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Unbounded quantifiers & strong axioms Michael Shulman The Question Two kinds of set theory Strong axioms Unbounded quantifiers and strong axioms Internalization The Answer in topos theory Stack semantics Logic in stack semantics Strong axioms Michael A. Shulman University of Chicago November 14, 2009 Unbounded quantifiers & strong axioms The motivating question Michael Shulman The Question Two kinds of set theory Strong axioms Internalization The Answer Stack semantics Logic in stack semantics What is the topos-theoretic counterpart of the Strong axioms strong set-theoretic axioms of Separation, Replacement, and Collection? Unbounded quantifiers & strong axioms Outline Michael Shulman The Question Two kinds of set theory Strong axioms 1 The Question Internalization The Answer Two kinds of set theory Stack semantics Logic in stack Strong axioms semantics Strong axioms Internalization 2 The Answer Stack semantics Logic in stack semantics Strong axioms Following Lawvere and others, I prefer to view this as an equivalence between two kinds of set theory. Unbounded quantifiers & strong axioms Basic fact Michael Shulman The Question Two kinds of set theory Strong axioms Internalization The Answer Stack semantics Elementary topos theory is equiconsistent with a Logic in stack semantics weak form of set theory. Strong axioms Unbounded quantifiers & strong axioms Basic fact Michael Shulman The Question Two kinds of set theory Strong axioms Internalization The Answer Stack semantics Elementary topos theory is equiconsistent with a Logic in stack semantics weak form of set theory. Strong axioms Following Lawvere and others, I prefer to view this as an equivalence between two kinds of set theory. Unbounded quantifiers & strong axioms “Membership-based” or Michael Shulman “material” set theory The Question Two kinds of set theory Strong axioms Data: Internalization The Answer • A collection of sets. Stack semantics Logic in stack • A membership relation 2 between sets. semantics Strong axioms Sample axioms: • Extensionality: if x 2 a () x 2 b for all x, then a = b. • Pairing: for all x and y, the set fx; yg exists. • Power set: for all sets a, the set P(a) = fx j x ⊆ ag exists. • ... Example: ZFC (Zermelo-Fraenkel set theory with Choice). Unbounded quantifiers & strong axioms “Structural” or “categorical” Michael Shulman set theory The Question Two kinds of set Data: theory Strong axioms • Internalization A collection of sets. The Answer • A collection of functions. Stack semantics Logic in stack semantics • Composition and identity operations on functions. Strong axioms Sample axioms: • Sets and functions form a category Set. • Set has finite limits and power objects. • Well-pointedness: The terminal set 1 is a generator. • ... Example: Lawvere’s ETCS (Elementary Theory of the Category of Sets). Note: An element of a set A is a function 1 ! A. NB: This has nothing to do with the “internal logic” (yet)! Unbounded quantifiers & strong axioms The basic comparison Michael Shulman Theorem (Cole, Mitchell, Osius) The Question Two kinds of set theory ETCS is equiconsistent with BZC. Strong axioms Internalization The Answer BZC = Bounded Zermelo set theory with Choice Stack semantics Logic in stack = ZFC without Replacement, and with Separation semantics Strong axioms restricted to formulas with bounded quantifiers (“8x 2 a” rather than “8x”) Proof (Sketch). 1 BZC)ETCS: The category of sets and functions in BZC satisfies ETCS. 2 ETCS)BZC: The collection of “pointed well-founded trees” in an ETCS-category satisfies BZC. Unbounded quantifiers & strong axioms The basic comparison Michael Shulman Theorem (Cole, Mitchell, Osius) The Question Two kinds of set theory ETCS is equiconsistent with BZC. Strong axioms Internalization The Answer BZC = Bounded Zermelo set theory with Choice Stack semantics Logic in stack = ZFC without Replacement, and with Separation semantics Strong axioms restricted to formulas with bounded quantifiers (“8x 2 a” rather than “8x”) Proof (Sketch). 1 BZC)ETCS: The category of sets and functions in BZC satisfies ETCS. 2 ETCS)BZC: The collection of “pointed well-founded trees” in an ETCS-category satisfies BZC. NB: This has nothing to do with the “internal logic” (yet)! Unbounded quantifiers & strong axioms Why do we care? Michael Shulman Why is this equivalence useful? The Question Two kinds of set 1 We can use either one as a foundation for theory Strong axioms mathematics, according to taste. Internalization The Answer 2 Some constructions are easier or more intuitive in one Stack semantics Logic in stack context or the other. semantics Strong axioms Material World Structural World forcing $ sheaves ultrapowers $ filterquotients realizability models $ realizability toposes ?? $ free toposes ?? $ glued toposes ?? $ exact completion V = L $ ?? Unbounded quantifiers & strong axioms Outline Michael Shulman The Question Two kinds of set theory Strong axioms 1 The Question Internalization The Answer Two kinds of set theory Stack semantics Logic in stack Strong axioms semantics Strong axioms Internalization 2 The Answer Stack semantics Logic in stack semantics Strong axioms • Replacement/Collection: 8a: (8x 2 a: 9y:'(x; y)) ) 9b: (8x 2 a: 9y 2 b:'(x; y)) ^ (8y 2 b: 9x 2 a:'(x; y)) What that really means: If for every x 2 a there exists a yx such that some property '(x; yx ) holds, then we can collect these y’s into a family of sets b = fyx gx2a. Unbounded quantifiers & strong axioms The strong axioms Michael Shulman The Question What’s missing from BZC? Two kinds of set theory • Strong axioms Unbounded Separation: For any set a and formula Internalization '(x), the set fx 2 a j '(x)g exists. The Answer Stack semantics Logic in stack NB: “Unbounded” means “not bounded by a set.” That semantics Strong axioms is, we allow quantifiers such as “for all sets” rather than “for all elements of the set a.” 8a: (8x 2 a: 9y:'(x; y)) ) 9b: (8x 2 a: 9y 2 b:'(x; y)) ^ (8y 2 b: 9x 2 a:'(x; y)) What that really means: If for every x 2 a there exists a yx such that some property '(x; yx ) holds, then we can collect these y’s into a family of sets b = fyx gx2a. Unbounded quantifiers & strong axioms The strong axioms Michael Shulman The Question What’s missing from BZC? Two kinds of set theory • Strong axioms Unbounded Separation: For any set a and formula Internalization '(x), the set fx 2 a j '(x)g exists. The Answer Stack semantics Logic in stack NB: “Unbounded” means “not bounded by a set.” That semantics Strong axioms is, we allow quantifiers such as “for all sets” rather than “for all elements of the set a.” • Replacement/Collection: What that really means: If for every x 2 a there exists a yx such that some property '(x; yx ) holds, then we can collect these y’s into a family of sets b = fyx gx2a. Unbounded quantifiers & strong axioms The strong axioms Michael Shulman The Question What’s missing from BZC? Two kinds of set theory • Strong axioms Unbounded Separation: For any set a and formula Internalization '(x), the set fx 2 a j '(x)g exists. The Answer Stack semantics Logic in stack NB: “Unbounded” means “not bounded by a set.” That semantics Strong axioms is, we allow quantifiers such as “for all sets” rather than “for all elements of the set a.” • Replacement/Collection: 8a: (8x 2 a: 9y:'(x; y)) ) 9b: (8x 2 a: 9y 2 b:'(x; y)) ^ (8y 2 b: 9x 2 a:'(x; y)) Unbounded quantifiers & strong axioms The strong axioms Michael Shulman The Question What’s missing from BZC? Two kinds of set theory • Strong axioms Unbounded Separation: For any set a and formula Internalization '(x), the set fx 2 a j '(x)g exists. The Answer Stack semantics Logic in stack NB: “Unbounded” means “not bounded by a set.” That semantics Strong axioms is, we allow quantifiers such as “for all sets” rather than “for all elements of the set a.” • Replacement/Collection: 8a: (8x 2 a: 9y:'(x; y)) ) 9b: (8x 2 a: 9y 2 b:'(x; y)) ^ (8y 2 b: 9x 2 a:'(x; y)) What that really means: If for every x 2 a there exists a yx such that some property '(x; yx ) holds, then we can collect these y’s into a family of sets b = fyx gx2a. Unbounded quantifiers & strong axioms Why unbounded separation? Michael Shulman The Question Two kinds of set theory Strong axioms • Any universal mapping property in a large category Internalization The Answer involves unbounded quantifiers. E.g. if fFx gx2a is a Stack semantics family of diagrams F : D ! C, then forming Logic in stack x semantics Strong axioms fx 2 a j Fx has a limitg uses unbounded separation. • To justify mathematical induction for some property ', we form the set fn j '(n)g and show that it is all of N. This only works if we have separation for '. Unbounded quantifiers & strong axioms Why replacement/collection? Michael Shulman • To construct objects recursively, such as free algebras: The Question for an operation T and a starting point X, we consider Two kinds of set theory the sequence Strong axioms Internalization 2 3 The Answer X; TX; T X; T X;::: Stack semantics Logic in stack semantics n Strong axioms BZC can construct each T X, but not the entire sequence or its limit. • In particular, cardinal numbers above @! are unreachable without replacement/collection. • In BZC we have to be careful when saying “Set is n (co)complete.” E.g. the family fP (N)gn2N may not have a coproduct (so we have to exclude it from the notion of “family”). • Even some “concrete” facts depend on replacement/collection, e.g. Borel Determinacy. Unbounded quantifiers & strong axioms Structural quantifiers Michael Shulman The Question Two kinds of set theory • In material set theory, unbounded quantifiers are very Strong axioms Internalization natural, while bounded ones feel like an ad hoc The Answer restriction.
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