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Introduction Modal graph theory Modal model theory Validities Varieties of potentialism Modal model theory as mathematical potentialism Joel David Hamkins Professor of Logic Sir Peter Strawson Fellow University of Oxford University

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Introduction Modal graph theory Modal model theory Validities Varieties of potentialism

Modal model theory as mathematical potentialism

Joel David Hamkins

Professor of Logic Sir Peter Strawson Fellow

University of Oxford University College

Oslo Potentialism Workshop Varieties of Potentialism 23 September 2020

Modal model theory as mathematical potentialism Joel David Hamkins

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Introduction Modal graph theory Modal model theory Validities Varieties of potentialism

This talk includes joint work with: Wojciech Aleksander Wołoszyn, Oxford University

Modal model theory as mathematical potentialism Joel David Hamkins

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Introducing modal model theory

In modal model theory, we consider a mathematical structure within the context of a class of similar structures.

Modal model theory as mathematical potentialism Joel David Hamkins

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Introducing modal model theory

In modal model theory, we consider a mathematical structure within the context of a class of similar structures. A potentialist system is a class of models W with an extension relation M ⊑ N, refining the substructure relation.

Modal model theory as mathematical potentialism Joel David Hamkins

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Introducing modal model theory

In modal model theory, we consider a mathematical structure within the context of a class of similar structures. A potentialist system is a class of models W with an extension relation M ⊑ N, refining the substructure relation. Define the modalities:

1 M thinks ϕ is possible, written M |

= ϕ, if there is an extension M ⊑ N with N | = ϕ.

2 M thinks ϕ is necessary, written M |

= ϕ, if every extension M ⊑ N has N | = ϕ.

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Focus on Mod(T)

A principal case for modal model theory is the class Mod(T) of all models of first-order theory T.

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Focus on Mod(T)

A principal case for modal model theory is the class Mod(T) of all models of first-order theory T. All graphs All groups All fields

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Focus on Mod(T)

A principal case for modal model theory is the class Mod(T) of all models of first-order theory T. All graphs All groups All fields Models of PA. Models of set theory.

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Illustrating the modal vocabulary

Every graph thinks “possibly the diameter is 2.”

Modal model theory as mathematical potentialism Joel David Hamkins

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Illustrating the modal vocabulary

Every graph thinks “possibly the diameter is 2.” Every group is possibly necessarily nonabelian.

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Illustrating the modal vocabulary

Every graph thinks “possibly the diameter is 2.” Every group is possibly necessarily nonabelian. Every field thinks possibly every element has a square root, but this is necessarily not necessary.

Modal model theory as mathematical potentialism Joel David Hamkins

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Distinguish several languages

1 L is language of structures in potentialist system.

Modal model theory as mathematical potentialism Joel David Hamkins

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Distinguish several languages

1 L is language of structures in potentialist system. 2

L closes under , and Boolean connectives.

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Distinguish several languages

1 L is language of structures in potentialist system. 2

L closes under , and Boolean connectives.

3 L

is full first-order modal language, closing under modal

perators, Boolean connectives and quantifiers.

Modal model theory as mathematical potentialism Joel David Hamkins

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Introduction Modal graph theory Modal model theory Validities Varieties of potentialism

Distinguish several languages

1 L is language of structures in potentialist system. 2

L closes under , and Boolean connectives.

3 L

is full first-order modal language, closing under modal

perators, Boolean connectives and quantifiers.

4 L ,@ extends with actuality operator @.

Modal model theory as mathematical potentialism Joel David Hamkins

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Introduction Modal graph theory Modal model theory Validities Varieties of potentialism

Distinguish several languages

1 L is language of structures in potentialist system. 2

L closes under , and Boolean connectives.

3 L

is full first-order modal language, closing under modal

perators, Boolean connectives and quantifiers.

4 L ,@ extends with actuality operator @. 5 P is propositional modal logic. Propositional variables,

Boolean connectives and modal operators.

Modal model theory as mathematical potentialism Joel David Hamkins

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Introduction Modal graph theory Modal model theory Validities Varieties of potentialism

Distinguish several languages

1 L is language of structures in potentialist system. 2

L closes under , and Boolean connectives.

3 L

is full first-order modal language, closing under modal

perators, Boolean connectives and quantifiers.

4 L ,@ extends with actuality operator @. 5 P is propositional modal logic. Propositional variables,

Boolean connectives and modal operators. L assertions are substitution instances of P assertions ϕ(p0, . . . , pn) by L sentences: ϕ(ψ0, . . . , ψn).

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Remarkable expressive power of modal graph theory

The language of modal graph theory has a remarkable expressive power. Let us illustrate this in several instances.

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Theorem 2-colorability is expressible in modal graph theory.

Modal model theory as mathematical potentialism Joel David Hamkins

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Theorem 2-colorability is expressible in modal graph theory. Proof. G is 2-colorable ⇐ ⇒ possibly, there are adjacent nodes r and b, such that every node is adjacent to exactly one of them and adjacent nodes are connected to them oppositely. G

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Theorem 2-colorability is expressible in modal graph theory. Proof. G is 2-colorable ⇐ ⇒ possibly, there are adjacent nodes r and b, such that every node is adjacent to exactly one of them and adjacent nodes are connected to them oppositely. G

→

G r b

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Theorem Connectivity is expressible in modal graph theory.

Modal model theory as mathematical potentialism Joel David Hamkins

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Theorem Connectivity is expressible in modal graph theory.

Proof. Vertex x connected with y ⇐ ⇒ necessarily, any c adjacent to x, with neighbors closed under adjacency, is adjacent to y. x y

→

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Theorem Connectivity is expressible in modal graph theory.

Proof. Vertex x connected with y ⇐ ⇒ necessarily, any c adjacent to x, with neighbors closed under adjacency, is adjacent to y. x y

→

x y c

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Theorem Connectivity is expressible in modal graph theory.

Proof. Vertex x connected with y ⇐ ⇒ necessarily, any c adjacent to x, with neighbors closed under adjacency, is adjacent to y. x y

→

x y c ∀c[(c ∼ x ∧ ∀u, v(c ∼ u ∧ u ∼ v ∧ v = c → c ∼ v)) → c ∼ y].

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Theorem Finiteness is expressible in modal graph theory.

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Theorem Finiteness is expressible in modal graph theory. Proof. G is finite ⇐ ⇒ possibly, there is n, whose neighbor graph is connected and all degree 2 except two vertices of degree 1, and all other nodes are adjacent to distinct neighbors of n.

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Theorem Finiteness is expressible in modal graph theory. Proof. G is finite ⇐ ⇒ possibly, there is n, whose neighbor graph is connected and all degree 2 except two vertices of degree 1, and all other nodes are adjacent to distinct neighbors of n. G

→

G n

start end

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Theorem Countability is expressible in modal graph theory.

Modal model theory as mathematical potentialism Joel David Hamkins

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Theorem Countability is expressible in modal graph theory. Proof. G is countable ⇐ ⇒ possibly, there is ω, with neighbor graph connected and all of degree 2 except one node, and all other nodes adjacent to distinct neighbors of ω.

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Theorem Countability is expressible in modal graph theory. Proof. G is countable ⇐ ⇒ possibly, there is ω, with neighbor graph connected and all of degree 2 except one node, and all other nodes adjacent to distinct neighbors of ω. G · · · → G · · · ω

start

· · ·

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Theorem Size at most continuum is expressible in modal graph theory.

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Theorem Size at most continuum is expressible in modal graph theory. Proof. G has size at most continuum ⇐ ⇒ if we can associate every node in the graph with a distinct subset of ω.

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Theorem Size at most continuum is expressible in modal graph theory. Proof. G has size at most continuum ⇐ ⇒ if we can associate every node in the graph with a distinct subset of ω.

x y ω

1 2 3 4 5 6 7 8 9 10 · · ·

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Much more is expressible in modal graph theory

Size ℵ1, ℵ2, . . .

Modal model theory as mathematical potentialism Joel David Hamkins

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Much more is expressible in modal graph theory

Size ℵ1, ℵ2, . . . Size ℵω, ω.

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Much more is expressible in modal graph theory

Size ℵ1, ℵ2, . . . Size ℵω, ω. Size of the least -fixed point.

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Much more is expressible in modal graph theory

Size ℵ1, ℵ2, . . . Size ℵω, ω. Size of the least -fixed point. The least -hyper-fixed point. Much more.

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Much more is expressible in modal graph theory

Size ℵ1, ℵ2, . . . Size ℵω, ω. Size of the least -fixed point. The least -hyper-fixed point. Much more. It turns out that a large fragment of set-theoretic truth is interpretable in modal graph theory.

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Modal model theory

Let us now begin to develop some of the elementary modal model theory. We focus on the case of Mod(T) for a fixed first-order theory T.

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Two natural accessibility notions in Mod(T)

Direct extension M ⊆ N, for possibility M | = ϕ.

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Two natural accessibility notions in Mod(T)

Direct extension M ⊆ N, for possibility M | = ϕ. natural from potentialist point of view

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Two natural accessibility notions in Mod(T)

Direct extension M ⊆ N, for possibility M | = ϕ. natural from potentialist point of view poor algebraic properties: not convergent, not directed

Modal model theory as mathematical potentialism Joel David Hamkins

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Two natural accessibility notions in Mod(T)

Direct extension M ⊆ N, for possibility M | = ϕ. natural from potentialist point of view poor algebraic properties: not convergent, not directed Embedded extension M ⊂

∼ N and possibility M |

= ϕ.

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Introduction Modal graph theory Modal model theory Validities Varieties of potentialism

Two natural accessibility notions in Mod(T)

Direct extension M ⊆ N, for possibility M | = ϕ. natural from potentialist point of view poor algebraic properties: not convergent, not directed Embedded extension M ⊂

∼ N and possibility M |

= ϕ. mathematically natural, better algebraic properties

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Introduction Modal graph theory Modal model theory Validities Varieties of potentialism

Two natural accessibility notions in Mod(T)

Direct extension M ⊆ N, for possibility M | = ϕ. natural from potentialist point of view poor algebraic properties: not convergent, not directed Embedded extension M ⊂

∼ N and possibility M |

= ϕ. mathematically natural, better algebraic properties Fortunately, the two modalities coincide in Mod(T):

Modal model theory as mathematical potentialism Joel David Hamkins

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Introduction Modal graph theory Modal model theory Validities Varieties of potentialism

Two natural accessibility notions in Mod(T)

Direct extension M ⊆ N, for possibility M | = ϕ. natural from potentialist point of view poor algebraic properties: not convergent, not directed Embedded extension M ⊂

∼ N and possibility M |

= ϕ. mathematically natural, better algebraic properties Fortunately, the two modalities coincide in Mod(T): Theorem M | = ϕ[a] ⇐ ⇒ M | = ϕ[a].

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Two natural accessibility notions in Mod(T)

Direct extension M ⊆ N, for possibility M | = ϕ. natural from potentialist point of view poor algebraic properties: not convergent, not directed Embedded extension M ⊂

∼ N and possibility M |

= ϕ. mathematically natural, better algebraic properties Fortunately, the two modalities coincide in Mod(T): Theorem M | = ϕ[a] ⇐ ⇒ M | = ϕ[a]. Sam Adam-Day proved the two potentialist systems bisimilar.

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L theory determines L theory

Key Lemma In Mod(T) for any first order theory M ≺L N if and only if M ≺

L N.

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L theory determines L theory

Key Lemma In Mod(T) for any first order theory M ≺L N if and only if M ≺

L N.

But it isn’t true for L .

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L theory determines L theory

Key Lemma In Mod(T) for any first order theory M ≺L N if and only if M ≺

L N.

But it isn’t true for L . Lemma M ≡L N if and only if M ≡

L N.

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L theory determines L theory

Key Lemma In Mod(T) for any first order theory M ≺L N if and only if M ≺

L N.

But it isn’t true for L . Lemma M ≡L N if and only if M ≡

L N.

Also not true for L .

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Theorem Every L formula ϕ is equivalent in Mod(T) to an infinitary disjunction of infinitary conjunctions of L-assertions.

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Theorem Every L formula ϕ is equivalent in Mod(T) to an infinitary disjunction of infinitary conjunctions of L-assertions. Proof. Let T be the set of L-theories T of a model M | = ϕ.

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Theorem Every L formula ϕ is equivalent in Mod(T) to an infinitary disjunction of infinitary conjunctions of L-assertions. Proof. Let T be the set of L-theories T of a model M | = ϕ. Since the L theory determines the L theory, ϕ ⇐ ⇒

T∈T
ψ∈T

ψ. It’s not true for L .

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T∈T
ψ∈T

ψ. It’s not true for L . Open Question Is every L assertion equivalent to an assertion of Lω1,ω?

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Quantifier elimination

A theory admits quantifier elimination when every L assertion is equivalent to a quantifier-free assertion.

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Quantifier elimination

A theory admits quantifier elimination when every L assertion is equivalent to a quantifier-free assertion. Theorem If T admits quantifier elimination, then also quantifier/modality elimination and modality trivialization.

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Quantifier elimination

A theory admits quantifier elimination when every L assertion is equivalent to a quantifier-free assertion. Theorem If T admits quantifier elimination, then also quantifier/modality elimination and modality trivialization. Modality elimination means that every modal assertion is equivalent to a modality-free assertion. Modality trivialization means ϕ is equivalent to ϕ.

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Modality trivialization

Modality trivialization means ϕ is equivalent to ϕ.

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Modality trivialization

Modality trivialization means ϕ is equivalent to ϕ. Theorem For any first-order theory T, the following are equivalent:

1 T admits modality trivialization over all assertions in L . 2 T admits modality trivialization over all assertions in

L.

3 T admits modality trivialization over all assertions in L. 4 T is model complete.

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Modality trivialization

Modality trivialization means ϕ is equivalent to ϕ. Theorem For any first-order theory T, the following are equivalent:

1 T admits modality trivialization over all assertions in L . 2 T admits modality trivialization over all assertions in

L.

3 T admits modality trivialization over all assertions in L. 4 T is model complete.

Theory T is model complete if submodels M ⊆ N are elementary M ≺ N.

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Actuality operator

Augment the modal language with an actuality operator @, which allows reference back to original world of evaluation.

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Actuality operator

Augment the modal language with an actuality operator @, which allows reference back to original world of evaluation. In graph theory, ∃x∀y (x ∼ y ↔ (@y ∧ @∀z ¬y ∼ z)) asserts that possibly, there a node adjacent to all and only the isolated nodes of the actual world.

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Actuality operator

Augment the modal language with an actuality operator @, which allows reference back to original world of evaluation. In graph theory, ∃x∀y (x ∼ y ↔ (@y ∧ @∀z ¬y ∼ z)) asserts that possibly, there a node adjacent to all and only the isolated nodes of the actual world. Iterated semantics allow for a notion of relative actuality.

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Modal graph theory with actuality

Appears stronger than modal graph theory

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Modal graph theory with actuality

Appears stronger than modal graph theory can express equinumerosity of neighbor sets can express well-foundedness of coded relations can interpret set-theoretic truth V, ∈.

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Modal graph theory with actuality

Appears stronger than modal graph theory can express equinumerosity of neighbor sets can express well-foundedness of coded relations can interpret set-theoretic truth V, ∈. Open Question Is actuality @ expressible in modal graph theory? We conjecture not, but have no proof.

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Modal validities

A modal assertion ϕ(p1, . . . , pn) is valid at world M in potentialist system W for an allowed language if all substitution instances ϕ(ψ1, . . . , ψn) arising for ψi in that language are true at M in W.

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Modal validities

A modal assertion ϕ(p1, . . . , pn) is valid at world M in potentialist system W for an allowed language if all substitution instances ϕ(ψ1, . . . , ψn) arising for ψi in that language are true at M in W. This is often sensitive to the allowed language of substitution instances, or whether parameters are allowed.

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Easy lower bounds

1 S4 is universally valid in potentialist systems.

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Easy lower bounds

1 S4 is universally valid in potentialist systems. 2 So is the converse Barcan:

∀x ϕ(x) = ⇒ ∀x ϕ(x)

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Easy lower bounds

1 S4 is universally valid in potentialist systems. 2 So is the converse Barcan:

∀x ϕ(x) = ⇒ ∀x ϕ(x)

3 If W is convergent, then S4.2 is valid for L -sentences.

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Easy lower bounds

1 S4 is universally valid in potentialist systems. 2 So is the converse Barcan:

∀x ϕ(x) = ⇒ ∀x ϕ(x)

3 If W is convergent, then S4.2 is valid for L -sentences. 4 If amalgamation, then S4.2 is valid with parameters.

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Easy lower bounds

1 S4 is universally valid in potentialist systems. 2 So is the converse Barcan:

∀x ϕ(x) = ⇒ ∀x ϕ(x)

3 If W is convergent, then S4.2 is valid for L -sentences. 4 If amalgamation, then S4.2 is valid with parameters. 5 If W is linearly pre-ordered, then S4.3 is valid with

parameters.

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Upper bounds via the control statement method

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Upper bounds via the control statement method

Switch: necessarily, s and ¬s.

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Upper bounds via the control statement method

Switch: necessarily, s and ¬s. Button: b.

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Upper bounds via the control statement method

Switch: necessarily, s and ¬s. Button: b. Ratchet: sequence of buttons, each implies previous; can push each without pushing next.

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Upper bounds via the control statement method

Switch: necessarily, s and ¬s. Button: b. Ratchet: sequence of buttons, each implies previous; can push each without pushing next. Railway switch: r and ¬r.

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Upper bounds via the control statement method

Switch: necessarily, s and ¬s. Button: b. Ratchet: sequence of buttons, each implies previous; can push each without pushing next. Railway switch: r and ¬r. Railyard: finite tree of railway switches.

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Upper bounds via the control statement method

Switch: necessarily, s and ¬s. Button: b. Ratchet: sequence of buttons, each implies previous; can push each without pushing next. Railway switch: r and ¬r. Railyard: finite tree of railway switches. Theorem

1 If independent switches, then validities contained in S5.

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Upper bounds via the control statement method

Switch: necessarily, s and ¬s. Button: b. Ratchet: sequence of buttons, each implies previous; can push each without pushing next. Railway switch: r and ¬r. Railyard: finite tree of railway switches. Theorem

1 If independent switches, then validities contained in S5. 2 If buttons+switches, then validities contained in S4.2.

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Upper bounds via the control statement method

Switch: necessarily, s and ¬s. Button: b. Ratchet: sequence of buttons, each implies previous; can push each without pushing next. Railway switch: r and ¬r. Railyard: finite tree of railway switches. Theorem

1 If independent switches, then validities contained in S5. 2 If buttons+switches, then validities contained in S4.2. 3 If long ratchets+switches, then validities contained in S4.3.

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Upper bounds via the control statement method

Switch: necessarily, s and ¬s. Button: b. Ratchet: sequence of buttons, each implies previous; can push each without pushing next. Railway switch: r and ¬r. Railyard: finite tree of railway switches. Theorem

1 If independent switches, then validities contained in S5. 2 If buttons+switches, then validities contained in S4.2. 3 If long ratchets+switches, then validities contained in S4.3. 4 If railyards, then validities are exactly S4.

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Validating S5

Theorem

1 Every model in Mod(T) can be extended to one in which

S5 is valid for L sentences.

2 If T is ∀∃ axiomatizable, then every model can be extended

to one validating S5 for L assertions with parameters.

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Validating S5

Theorem

1 Every model in Mod(T) can be extended to one in which

S5 is valid for L sentences.

2 If T is ∀∃ axiomatizable, then every model can be extended

to one validating S5 for L assertions with parameters. Chains of models argument.

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Theorem A countable graph G validates S5 for L with parameters ϕ(¯ a) → ϕ(¯ a)

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Theorem A countable graph G validates S5 for L with parameters ϕ(¯ a) → ϕ(¯ a) if and only if G is the countable random graph.

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Theorem A countable graph G validates S5 for L with parameters ϕ(¯ a) → ϕ(¯ a) if and only if G is the countable random graph. Theorem A graph G validates S5 for ϕ in L with parameters iff it satisfies the theory of the countable random graph.

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Theorem A countable graph G validates S5 for L with parameters ϕ(¯ a) → ϕ(¯ a) if and only if G is the countable random graph. Theorem A graph G validates S5 for ϕ in L with parameters iff it satisfies the theory of the countable random graph. Theorem G validates S5 for sentences iff G is universal for finite graphs.

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Validities in graphs

Theorem Every graph G validates (for L assertions with parameters) either exactly S4.2 or exactly S5.

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Validities in graphs

Theorem Every graph G validates (for L assertions with parameters) either exactly S4.2 or exactly S5. If it has the finite pattern property, get S5. If not, there are independent buttons and switches, so S4.2.

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General case Mod(T)

Theorem A model M | = T validates S5 for L with parameters ϕ(¯ a) → ϕ(¯ a)

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General case Mod(T)

Theorem A model M | = T validates S5 for L with parameters ϕ(¯ a) → ϕ(¯ a) if and only if M is existentially closed in Mod(T).

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General case Mod(T)

Theorem A model M | = T validates S5 for L with parameters ϕ(¯ a) → ϕ(¯ a) if and only if M is existentially closed in Mod(T). This result explains what was important about the countable random graph.

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Universal S5 is impossible

Theorem If every model in Mod(T) validates S5 for L assertions with parameters, then T is model complete and consequently admits modality trivialization.

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Universal S5 is impossible

Theorem If every model in Mod(T) validates S5 for L assertions with parameters, then T is model complete and consequently admits modality trivialization. So p ↔ p also is valid, and this is not part of S5. Conclusion The validities of Mod(T) cannot be exactly S5.

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Varieties of potentialism

The modal language enables us to express sweeping general principles describing the nature of our potentialist conception.

M M1 M2 M3

Linear inevitability S4.3

M′ M′′

Directed convergence S4.2

M0 M1 M11 M10

Branching possibility S4

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Thank you.

Article is now available: Joel David Hamkins and Wojciech Aleksander Wołoszyn, “Modal model theory,” under review. http://jdh.hamkins.org/modal-model-theory. These slides are available at http://jdh.hamkins.org/modal- model-theory-as-mathematical-potentialism. Joel David Hamkins Oxford

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References

Joel David Hamkins. “A simple maximality principle”. Journal of Symbolic Logic 68.2 (2003),

pp. 527–550. ISSN: 0022-4812. DOI: 10.2178/jsl/1052669062.

arXiv:math/0009240[math.LO]. http://wp.me/p5M0LV-2v. Joel David Hamkins. “The modal logic of arithmetic potentialism and the universal algorithm”. ArXiv e-prints (2018). Under review, pp. 1–35. arXiv:1801.04599[math.LO]. http://wp.me/p5M0LV-1Dh. Joel David Hamkins and Benedikt Löwe. “The modal logic of forcing”. Trans. AMS 360.4 (2008),

pp. 1793–1817. ISSN: 0002-9947. DOI: 10.1090/S0002-9947-07-04297-3.

arXiv:math/0509616[math.LO]. http://wp.me/p5M0LV-3h. Joel David Hamkins and Benedikt Löwe. “Moving up and down in the generic multiverse”. Logic and its Applications, ICLA 2013 LNCS 7750 (2013). Ed. by Kamal Lodaya, pp. 139–147. DOI: 10.1007/978-3-642-36039-8_13. arXiv:1208.5061[math.LO]. http://wp.me/p5M0LV-od. Joel David Hamkins and Øystein Linnebo. “The modal logic of set-theoretic potentialism and the potentialist maximality principles”. Review of Symbolic Logic (2019). DOI: 10.1017/S1755020318000242. arXiv:1708.01644[math.LO]. http://wp.me/p5M0LV-1zC. Joel David Hamkins, George Leibman, and Benedikt Löwe. “Structural connections between a forcing class and its modal logic”. Israel Journal of Mathematics 207.2 (2015), pp. 617–651.

ISSN: 0021-2172. DOI: 10.1007/s11856-015-1185-5. arXiv:1207.5841[math.LO].

http://wp.me/p5M0LV-kf. Modal model theory as mathematical potentialism Joel David Hamkins

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References Joel David Hamkins and W. Hugh Woodin. “The universal finite set”. ArXiv e-prints (2017). Manuscript under review, pp. 1–16. arXiv:1711.07952[math.LO]. http://jdh.hamkins.org/the-universal-finite-set. Joel David Hamkins and Kameryn J. Williams. “The Σ1-definable universal finite sequence”. ArXiv e-prints (2019). Under review. arXiv:1909.09100[math.LO]. Modal model theory as mathematical potentialism Joel David Hamkins