Symbiosis Reflection LST Sort logic

Set theory and model theory: a symbiosis

Jouko Väänänen

Helsinki, Finland

Montseny, November 2018

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Model theory

Model theory (in this talk) is the study of classes of models, closed under isomorphisms, on the basis of the formal languages used to define the classes.

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The basic equivalence

• A model M satisfies the sentence ϕ of a logic L∗ iffa certain formula Φ(x) of set theory is true for x = M. • M |= ϕ ⇐⇒ Φ(M)

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The basic equivalence, spelled out

• M |= ϕ if an only if there is a model N of set theory such that M, ϕ ∈ N, in N the first order sentence “M |= ϕ” is true, and N satisfies some absoluteness criteria. • Φ(M) if and only if there is a binary predicate E on the universe of M, augmented perhaps with some new sorts, such that E satisfies some set theory, some absoluteness criteria, and in the sense of E the set-theoretical formula Φ(x) is satisfied by an element x = m of the universe such that the structure that E thinks m is, is isomorphic to M.

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An early example: simple theory of types

• x, y, z,... for individuals. • X, Y , Z ,... for sets. • X , Y, Z,... for sets of sets. • Etc...

• First order logic: x = y, R(x1,..., xn) • Second order logic: X(y) • Third order logic: X(y), X (Y ) • Etc...

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Gödel "On the present situation in the foundations of mathematics" (*1933o):1 "...but it turns out that this system [set theory] is nothing else but a natural generalization of the theory of types, or rather, it is what becomes of the theory of types if certain superfluous restrictions are removed."

(Hodges, "Tarski’s Theory of Definition", New Essays on Tarski and Philosophy, 2008, D. Patterson, ed.) "The deductive theories in question (such as RCF) are formulated in simple type theory; by 1935 the axioms for RCF is regarded as a definition within set theory."

1Unpublished manuscript. 7 / 53 Symbiosis Reflection LST Sort logic

Model theory Set theory

Sort logic First order logic

Second order logic ∆2 in Levy hierarchy

First order logic with ∆1 in Levy hierarchy the game quantifier

KP Infinitary logic LHYP ∆1 in Levy hierarchy

KPU− First order logic ∆1 in Levy hierarchy

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• First order logic Lωω,

• Infinitary logics Lκλ,

• Logic with the generalized quantifier Lωω(Qα),

• Härtig-quantifier logic Lωω(I). • Second order logic L2. cf • Cofinality logic Lωω(Qω ). • Stationary logic Lωω(aa).

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The map of logics

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Two key auxiliary concepts

Definition (V. 78) Suppose R is a finite set of predicates of set theory and L∗ is a logic.

1. ∆1(R) means ∆1 in the extended language {∈, R}. 2. ∆(L∗) is the logic the definable model classes of which are such K that both K and −K are reducts of L∗-definable model classes.

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Symbiosis

Definition A (finite set of) n-ary predicates R and a logic L∗ are symbiotic if the following conditions are satisfied: ∗ 1. Every L -definable model class is ∆1(R)-definable. 2 ∗ 2. Every ∆1(R)-definable model class is ∆(L )-definable.

2 It is enough that a particular ∆1(R)-definable model class i.e. a particular generalized quantifier, derived from R, is ∆(L∗)-definable. 12 / 53 Symbiosis Reflection LST Sort logic

The following pairs (R, L∗) are symbiotic. 1. R: Cd, i.e. the predicate “x is a cardinal". ∗ L : Lωω(I), where Ixyϕ(x)ψ(y) ↔ |ϕ| = |ψ| is the Härtig quantifier. 2. R: Cd ∗ L : Lωω(R), where Rxyϕ(x)ψ(y) ↔ |ϕ| ≤ |ψ| is the Rescher quantifier. 3. R: Cd ∗ Cd Cd L : Lωω(W ), where W xyϕ(x, y) ↔ ϕ(·, ·) is a well-ordering of the order-type of a cardinal. 4. R: Cd, WI ∗ WI WI L : Lωω(I, W ), where W xϕ(x) ↔ |ϕ(·)| is weakly-inaccessible. 5. R: Rg, i.e. the predicate “x is a regular cardinal" ∗ Rg Rg L : Lωω(I, W ), where W xyϕ(x, y) ↔ ϕ(·, ·) has the order-type of a regular cardinal.

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1. R: Cd, WC ∗ L : Lωω(I, QBr ), where QBr xyϕ(x, y) ↔ ϕ(·, ·) is a tree order of height some α and has no branch of length α. 2. R: Cd, WC ∗ ¯ ¯ L : Lωω(I, QBr ), where QBr xyuvϕ(x, y)ψ(u, v) ↔ ϕ(·, ·) is a partial order with a chain of order-type ψ(·, ·). 3. R: Pw i.e. the predicate {(x, y): y = P(x)} L∗: The second order logic L2.

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Cd and Lωω(I) are symbiotic.

Theorem TFAE:

1. K is ∆(Lωω(I))-definable.

2. K is ∆1(Cd)-definable in set theory

µ 2 Note: If V = L (or L ), then ∆(Lωω(I)) = ∆(L ).

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Question Is there a natural canonical inner model for set theory in the vocabulary {∈, Cd}? Is there a minimal transitive class model with true cardinals?

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Pw and the L2 are symbiotic.

Let Pw be the relation {(x, y): y = P(x)}. ∆1(Pw) = ∆2. Theorem TFAE: 1. K is ∆(L2)-definable.

2. K is ∆2-definable in set theory

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Structural reflection

Let R be a finite set of predicates or relations. Definition (Bagaria-V. 2016)

(SR)R(κ): If K is a Σ1(R) class of models, then for every A ∈ K, there exist B ∈ K of cardinality less than κ and an elementary embedding e : B  A.

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] • The principle (SR)Cd implies 0 , and much more, e.g. failure of square at all cardinals from κ on (Magidor and V. 2011, see later).

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Weaker forms of structural reflection

−− (SR)R : If K is a non-empty Σ1(R) class of models, then there exists A ∈ K of cardinality less than κ.

Proposition (Hanf) −− ZFC ` ∃κ((SR)R holds for κ). Proof: Let {Kn : n < ω} list all non-empty Σ1(R) model classes. Pick An ∈ Kn, for each n < ω. Let κ be the supremum of all the + cardinalities of the An, for n < ω. Then κ is as required.

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− (SR)R : If K is a Σ1(R) class of models and A ∈ K has cardinality κ, then there exists B ∈ K of cardinality less than κ and an elementary embedding e : B  A.

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Theorem (Stavi-V. 2002) − If (SR)Cd holds for κ, then there exists a weakly inaccessible cardinal λ ≤ κ.

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Let Rg be the predicate “x is a regular ordinal". Theorem (Bagaria-V. 2016) − If κ satisfies (SR)Rg, then there exists a weakly Mahlo cardinal λ ≤ κ.

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• We cannot hope to get from (SR)Rg more than one weakly Mahlo cardinal ≤ κ. Indeed, starting from a weakly Mahlo

cardinal one obtains a model of set theory in which (SR)Rg holds for the least weakly Mahlo cardinal.

• We cannot hope either to obtain from (SR)Rg that κ is strongly inaccessible, for it can be shown that one can ℵ0 have (SR)Rg for κ = 2 .

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− − There is a condition between (SR)Cd and (SR)Rg, namely − (SR)Cd,WI, where WI is the predicate “x is weakly inaccessible". Proposition (Bagaria-V. 2016) − If κ satisfies (SR)Cd,WI, then there exists a 2-weakly inaccessible cardinal λ ≤ κ.

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• We may also consider predicates α-WI, for α an ordinal. That is, the predicate “x is α-weakly inaccessible". • Then, similar arguments would show that the principle − (SR)Cd, α - WI implies that there is an (α + 1)-weakly inaccessible cardinal ≤ κ.

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Let WC(x, α) be the Π1 relation “α is a limit ordinal and x is a partial ordering with no chain of order-type α". Theorem (Bagaria-V. 2016) − If κ satisfies (SR)Cd,WC, then there exists a weakly compact cardinal λ ≤ κ.

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Recap The following pairs (R, L∗) are symbiotic. • R: Cd ∗ L : Lωω(I), where Ixyϕ(x)ψ(y) ↔ |ϕ| = |ψ| is the Härtig quantifier. • R: Cd ∗ L : Lωω(R), where Rxyϕ(x)ψ(y) ↔ |ϕ| ≤ |ψ| is the Rescher quantifier. • R: Cd ∗ Cd Cd L : Lωω(W ), where W xyϕ(x, y) ↔ ϕ(·, ·) is a well-ordering of the order-type of a cardinal. • R: Cd, WI ∗ WI WI L : Lωω(I, W ), where W xϕ(x) ↔ |ϕ(·)| is weakly-inaccessible. • R: Rg ∗ Rg Rg L : Lωω(I, W ), where W xyϕ(x, y) ↔ ϕ(·, ·) has the order-type of a regular cardinal.

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Recap

• R: Cd, WC ∗ L : Lωω(I, QBr ), where QBr xyϕ(x, y) ↔ ϕ(·, ·) is a tree order of height some α and has no branch of length α. • R: Cd, WC ∗ ¯ ¯ L : Lωω(I, QBr ), where QBr xyuvϕ(x, y)ψ(u, v) ↔ ϕ(·, ·) is a partial order with a chain of order-type ψ(·, ·). • R: Pw L∗: The second order logic L2.

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Structural reflection as a Downward Löwenheim Skolem Theorem

Theorem (Bagaria-V. 2016) Suppose L∗ and R are symbiotic. Then the following are equivalent:

(i) (SR)R holds for κ. (ii) LST (L∗) holds for κ i.e. for any ϕ ∈ L∗ and any A that satisfies ϕ, there exists B ⊆ A of cardinality less than κ that also satisfies ϕ.

The smallest κ such that LST (L) (if any) is called the Löwenheim-Skolem-Tarski number of the logic L.

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Writing ≡ to indicate that the corresponding cardinals are the same, assuming they exist, we have:

1. (SR)Cd ≡ LST (Lωω(I)). WI 2. (SR)Cd,WI ≡ LST (Lωω(I, W )). Rg 3. (SR)Rg ≡ LST (Lωω(W )).

4. (SR)Cd,WC ≡ LST (Lωω(I, QBr )).

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Theorem (Magidor-V. 2011) 1. Relative to the consistency of a supercompact cardinal, it is consistent that LST (Lωω(I)) holds for the first weakly inaccessible cardinal. 2. Relative to the consistency of a supercompact cardinal, it is consistent that the first cardinal with LST (Lωω(I)) is the first supercompact cardinal. 3. If LST (Lωω(I)) holds for κ, then λ fails for all λ ≥ κ, λ,λ fails for ω-cofinal λ ≥ κ, and SCH holds above κ.

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Theorem (Magidor 1971) The LST-number of L2, if exists, is the first supercompact cardinal.

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Weaker Löwenheim-Skolem-type properties

Definition • We say that a cardinal κ has the Löwenheim-Skolem-property for L∗, written LS(L∗), if whenever ϕ ∈ L∗ is such that A |= ϕ for some A, then B |= ϕ for some B such that |B| < κ. • We say that a cardinal κ has the strict Löwenheim-Skolem- Tarski property for L∗, written SLST (L∗), if whenever A is a model and ϕ ∈ L∗ is such that A |= ϕ, and |A| = κ, then there is B ⊆ A such that B |= ϕ and |B| < κ.

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Theorem (Bagaria-V. 2016) If R and L∗ are symbiotic, then ∗ −− 1. κ has the LS(L ) property if and only (SR)R holds for κ. ∗ − 2. κ has the SLST (L ) property if and only (SR)R holds for κ.

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We now have the following: − 1. (SR)Cd ≡ SLST (Lωω(I)). − WI 2. (SR)Cd,WI ≡ SLST (Lωω(I, W )). − Rg 3. (SR)Rg ≡ SLST (Lωω(W )). − 4. (SR)Cd,WC ≡ SLST (Lωω(I, QBr )).

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Theorem (A. Pinus 1978)

If κ is weakly inaccessible, then SLST (Lωω(I)) holds at κ. Corollary

SLST (Lωω(I)) = κ if and only if κ is the first weakly inaccessible cardinal. Theorem (Bagaria-V. 2016) WI If κ is 2-weakly inaccessible, then SLST (Lωω(I, W )) holds at WI κ. SLST (Lωω(I, W )) = κ if and only if κ is the first 2-weakly inaccessible cardinal.

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Theorem (Bagaria-V. 2016) Rg If κ is weakly Mahlo, then SLST (Lωω(I, W )) holds for κ. Rg SLST (Lωω(I, W )) = κ if and only if κ is the first weakly Mahlo cardinal.

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Theorem (Bagaria-V. 2016)

If κ is weakly compact, then SLST (Lωω(I, QBr )) holds for κ. SLST (Lωω(I, QBr )) = κ if and only if κ is the first weakly compact cardinal.

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Theorem (Scott 1961) If κ is a measurable cardinal, then SLST (L2) holds for κ.

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A hierarchy of strong logics

Definition Sort logic is the smallest extension of second order logic which is closed under taking reducts of definable model classes. Iterating taking reducts and complements gives rise to the ∆n-fragment of sort logic.

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A hierarchy of strong logics

Example

• ∆1 is first order logic.

• ∆2 is essentially second order logic.

• ∆3 is essentially the extension of second order logic by the quantifier “there are supercompactly many" (assuming there are a proper class of supercompact cardinals).

• ∆4 is essentially the extension of second order logic by the quantifier “there are extendibly many" (assuming there are a proper class of extendible cardinals).

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Theorem (V. 78) The following conditions are equivalent for any model class K and for any n > 1:

(1) K is definable in the level ∆n of sort logic.

(2) K is ∆n-definable in the Levy-hierarchy.

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The strongest logic

In class theory: Corollary Every (set-theoretically definable) model class is definable in sort logic. Sort logic is therefore the strongest logic.

The logics ∆n, n = 2, 3,..., provide a sequence of stronger and stronger logics, covering eventually every other logic.

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Model theory Set theory

Sort logic First order logic

Second order logic ∆2 in Levy hierarchy ∆2 in sort logic

First order logic with ∆1 in Levy hierarchy the game quantifier

KP Infinitary logic LHYP ∆1 in Levy hierarchy

KPU− First order logic ∆1 in Levy hierarchy

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Definition (Bagaria 2012) (n) Let C be the club class of ordinals α such that Rα ≺n V .

C(n)-supercompact, C(n)-extendible, C(n)-huge, C(n)-superhuge, etc.

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Theorem (V. 1978)

1. The Hanf-numberh n of the logic ∆n is the supremum of Σn-definable ordinals. 3 2. The Löwenheim number `n of the logic ∆n is the supremum of Πn-definable ordinals.

3. The decision problem of the logic ∆n is the complete Πn-definable set of natural numbers.

4. `n+1 = hn for all n > 1. (n) 5. `n < tn < hn for all n > 1, wheret n = min(C ).

3I.e. least cardinal with the Löwenheim-Skolem property 48 / 53 Symbiosis Reflection LST Sort logic

Definition Q(n)xϕ(x) ⇐⇒ |{x : ϕ(x)}| ∈ C(n).

Theorem The following conditions are equivalent for any model class K and for any n > 1: (n) (1) K is definable in the logic ∆(Lωω(I, Q )).

(2) K is ∆n+1-definable in the Levy-hierarchy.

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Proposition (n) LST (∆n+1) holds for every C -supercompact cardinal.

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• Model theory (in this talk) is based on logics, such as first order logic, second order logic, infinitary logic, sort logic, etc. • Logics (however strong) treat mathematical structures up to isomorphism only, there is no preference of one construction of a structure over another, and this is in line with common mathematical thinking about structures.

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• Model theory of strong logics gives rise to interesting reflection principles via the Löwenheim-Skolem-Tarski properties. • Model theory of strong logics provides a natural model theoretic scale for investigating complicated set theoretical properties of models, without going into the nuts and bolts of the constructions of specific structures. • Ultimately, strong logics, and thereby also model theory, and set theory live in a symbiosis.

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Happy Birthday, Joan!

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