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 ϕ, the set {x | ϕ(x)} 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 := {x | x ∈/ x} NGB

MK Then K ∈ K ↔ K ∈/ 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 {x | ϕ(x)} 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 ∈. 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

KP von Neumann-G¨odel-Bernays (NBG)

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 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 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

KP Morse-Kelley (MK)

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

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)) |= NGB while IZF / CZF (Vκ, P(Vκ)) |= 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

KP Kripke-Platek (KP)

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

KP Quine’s New Foundations (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

Yurii A sentence φ in the language of set theory (only = and ∈ 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 ∈ y” in φ, the type of y is one Other higher than the type of x.

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

KP Stratiﬁed comprehension scheme:

AFA {x | ϕ(x)} exists

IZF / CZF

Other for every stratiﬁed formula ϕ.

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: {x | ⊥} NGB

MK The singleton set exists: {x | x = y} KP The union of a set a exists: {x | ∃y ∈ a (x ∈ y)} NF ... AFA as well as other “non-ZFC-ish” axioms, e.g.: IZF / CZF

Other The universe exists: {x | >} 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 Khomskii NF is weird!

Introduction NGB V := {x | >}, the universe of all sets, exists, and V ∈ V . MK Therefore: KP · · · ∈ V ∈ V ∈ 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 ∀x∀y (x 6= ∅ ∧ y 6= ∅ ∧ ∀z (z ∈ x ↔ z ∈ 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

KP Non-well-founded set theory

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 ∈ V holds in NF. MK

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 ∈ x”, e.g.:

NGB

KP ◦

NF AFA } ! IZF / CZF ◦ ◦

Other ◦

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 ∈ x”, e.g.:

NGB

KP ◦

NF AFA ~ ! IZF / CZF ∅ ◦

Other ◦

Yurii Khomskii Alternative Set Theories Pictures of sets

Alternative Set Theories

Yurii Khomskii Idea: sets can be pictured using graphs, with “x → y” representing “y ∈ x”, e.g.: Introduction

NGB

KP ◦

NF AFA ~ ! ◦ IZF / CZF ∅

Other ∅

Yurii Khomskii Alternative Set Theories Pictures of sets

Alternative Set Theories

Yurii Khomskii Idea: sets can be pictured using graphs, with “x → y” representing “y ∈ x”, e.g.: Introduction

NGB

KP ◦

NF AFA ~ 1 IZF / CZF ∅

Other ∅

Yurii Khomskii Alternative Set Theories Pictures of sets

Alternative Set Theories

Yurii Khomskii Idea: sets can be pictured using graphs, with “x → y” representing “y ∈ x”, e.g.: Introduction

NGB

KP 2

NF AFA IZF / CZF ∅ 1

Other ∅

Yurii Khomskii Alternative Set Theories Pictures of sets

Alternative Set Theories Idea: sets can be pictured using graphs, with “x → y” Yurii representing “y ∈ x”, e.g.: Khomskii

Introduction ◦ NGB

NF AFA ◦Õ ◦ & ◦ IZF / CZF

Other ◦ ◦} ! ◦

◦

Yurii Khomskii Alternative Set Theories Pictures of sets

Alternative Set Theories Idea: sets can be pictured using graphs, with “x → y”

Yurii representing “y ∈ x”, e.g.: Khomskii

Introduction ◦ NGB

NF AFA Õ & ∅ ◦ ◦ IZF / CZF

Other ~ ∅ ∅ ◦

∅

Yurii Khomskii Alternative Set Theories Pictures of sets

Alternative Set Theories Idea: sets can be pictured using graphs, with “x → y”

Yurii representing “y ∈ x”, e.g.: Khomskii

Introduction ◦ NGB

NF AFA Ö & ∅ 1 ◦ IZF / CZF

Other ~ ∅ ∅ 1

∅

Yurii Khomskii Alternative Set Theories Pictures of sets

Alternative Set Theories Idea: sets can be pictured using graphs, with “x → y”

Yurii representing “y ∈ x”, e.g.: Khomskii

Introduction ◦ NGB

NF AFA Ö & ∅ 1 2 IZF / CZF

Other ∅ ∅ 1

∅

Yurii Khomskii Alternative Set Theories Pictures of sets

Alternative Set Theories Idea: sets can be pictured using graphs, with “x → y”

Yurii representing “y ∈ x”, e.g.: Khomskii

Introduction 3 NGB

NF AFA Ö & ∅ 1 2 IZF / CZF

Other ∅ ∅ 1

∅

Yurii Khomskii Alternative Set Theories You can write this as: Ω = {Ω}. But also as Ω = {{Ω}}. And also as Ω = {Ω, {Ω}}.

Pictures of sets

Alternative Set Theories

Yurii Khomskii Every graph without inﬁnite paths or cycles corresponds to a Introduction unique set (Mostowski Collapse). NGB MK Aczel: now look at non-well-founded graphs! KP

NF AFA ◦ Ô IZF / CZF

Other

Yurii Khomskii Alternative Set Theories But also as Ω = {{Ω}}. And also as Ω = {Ω, {Ω}}.

Pictures of sets

Alternative Set Theories

Yurii Khomskii Every graph without inﬁnite paths or cycles corresponds to a Introduction unique set (Mostowski Collapse). NGB MK Aczel: now look at non-well-founded graphs! KP

NF AFA ◦ Ô IZF / CZF Other You can write this as: Ω = {Ω}.

Yurii Khomskii Alternative Set Theories And also as Ω = {Ω, {Ω}}.

Pictures of sets

Alternative Set Theories

Yurii Khomskii Every graph without inﬁnite paths or cycles corresponds to a Introduction unique set (Mostowski Collapse). NGB MK Aczel: now look at non-well-founded graphs! KP

NF AFA ◦ Ô IZF / CZF Other You can write this as: Ω = {Ω}. But also as Ω = {{Ω}}.

Yurii Khomskii Alternative Set Theories Pictures of sets

Alternative Set Theories

Yurii Khomskii Every graph without inﬁnite paths or cycles corresponds to a Introduction unique set (Mostowski Collapse). NGB MK Aczel: now look at non-well-founded graphs! KP

NF AFA ◦ Ô IZF / CZF Other You can write this as: Ω = {Ω}. But also as Ω = {{Ω}}. And also as Ω = {Ω, {Ω}}.

Yurii Khomskii Alternative Set Theories It seems that you can write this as X = {Y } and Y = {X }. But then X = {{X }}. So in fact X and Y are the same set Ω = {Ω}.

Pictures of sets

Alternative Set Theories

Yurii Khomskii Now consider this graph:

Introduction

NGB

MK ◦ ` KP NF AFA ◦

IZF / CZF

Other

Yurii Khomskii Alternative Set Theories But then X = {{X }}. So in fact X and Y are the same set Ω = {Ω}.

Pictures of sets

Alternative Set Theories

Yurii Khomskii Now consider this graph:

Introduction

NGB

MK ◦ ` KP NF AFA ◦

IZF / CZF It seems that you can write this as X = {Y } and Y = {X }. Other

Yurii Khomskii Alternative Set Theories So in fact X and Y are the same set Ω = {Ω}.

Pictures of sets

Alternative Set Theories

Yurii Khomskii Now consider this graph:

Introduction

NGB

MK ◦ ` KP NF AFA ◦

IZF / CZF It seems that you can write this as X = {Y } and Y = {X }. Other But then X = {{X }}.

Yurii Khomskii Alternative Set Theories Pictures of sets

Alternative Set Theories

Yurii Khomskii Now consider this graph:

Introduction

NGB

MK ◦ ` KP NF AFA ◦

IZF / CZF It seems that you can write this as X = {Y } and Y = {X }. Other But then X = {{X }}. So in fact X and Y are the same set Ω = {Ω}.

Yurii Khomskii Alternative Set Theories Pictures of sets

Alternative Set Theories

Yurii Khomskii Now consider this graph:

Introduction

NGB

MK ◦ ` KP NF AFA ◦

IZF / CZF It seems that you can write this as X = {Y } and Y = {X }. Other But then X = {{X }}. So in fact X and Y are the same set Ω = {Ω}.

Yurii Khomskii Alternative Set Theories More pictures of Ω

Alternative Set Theories

Yurii Khomskii More pictures of the set Ω:

Introduction ◦ ◦ ◦ ◦ NGB 9 MK

KP ◦ ◦ ◦ ◦Õ ◦ NF 9 ` AFA IZF / CZF ◦ ◦ ◦ ◦ Other Y Y . .

Yurii Khomskii Alternative Set Theories AFA

Alternative Set Theories

Yurii Khomskii

Introduction

NGB The anti-foundation axiom AFA is (roughly speaking): MK Every connected pointed graph represents a unique KP set. NF

AFA

IZF / CZF AFA is consistent with ZF − Foundation.

Other

Yurii Khomskii Alternative Set Theories Alternative Set Theories

Yurii Khomskii

Introduction

NGB

KP Intuitionistic/Constructive Set Theory

AFA

IZF / CZF

Other

Yurii Khomskii Alternative Set Theories Intuitionistic and Constructive Set Theory

Alternative Set Theories

Yurii Khomskii

Introduction

NGB IZF (the simplest variant)

KP Suppose we take all the ZFC axioms exactly as they are, but

NF change the logic from classical logic to intuitionistic logic?

AFA I.e., φ is a theorem of the system iﬀ ZFC ` φ in intuitionistic IZF / CZF predicate logic. Other

Yurii Khomskii Alternative Set Theories Axiom of Choice

Alternative Set Theories Axiom of Choice Yurii Khomskii Let φ be some formula. Consider:

Introduction A := {n ∈ {0, 1} | n = 0 ∨ (n = 1 ∧ φ)} NGB B := {n ∈ {0, 1} | n = 1 ∨ (n = 0 ∧ φ)} MK

KP Since A and B are non-empty, by AC, there is a choice function

NF f : {A, B} → {0, 1}. AFA Since equality of natural numbers is intuitionistically decidable, IZF / CZF f (A) = f (B) or f (A) 6= f (B). Other If f (A) = f (B) = 0 then 0 ∈ B, hence φ. Similarly if f (A) = f (B) = 1.

Suppose f (A) 6= f (B). Towards contradiction, suppose φ. Then, by extentionality, A = B, hence f (A) = f (B): contradiction! Hence ¬φ.

(Recall that (φ → ⊥) → ¬φ is valid, only (¬φ → ⊥) → φ is invalid).

Yurii Khomskii Alternative Set Theories Axiom of Choice

Alternative Set Theories

Yurii Khomskii

Introduction

NGB MK Thus AC (formulated as above) implies φ ∨ ¬φ for any formula KP φ. We have obtained the Law of Excluded Middle, thus the NF ZFC axioms with intuitionistic logic is just normal ZFC. AFA

IZF / CZF

Other

Yurii Khomskii Alternative Set Theories Let A := {n ∈ {0, 1} | n = 1 ∨ (n = 0 ∧ φ)}.

A is non-empty so there is a y ∈ A which is ∈-minimal. By deﬁnition of A, either y = 1 or y = 0 ∧ φ. But the former case implies that 0 ∈/ A, hence ¬φ.

Hence, in either case, φ ∨ ¬φ.

Again we have proved the Law of Excluded Middle (from a seemingly harmless statement).

Foundation

Alternative Set Theories You may think AC is inherently non-constructive, and thus Yurii Khomskii shouldn’t even be taken into account, etc.

Introduction But consider Foundation: ∀X (∃y ∈ X → ∃y ∈ X ∀z ∈ X (z ∈/ y)). NGB

AFA

IZF / CZF

Other

Yurii Khomskii Alternative Set Theories A is non-empty so there is a y ∈ A which is ∈-minimal. By deﬁnition of A, either y = 1 or y = 0 ∧ φ. But the former case implies that 0 ∈/ A, hence ¬φ.

Hence, in either case, φ ∨ ¬φ.

Again we have proved the Law of Excluded Middle (from a seemingly harmless statement).

Foundation

Alternative Set Theories You may think AC is inherently non-constructive, and thus Yurii Khomskii shouldn’t even be taken into account, etc.

Introduction But consider Foundation: ∀X (∃y ∈ X → ∃y ∈ X ∀z ∈ X (z ∈/ y)). NGB

MK Let A := {n ∈ {0, 1} | n = 1 ∨ (n = 0 ∧ φ)}. KP

AFA

IZF / CZF

Other

Yurii Khomskii Alternative Set Theories Foundation

Alternative Set Theories You may think AC is inherently non-constructive, and thus Yurii Khomskii shouldn’t even be taken into account, etc.

Introduction But consider Foundation: ∀X (∃y ∈ X → ∃y ∈ X ∀z ∈ X (z ∈/ y)). NGB

MK Let A := {n ∈ {0, 1} | n = 1 ∨ (n = 0 ∧ φ)}. KP NF A is non-empty so there is a y ∈ A which is ∈-minimal. By deﬁnition of A, AFA either y = 1 or y = 0 ∧ φ. But the former case implies that 0 ∈/ A, hence IZF / CZF ¬φ.

Other Hence, in either case, φ ∨ ¬φ.

Again we have proved the Law of Excluded Middle (from a seemingly harmless statement).

Yurii Khomskii Alternative Set Theories Set Induction

Alternative Set Theories

Yurii Khomskii

Introduction Set Induction is the following axiom schema for all φ: NGB MK ∀x [(∀y ∈ x φ(y)) → φ(x)] → ∀x φ(x) KP

NF This principle does not imply Excluded Middle.

AFA

IZF / CZF Likewise, there are variants of AC which do not imply Excluded

Other Middle and are compatible with an intuitionistic system.

Yurii Khomskii Alternative Set Theories IZF

Alternative Set Theories

Yurii Khomskii

Introduction IZF is the theory consisting of the ZF axioms without Choice, NGB but with Foundation replaced by Set Induction, and MK Replacement by a stronger principle called Collection. KP NF Other theories, most notably CZF, implements other changes AFA as well (mostly based on conceptual/philosophical IZF / CZF justiﬁcations). Other

Yurii Khomskii Alternative Set Theories Alternative Set Theories

Yurii Khomskii

Introduction

NGB

KP Other Logics

AFA

IZF / CZF

Other

Yurii Khomskii Alternative Set Theories Paraconsistent set theory?

Other logics

Alternative Set Theories

Yurii Khomskii Set Theories based on other logics?

Introduction Modal Set Theory would be some form of ZF or ZFC done in NGB some form of modal predicate logic. There is no universal MK consensus on what should count as modal predicate logic — let KP alone for modal ZF or ZFC. Therefore this is a highly NF experimental subject. AFA

IZF / CZF Robert Passmann is currently writing his Master Thesis on Other modal set theory, so I hope he can tell us something about it!

Yurii Khomskii Alternative Set Theories Other logics

Alternative Set Theories

Yurii Khomskii Set Theories based on other logics?

IZF / CZF Robert Passmann is currently writing his Master Thesis on Other modal set theory, so I hope he can tell us something about it!

Paraconsistent set theory?

Yurii Khomskii Alternative Set Theories Alternative Set Theories

Alternative Set Theories

Yurii Khomskii

1 Introduction NGB (von Neumann-G¨odel-Bernays) NGB 2 MK (Morse-Kelley) MK 3 NF (New Foundations) KP 4 NF 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