · · ∈ V ∈ V ∈ V . NF NF ⊢ ¬ AC. AFA Therefore: NF ⊢ Infinity. IZF / CZF 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 Ronald Jensen considered a weakening of NF called NFU (New Introduction Foundations with Urelements), weakening Extensionality to NGB MK ∀ x ∀ y ( x � = ∅ ∧ y � = ∅ ∧ ∀ z ( z ∈ x ↔ z ∈ y ) → x = y ) KP NF AFA NFU was known to be consistent for a long time, NFU �⊢ ¬ AC IZF / CZF and NFU �⊢ Infinity. So NFU+ Infinity +AC is consistent. Other Yurii Khomskii Alternative Set Theories
Alternative Set Theories Yurii Khomskii Introduction NGB MK Non-well-founded set theory KP NF AFA IZF / CZF Other Yurii Khomskii Alternative Set Theories
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