aa r X i v : . [ m a t h . L O ] S e p MODAL MODEL THEORY
JOEL DAVID HAMKINS AND WOJCIECH ALEKSANDER WO LOSZYN
Abstract.
We introduce the subject of modal model theory, where one stud-ies a mathematical structure within a class of similar structures under anextension concept that gives rise to mathematically natural notions of possi-bility and necessity. A statement ϕ is possible in a structure (written ϕ ) if ϕ is true in some extension of that structure, and ϕ is necessary (written ϕ )if it is true in all extensions of the structure. A principal case for us will bethe class Mod( T ) of all models of a given theory T —all graphs, all groups,all fields, or what have you—considered under the substructure relation. Inthis article, we aim to develop the resulting modal model theory. The classof all graphs is a particularly insightful case illustrating the remarkable powerof the modal vocabulary, for the modal language of graph theory can expressconnectedness, k -colorability, finiteness, countability, size continuum, size ℵ , ℵ , ℵ ω , i ω , first i -fixed point, first i -hyper-fixed-point and much more. Agraph obeys the maximality principle ϕ ( a ) → ϕ ( a ) with parameters if andonly if it satisfies the theory of the countable random graph, and it satisfiesthe maximality principle for sentences if and only if it is universal for finitegraphs. Contents
1. Introduction 12. The remarkable expressive power of modal graph theory 23. Some elementary modal model theory 74. A quibble about the accessibility relation 145. Modal validities 166. Maximality principle 197. The interpretative power of modal graph theory 268. Actuality operator @ 329. Set-theoretic and meta-mathematical issues 34References 351.
Introduction
In modal model theory, we consider a mathematical structure within the contextof a class of similar structures, investigating the nature of possibility and necessityas one extends to larger structures. In the general case, we have a potentialistsystem , which is a class W of models in a common language together with anextension relation M ⊑ N refining the substructure relation, and we define thenatural modal operators: Commentary can be made about this article on the first author’s blog athttp://jdh.hamkins.org/modal-model-theory. (1) A model M in W thinks ϕ is possible , written M | = ϕ , if there is anextension M ⊑ N with N | = ϕ .(2) A model M in W thinks ϕ is necessary , written M | = ϕ , if every extension M ⊑ N has N | = ϕ .In modal model theory, we focus particularly on the case of W = Mod( T ), thepotentialist system consisting of all the models of a given first-order theory T ,considered under the submodel relation. We shall consider all graphs, all groups, allfields, or what have you, considering each model in the context of all models of thecorresponding theory. By augmenting the languages with these modal operators,every first-order language and theory thus extends canonically to a modal languageand theory, whose basic model theory and expressive power we aim to investigate.To illustrate the modal vocabulary, observe that in the class of all graphs, everygraph thinks “possibly the diameter is 2,” since any graph can be extended toinclude a new vertex having suitable edges with the previous nodes so as to makethe diameter of the larger graph exactly 2; in the class of all groups, every groupis possibly necessarily non-abelian, since every group is a subgroup of a nonabeliangroup, all of whose further extensions will be non-abelian; and in fields, possiblyevery element is a square, but this statement is necessarily not necessary.For clarity let us be precise about the various distinct but closely related lan-guages we shall treat in this article.(1) We denote by L the language of the structures in the potentialist system W . In the case of Mod( T ), this is the language of the theory T .(2) L is the closure of L under the modal operators , and Boolean con-nectives (but not quantifiers).(3) L is the full first-order modal language, closing L under modal operators,Boolean connectives and quantifiers.(4) L , @ extends the full modal language with the actuality operator @, ex-plained in section 8.(5) P is the language of propositional modal logic, with propositional variables p , q , r , and so on, closed under Boolean connectives and modal operators.The assertions of L are exactly those assertions of L in which no modal operatorfalls under the scope of a quantifier; these are the same as the substitution instances ϕ ( ψ , . . . , ψ n ), where ϕ ( p , . . . , p n ) is a propositional modal assertion in P and each ψ i is an assertion of the first-order language L . Our results in section 3 will showthat much of the classical model theory extends from the base language L to L ,but not to L , which is why we separate out this language fragment. For example,theorem 20 shows that L obeys the L¨owenheim-Skolem theorem in Mod( T ), but L does not in general.The potentialist system terminology was introduced in [HL19], with additionalrelated work in [HW17, HW19, Ham18] analyzing the modal logic of models ofarithmetic and set theory. The modal analysis of forcing extensions in set theoryhad begun earlier with [Ham03] and continued with [HL08, HL13, HLL15].2. The remarkable expressive power of modal graph theory
Let us begin our project by illustrating the remarkable expressive power of themodal language of graph theory. We work in the class of all graphs, where a graphis a set of vertices and an irreflexive, symmetric binary edge relation ∼ . (So, there ODAL MODEL THEORY 3 are no self-edges and no parallel edges.) Let L ∼ be the first-order language of graphtheory, which has only the binary edge relation ∼ . Theorem 1.
In the class of graphs, -colorability is expressible in the modal lan-guage of graph theory. There is a sentence ρ ∈ L ∼ such that for any graph G , G | = ρ if and only if G is -colorable.Similarly, k -colorability is expressible by a sentence ρ k for any finite k .Proof. We claim that a graph G is 2-colorable if and only if possibly, there areadjacent nodes r and b , such that every node is adjacent to exactly one of themand adjacent nodes are connected to them oppositely. G → Gr b
If we think of the neighbors of r as red and the neighbors of b as blue, any such graphextension will be 2-colorable. (Note that this coloring assignment will actually color r blue and b red, since they are neighbors of each other and not of themselves.)Conversely, if the graph is 2-colorable, then we can add new vertices r and b andjoin with edges according to the coloring. A similar idea works for k -colorings withany finite number k . (cid:3) One might notice that in the particular figure of the proof, we needn’t actuallyhave added any nodes at all, since we could have used the bottom two nodes as b and r to realize the property already in the original graph. Theorem 2.
In the class of graphs, connectivity of nodes is expressible in the modallanguage of graph theory. There is a formula χ ( x, y ) in L ∼ expressing that vertex x is connected to vertex y . G | = χ ( x, y ) if and only if x is connected to y in G. Similarly, there is a sentence in L ∼ expressing that the graph as a whole is con-nected.Proof. Vertex x is connected with y if and only if there is a finite path in the graphfrom x to y . This is equivalent to asserting in the modal language that necessarily,any vertex c that is adjacent to x and whose neighbors are closed under adjacencyis also adjacent to y . This can be expressed in the modal language of graph theoryby the formula χ ( x, y ) as follows: ∀ c [( c ∼ x ∧ ∀ u, v ( c ∼ u ∧ u ∼ v ∧ v = c → c ∼ v )) → c ∼ y ] . JOEL DAVID HAMKINS AND WOJCIECH ALEKSANDER WO LOSZYN
To illustrate, consider the graph pictured here at left in black, where vertex x is infact connected to vertex y . x y → x yc Consider an extension as at right with a vertex c that is adjacent to x , whoseneighbors are closed under adjacency. Inductively c will be adjacent to every vertexon the path from x to y , and consequently it will be adjacent to y . Conversely, inany graph where x is not connected to y , we may extend it to a graph with a newnode c adjacent to every vertex in the connected component of x and to no others.This node will be adjacent to x and its neighbors will be closed under adjacency,but it will not be adjacent to y . So the formula will hold in a graph precisely when x is connected to y . For a nonempty graph as a whole to be connected, it shouldsatisfy ∀ x ∀ y χ ( x, y ). (cid:3) Notice that by relativizing the expression, we can express the connectivity of anyparticular definable subgraph of a graph, such as the set of neighbors of a givennode.It is a standard model-theoretic exercise to show that neither 2-colorability norconnectivity for graphs is expressible in the ordinary language of graph theory, andso these observations show that the expressive power of modal graph theory L ∼ strictly exceeds first-order graph theory. Let us continue—far more is expressiblewith the modal vocabulary. Theorem 3.
In the class of graphs, finiteness is expressible in the modal languageof graph theory. There is a sentence φ ∈ L ∼ such that for any graph G , G | = φ if and only if G is finite.Proof. A graph G is finite if and only if possibly, there is a point n , whose neighborgraph is connected, with all vertices of degree 2 within that neighbor set, exceptexactly two vertices of degree 1 in that neighbor set—a starting node and an endingnode—and all other nodes of the graph are adjacent to exactly one neighbor of n in a bijective correspondence. The neighbors of n will form a finite chain from thestarting vertex to the ending vertex, and the bijection will show that the graph isfinite. G → G n start end
Since the neighbor graph of vertex n is connected, it cannot have an infinite numberof neighbors, for when proceeding successively from the start node through the in-termediate nodes of degree 2, one must come to the end node in finitely many steps, ODAL MODEL THEORY 5 for otherwise the start and end nodes will occupy different connected components,contrary to assumption. So n really does have only finitely many neighbors underthose assumptions, and so the graph will be finite.Note that when employing the “possibly” operator, the original graph also mayget larger, and so the bijection is not necessarily talking only about the originalgraph in the extension with n , but possibly an extension of the original graph. G n start end
But since a set is finite if and only if it has a finite extension, this is not a problemfor the assertion. (cid:3)
Corollary 4.
In the class of graphs, for a vertex to have finite degree is a propertyexpressible in the modal language of graph theory. There is a formula η ( x ) in L ∼ that holds of a node x in a graph G if and only if x has finite degree in G .Proof. This is a consequence of the previous theorem, by relativizing to the sub-graph consisting of the neighbors of x . That is, η ( x ) asserts that possibly, thereis a node n whose neighbors are connected and all of degree 2 except a startingnode and an ending node, such that every neighbor of x is adjacent to a distinctneighbor of n . (cid:3) Theorem 5.
In the class of graphs, countability is expressible in the modal languageof graph theory. There is a sentence σ ∈ L ∼ such that for any graph G , G | = σ if and only if G is countable.Proof. A graph G is countable if and only if possibly, there is a point ω , whoseneighbor graph is connected, with all vertices of degree 2 except exactly one startingnode with degree 1 (amongst the neighbors of ω )—so that these neighbor verticesform an infinite linked chain from the starting node—and furthermore, all othernodes in the graph are adjacent to distinct neighbors of ω . G · · · → G · · · ω start · · · Note that the “possibly” clause includes the finite graphs. (cid:3)
One can express “countably infinite” by saying that the graph is countable butnot finite. And uncountability is expressible simply as “not countable.” Let usextend this by showing also that having size at most continuum is expressible.
JOEL DAVID HAMKINS AND WOJCIECH ALEKSANDER WO LOSZYN
Theorem 6.
In the class of graphs, the property of having size at most continuum c is expressible in the modal language of graph theory. There is a sentence ψ ≤ c ∈ L ∼ such that for any graph G , G | = ψ ≤ c if and only if | G | ≤ c . Proof.
A graph G has size at most continuum if and only if we can associate everynode in the graph with a distinct subset of ω . This is expressible in the modallanguage of graph theory as follows. The sentence ψ ≤ c asserts that possibly, thereis a node ω whose neighbors form a connected subgraph in which every node hasdegree 2 except one initial node of degree 1 (within that neighbor graph), suchthat all other nodes are adjacent to distinct subsets of the neighbors of ω . In otherwords, for any two distinct nodes x and y not among ω and its neighbors, there isa neighbor n of ω such that x is adjacent to n if and only if y is not adjacent to n ,as indicated in the figure here: x yω · · · n In this way, every node of the original graph G is associated with a distinct set ofthe neighbors of ω , and so G will have size at most continuum. (cid:3) Theorem 7.
In the class of graphs, the property of having size exactly continuum c is expressible in the modal language of graph theory. There is a sentence ψ c ∈ L ∼ such that for any graph G , G | = ψ c if and only if | G | = c . Proof.
This is a little subtler than one might expect—the difficulty is that whenone uses the possibility operator, new vertices can be added to the original graph,and there is no way of telling which points are new and which are original. (This isthe main point of the actuality operator @ discussed in section 8.) Nevertheless, wecan express the property as follows. A graph G has size continuum if and only if ithas size at most continuum and necessarily, if the graph consists of a node A and itsneighbors (as in red below), then necessarily, in any extension having two connectedcomponents (the other being a node B and its neighbors, as in brownish red), suchthat the union has size at most continuum, then necessarily, in any extension inwhich that is exhibited by an association as above of nodes with subsets of ω , thenpossibly, in a further extension in which that remains true, there is another copyof ω and a new association of the neighbors of A with distinct subsets of it, in such ODAL MODEL THEORY 7 a way that every pattern for that copy of ω is realized by a node adjacent to A . G A B
The point is that if G does have size continuum, then we can find the second desiredbijection with the subsets of ω . And if G does not have size continuum, then wemay extend G by adding a node A adjacent to every node in G and then add anew node B along with continuum many neighbors, associating these with subsetsof ω in such a way so as to use up all the possible subsets—this will prevent A from getting any new neighbors in an extension in which the neighbors of A and B are still associated with distinct subsets, since all the patterns are used up. Andsince there are fewer than continuum many neighbors of A , they cannot make anassociation with subsets of a second copy of ω in such a way that every pattern isrealized. (cid:3) We have made a good start in this section on illustrating the expressive powerof modal graph theory, but we shall actually extend these results much further insection 7, showing that having size ℵ ω or i ω is expressible, as is having size the first i -fixed point or the next or indeed the first i -hyperfixed-point and much more.3. Some elementary modal model theory
Let us now begin to develop some of the more general elementary theory of modalmodel theory, working in Mod( T ) for an arbitrary first-order theory T , beginningwith the fact that L -isomorphisms preserve L truths. Renaming lemma 8.
For any first-order theory T , isomorphisms of models of T as L -structures preserve L truth in Mod( T ) . That is, if π : M ∼ = N is anisomorphism of models of T , then for any modal assertion ϕ ∈ L , we have M | = ϕ [ a ] ⇐⇒ N | = ϕ [ π ( a )] . Proof.
The claim of the lemma is clearly true for assertions ϕ in the base language L . And it is clearly preserved by Boolean combinations and also quantifiers. Whatremains is to check the preservation by modal operators. But any extension of M can be translated to a corresponding isomorphic extension of N , preservingand extending the isomorphism. And so by induction the lemma holds for allassertions ϕ . (cid:3) Next, we show that L -elementarity is the same as L -elementarity. Observation21 will show that this isn’t generally true for L -elementarity. Key Lemma 9.
In the class of models
Mod( T ) of a first-order theory T , M ≺ L N if and only if M ≺ L N. Proof.
We shall show by induction on formulas ϕ that if M ≺ L N , then M | = ϕ [ a ] ⇐⇒ N | = ϕ [ a ] JOEL DAVID HAMKINS AND WOJCIECH ALEKSANDER WO LOSZYN for every ϕ ∈ L , for all models. This equivalence is immediate when ϕ is anassertion in the base language L , and it is clearly preserved through Boolean com-binations. All that remains for ϕ ∈ L is preservation through modal operators.The downward preservation of possibility is immediate, since if N | = ϕ [ a ], thenalso M | = ϕ [ a ], because M ⊆ N and so any extension of N also extends M . Toshow the upward preservation of possibility, suppose that M | = ϕ . So there is anextension M ⊆ H with H | = ϕ . Consider the theory ∆( H ) ∪ ∆ ( N ) consisting ofthe elementary diagram of H together with the atomic diagram of N . That is, wetake every L -assertion ϕ ( a , . . . , a n ) true in H , using constants for every element of H , together with the atomic and negated atomic truths ϕ ( b , . . . , b m ) of N , againusing constants for every element of N . This theory is finitely consistent, we claim,using the fact that M ≺ N . To see this, observe that if ϕ ( a , . . . , a n ) is true in H and ϕ ( b , . . . , b m ) is true in N , where ϕ is a conjunction of atomic and negatedatomic assertions, then M | = ∃ ¯ x ϕ (¯ x ) and so we may interpret the constants b i inside M and hence inside H , thereby realizing the conjunction ϕ (¯ a ) ∧ ϕ (¯ b ) in H .So the theory is consistent, and therefore we have a model H ′ | = ∆( H ) ∪ ∆ ( N ).It follows that H ′ is an elementary extension of an embedded copy of H , and adirect extension of a copy of N . We may assume H ≺ L H ′ , and so by the inductionhypothesis we know H ′ | = ϕ [ a ]. Since N embeds into H ′ while fixing a , we thereforealso have N | = ϕ [ a ]. (cid:3) We can use the key lemma to prove a version of it for mere elementary equiva-lence, as follows:
Theorem 10.
In the class of models
Mod( T ) of a first-order theory T in language L , the L -theory of a model determines its L -theory: M ≡ L N if and only if M ≡ L N. Proof.
Suppose that M ≡ L N for models of T . It follows that the two modelshave a common elementary extension, a model M + into which both M and N embed L -elementarily. By the key lemma and the renaming lemma, both of theseembeddings are also L -elementary, and so the L -theories of M and N must bethe same. (cid:3) Corollary 11. If T is a complete first-order theory in language L , then in Mod( T ) every L sentence σ is either true in all models of T or none. In this case everytrue L sentence σ is necessarily true: σ ↔ σ .Proof. If T is complete, then all models in Mod( T ) are L -elementarily equivalent,and so by theorem 10 they are also L -elementarily equivalent. So any L sentenceeither holds in all models of T or none. (cid:3) We can prove a version of theorem 10 for types.
Theorem 12.
In the class of models
Mod( T ) of a first-order theory T in language L , the L -type of individuals ¯ a = ( a , . . . , a n ) in a model M determines the L -typeof those individuals in that model.Proof. Suppose that ¯ a and ¯ b have the same L -type in a model M | = T . We mayfind an elementary extension M ≺ M for which M is sufficiently homogeneous, sothat some automorphism of M moves ¯ a to ¯ b . By the key lemma, the L -type of¯ a in M is the same as in M , which by the renaming lemma is the same as the ODAL MODEL THEORY 9 L -type of ¯ b in M , which by the key lemma again is the same as the L -type of¯ b in M . (cid:3) Since theorem 10 shows that the L theory of a model is determined completelyby its L -theory, one might take this to suggest that L assertions are actually L -expressible. But this isn’t true in general, since we have already seen in theorem 1that L can express 2-colorability of graphs, whereas this concept is not expressiblein the base language. Nevertheless, the 2-colorability of a graph is a consequenceof the theory of the graph, since it is equivalent to the nonexistence of odd-lengthcycles. In the general case what we get is the following: Theorem 13.
In the class
Mod( T ) of all models of a first-order theory T in lan-guage L , every L formula ϕ ( x ) is equivalent to an infinitary disjunction of in-finitary conjunctions of L -assertions.Proof. Let us first consider the case that ϕ is a L sentence. Consider all thevarious models M | = T in which ϕ holds. The theorem shows that this dependsonly on the L -theory of M , since any other model with that same theory must alsosatisfy ϕ . That is, ϕ holds in a model N if and only if the L -theory of N is one ofthe theories of a model in which ϕ holds. Let T be the set of all the L -theories T that are the L -theories of such a model M , one in which T + ϕ holds. By theorem10, the sentence ϕ is equivalent to the disjunction of these theories T , like this: ϕ ⇐⇒ _ T ∈T ^ ψ ∈ T ψ. This is an infinitary disjunction of infinitary conjunctions of L sentences.Now suppose that ϕ ( x ) is a formula, with free variables. Let L + be the languagewhere x is considered a constant symbol rather than a variable symbol. Everymodel of T in L has diverse expansions to L + simply by interpreting x by anelement. Evaluating a formula with a free variable x under a valuation of thatvariable amounts to the same thing as interpreting the corresponding sentence ina model in which x is regarded as a constant symbol. So the models of Mod( T ) in L + are the same thing as models in the language L under a valuation assigning x to an individual of the model. But in Mod( T ) considered in the language L + , wemay apply the analysis of the previous paragraph to realize ϕ ( x ) as an infinitarydisjunction of infinitary conjunctions of L + -assertions. So ϕ ( x ) is equivalent to adisjunction of conjunctions of formulas of L . (cid:3) Perhaps part of what is going on in theorem 13 is the following:
Theorem 14. If T is a theory in a first-order language L having only nonemptymodels and L + is an expansion of L , then for any modal assertion ϕ in L , a model M of T satisfies ϕ ( a ) in Mod L ( T ) if and only if every expansion M + of it also doesso in Mod L + ( T ) . In this theorem, we have two different versions of the potentialist system Mod( T ),depending on whether you consider L -structures or L + -structures. The point is thatthe two systems have the same modal truths for assertions in the unexpanded modallanguage L . Proof.
Every L -model M | = T has expansions to the language L + , and what weclaim is that if M + is an expansion of M to the language L + , then M | = ϕ ( a ) ⇐⇒ M + | = ϕ ( a ) . This we prove by induction on ϕ . The claim is clear for L assertions, since ϕ doesnot mention any of the new vocabulary of L + . And it is preserved by Booleancombinations and quantifiers, since the domains of M and M + are the same. Whatremains is the modal operator. If M | = ϕ ( a ), then there is an extension M ⊆ N with N | = ϕ ( a ). We may find an expansion of N to a model N + extending M + , simply by interpreting the extra structure first on M as in M + , and theninterpreting it on the rest of N . By the induction hypothesis, N + | = ϕ ( a ), andso M + | = ϕ ( a ). Conversely, if M + | = ϕ ( a ), then there is some extension N + of M + satisfying ϕ ( a ). The reduct N of N + to the L language will thereforesatisfy ϕ ( a ) by the induction hypothesis, and this is an extension of M . So M | = ϕ ( a ). (cid:3) In a countable language L , there are at most continuum many theories T com-pleting T , and so theorem 13 provides an at-most-continuum size disjunction ofcountable conjunctions of L sentences. By considering the negation, we would alsoget a representation of ϕ as a size-continuum conjunction of countable disjunctionsof L -sentences. Can these infinitary disjunctions and conjunctions be reduced tocountable? We suspect not in general. Question 15.
In the class
Mod( T ) for a first-order theory T in a countable lan-guage L , is every L assertion equivalent to an assertion of L ω ,ω ? But there are some important cases where the answer is actually positive, in avery strong way. Recall that a theory T admits quantifier elimination with respectto a language if every assertion in that language is equivalent in Mod( T ) to aquantifier-free formula. Similarly, a theory T admits modality elimination withrespect to a language, if every assertion in that language is equivalent in Mod( T )to a modality-free assertion. A stronger notion would be the modality trivialization ,which holds over a language if ϕ ( x ) is equivalent specifically to ϕ ( x ) in Mod( T )for every ϕ in that language. Theorem 16.
If a theory T admits quantifier elimination with respect to its lan-guage L , then it admits simultaneous quantifier and modality elimination—everyassertion in L is equivalent in Mod( T ) to a Boolean combination of atomic formu-las. Furthermore, the theory admits modality trivialization over L , so that ϕ ( x ) is equivalent to ϕ ( x ) for any assertion ϕ ∈ L .Proof. Let us prove by induction on formulas that every formula ϕ ( x ) in the fullmodal language L is equivalent in Mod( T ) to a quantifier/modality-free assertion.Since T admits quantifier elimination, we know this is true for the L formulas. Andthis property is preserved by Boolean connectives, as well as by quantifiers, whichcan subsequently be eliminated. So suppose that ϕ ( x ) is quantifier/modality-freeand consider ϕ ( x ). This holds in a model M if and only if there is an extension M ⊆ N | = T with N | = ϕ [ a ]. But since ϕ is quantifier/modality-free, this will beabsolute back to the original model, so M | = ϕ [ a ], as desired.It now follows generally that ϕ ( x ) is equivalent to ϕ ( x ), since if ψ ( x ) is thequantifier/modality-free formula equivalent to ϕ ( x ), then ϕ ( x ) is equivalent to ODAL MODEL THEORY 11 ψ ( x ), which is equivalent to ψ ( x ) since ψ is quantifier/modality-free, and this isequivalent to ϕ ( x ). So ϕ ( x ) is equivalent to ϕ ( x ) for any ϕ ( x ) in L . (cid:3) A theory T is model complete if every instance of the submodel relation M ⊆ N for models of T is an elementary submodel M ≺ N . This is a weakening of quantifiereliminability, for it is equivalent to the property that every formula in L is equivalentin models of T to a universal assertion [Hod93, theorem 8.3.1]. Theorem 17.
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.Proof. (1 → →
3) Immediate.(3 →
4) Assume that modalities trivialize in Mod( T ) over assertions in L . Sup-pose that M ⊆ N are models of T . If N | = ϕ ( a ) for some a ∈ M , then M | = ϕ ( a ),which by modality trivivialization means that M | = ϕ ( a ), and so M ≺ N . So thetheory is model complete.(4 → T is model complete. We shall show by induc-tion on formulas that ϕ ( x ) is equivalent in Mod( T ) to ϕ ( x ) for every assertion ϕ ∈ L . Suppose that the claim is true for ϕ and all subformulas of ϕ . It followsthat ϕ ( x ) is equivalent to the modality-free assertion ϕ ∗ ( x ), obtained from ϕ bydeleting all its modal operators. Thus, ϕ ∗ is an assertion of L . If M | = ϕ ( a ), thenthere is some extension M ⊆ N with N | = ϕ ( a ), which by our assumption meansthat N | = ϕ ∗ ( a ). By the model completeness of T , it follows that M | = ϕ ∗ ( a ), andconsequently also M | = ϕ ( a ). So the modality has trivialized, as desired. (cid:3) The following example shows that modality elimination is a strictly weaker prop-erty than modality trivialization.
Observation 18.
There is a theory T that is not model complete and consequentlydoes not admit modality trivialization, but it does admit modality elimination—every assertion in L is equivalent in Mod( T ) to an assertion in L .Proof. Let T be the theory of a dense linear order having a least and greatest ele-ment. This is not model complete, because one can have a suborder M ⊆ N whichchanges the particular element that is least or greatest, and so this will not be anelementary submodel. So by theorem 17 the theory does not admit modality trivial-ization. But we claim that the theory does admit modality elimination. Indeed, weclaim that every assertion in L is equivalent to a Boolean combination of atomicassertions and assertions that particular variables are least or greatest. These kindsof assertions are closed under Boolean combinations, and one can handle the quan-tifier case just as in the elimination of quantifiers argument for endless dense linearorders. Finally, consider ϕ ( x , . . . , x n ), where ϕ is such a Boolean combination.By putting ϕ into disjunctive normal form and then distributing the over thedisjunction, we may assume that ϕ is a conjunction of atomic or negated atomicassertions and assertions that particular variables x i are or are not least or greatest.But true atomic facts are necessarily true; an element x is possibly least if and onlyif it is actually least; it is possibly greatest if and only if it is actually greatest; andevery x is possibly not least and possibly not greatest. These observations allow usto express ϕ without the modal operator. (cid:3) Theorem 17 shows that modality trivialization over L is equivalent to trivial-ization over L and L . The corresponding fact for modality elimination is thefollowing: Theorem 19.
Every first-order theory T admits modality elimination for assertionsin L if and only if it does so for L .Proof. The converse implication is immediate. For the forward implication, supposethat T admits modality elimination for L -assertions. We now prove inductivelythat every assertion ϕ ( x ) of L is equivalent in Mod( T ) to an L -assertion ψ ( x ). Thisis true for the L -assertions themselves, and it is preserved by Boolean combinationsand quantifiers. Also, it is preserved by the modal operator , precisely because ψ ( x ) is an assertion of L , if ψ ∈ L , and we have assumed that T eliminatesmodalities for L assertions. (cid:3) The
L¨owenheim number of a language with respect to a class of models is thesmallest cardinal κ such that any assertion ϕ of that language that is satisfiable ina model of the class is satisfiable in such a model of size at most κ . Theorem 20.
For any first-order theory T , the upward and downward L¨owenheim-Skolem theorems hold in Mod( T ) with respect to L , using the same cardinals asfor L . That is, every model M | = T has L -elementarity extensions and sub-models of all the same cardinalities as it does for L -elementarity. Consequently theL¨owenheim number of L with respect to Mod( T ) is the same as for L with respectto Mod( T ) .Proof. This is immediate from the key lemma and the classical L¨owenheim-Skolemtheorem, since the key lemma shows that L -elementarity is the same as L -elementarity. So any L -elementary substructure or extension is also L -elementary. (cid:3) Let us now begin to observe that many of the properties we have proved for theintermediate modal language L do not extend to the full modal language L .The main lesson seems to be that L is in many respects closer to L than it is to L . Nevertheless, some classical model theoretic principles fail even for L . Observation 21.
The equivalence stated in the key lemma does not hold generallyfor elementarity in the full modal language L . In modal graph theory, there can beelementary substructures that are not L -elementary and not even L -elementarilyequivalent.Proof. Consider the class of all graphs. By the L¨owenheim-Skolem theorem, wecan have M ≺ L N , where M is a countable graph and N is uncountable. Butby theorem 5, countability is L ∼ expressible for graphs, showing the failure of M ≺ L ∼ N and indeed of M ≡ L ∼ N . (cid:3) The same example establishes the following:
Observation 22.
The equivalence stated in theorem 10 does not generally extendto equivalence in the full modal language L . In modal graph theory, there can beelementarily equivalent models that are not L -elementarily equivalent. We are not sure whether corollary 11 extends to the full modal language L . ODAL MODEL THEORY 13
Question 23.
Is there a complete first-order theory T for which not all models in Mod( T ) have the same L theory? One candidate may be to take T as the theory of true arithmetic, and then arguesomehow that in a nonstandard model of T one can define the standard cut inthe modal language and be thereby enabled to make assertions about the standardsystem of the model in L . But we have not yet been able to make this idea work. Observation 24.
Theorem 12 does not hold generally for types in the full modallanguage L —in modal graph theory, the L -type of an individual in a model of T does not necessarily determine the L type of that individual in Mod( T ) .Proof. Consider the class of graphs. Let G be a graph consisting of two stars,one with center vertex a having countably infinitely many neighbors and one withcenter vertex b having uncountably many neighbors, and no other vertices or edges. a b The types of a and b are the same in the language of graph theory, but they are notthe same in the language of modal graph theory, which can express the fact that a has only countably many neighbors, while b has uncountably many. (cid:3) Observation 25.
The statement of theorem 13 does not generally hold for asser-tions in the full modal language L . In modal graph theory there are L -expressibleassertions that are not equivalent to any infinitary Boolean combination of L as-sertions.Proof. Theorem 5 shows that countability is expressible in modal graph theory, butthe assertion that the graph is uncountable cannot be equivalent to any infinitaryBoolean combination of sentences in the language of graph theory, since the truth ofsuch assertsion is preserved from an uncountable model to a countable elementarysubstructure. (cid:3)
Observation 26.
The conclusion of theorem 20 does not hold generally for the fullmodal language L . In modal graph theory, no uncountable graph has a countable L -elementary substructure, and the L¨owenheim number for modal graph theory inthe language L ∼ , if it exists, is enormous.Proof. The first part of this is an immediate consequence of the fact that count-ability is expressible in L , as shown in theorem 5. Every countable submodel willsatisfy the countability assertion, but the whole uncountable graph will not.The L¨owenheim number of modal graph theory must be at least as large as everycardinal κ whose size is expressible in modal graph theory. By the results of section2, these cardinals reach to the continuum, and in section 7 we shall prove that theyexceed the first i -fixed point and indeed the first i -hyper-fixed point. (cid:3) Observation 27.
The compactness property can fail for L -theories (and hencealso for L theories). Specifically, there is a theory T in the language of modal graphtheory L ∼ that is finitely realized in the class of graphs but not fully realized. Proof.
Let T assert that the graph is not 2-colorable, but that there are no cyclesof length 1, none of length 2, none of length 3, and so on—there are no cycles ofany particular finite length n . This theory is expressible in L ∼ , since theorem 1shows that 2-colorability is expressible in L ∼ , and the assertion that there is no n -cycle is expressible in L ∼ as a separate assertion for each n .This theory is finitely satisfiable, because if only finitely many sizes of cycles areruled out, we can satisfy the assertion with a large enough odd cycle, which willnot be 2-colorable. The theory altogether, however, is not satisfiable, since everycycle-free graph is 2-colorable. One can color every node by the parity of the lengthof the shortest path to a given fixed node in each connected component. (cid:3) One can also naturally construct a violation of compactness using the fact thatfiniteness is expressible, since one can consider the theory asserting that the graph isfinite, but has size at least n for every particular finite n . This theory would indeedbe finitely realizable and not realizable in the class of graphs. Since finiteness isexpressible only in L ∼ , however, this way of arguing would get the violation ofcompactness only in this stronger language, rather than in the weaker fragment L as we have above.Although we have presented these examples as violations of compactness, thereis another sense in which we have not violated compactness here. Namely, tosatisfy an assertion of the modal language does not mean just to provide a model M in which it is true in the potentialist system Mod( T ), but rather, to provide awhole new potentialist system, a class of models with an accessibility relation, notnecessarily Mod( T ) under direct extensions. The observation above, in contrast, isabout realizing a theory in a graph of Mod( T ), but always working in this sameKripke model. For this reason, one might look upon the counterexamples as akin toviolations of saturation in this Kripke model, rather than violations of compactnessfor the modal logic. Observation 28.
The Lo´s theorem for ultraproducts can fail for assertions inthe modal language L . Specifically, in modal graph theory there is a statement ϕ ∈ L ∼ that holds in some graphs G n , but fails in every nonprincipal ultraproduct Q n G u /U .Proof. Let the graphs G n be odd-length cycles of increasing size. None of thesegraphs is 2-colorable, but the ultraproduct consists of uncountably many discon-nected Z -chains, and this is 2-colorable. (cid:3) A quibble about the accessibility relation
We should like to dispense with a certain issue about the accessibility relation inMod( T ), for there are actually two natural accessibility relations for Mod( T ). Onthe one hand, it is natural to define possibility as we have above, as direct extensionpossibility , using the submodel or direct extension relation M ⊆ N , defining that M | = ϕ [ a ] if and only if there is a model N with M ⊆ N and N | = ϕ [ a ].This accessibility relation is very natural from the perspective of potentialism, bywhich one views each of the various models as a fragment approximating the largeruniverse to which they are building. On this account, individuals come into actualexistence in a world and then persist through all subsequent larger worlds.On the other hand, it is also natural to define a variant of this possibility notionusing embeddability M ⊂ ∼ N in place of direct extensions M ⊆ N . Namely, the ODAL MODEL THEORY 15 embedded extension possibility operator is defined by: M | = ϕ [ a ] if and only ifthere is an embedded extension M ⊂ ∼ N , that is, a model N with an embedding j : M → N , for which N | = ϕ [ j ( a )]. Thus, every individual in M has a counterpartin N via the embedding. Notice that there may be more than one embedding j of M into N , and so the models M and N alone do not necessarily determine the coun-terpart correspondence of their individuals, but rather the particular embedding j : M → N .As an accessibility relation, the embedded extension relation M ⊂ ∼ N exhibitsfar better algebraic closure properties than direct extension M ⊆ N . For example,in nontrivial cases the direct extension relation M ⊆ N is not directed, and neitherdoes it exhibit amalgamation or convergence (see definitions in section 5), since twodifferent extensions might happen to use the same individual object in incompatibleways. But the embedded extension relation often has all of these features, sincethe particular identity of individuals no longer matters when they are embeddedinto another common structure. From this perspective, the embedded extensionrelation is often more robust algebraically.In terms of modal assertions, we are glad to say that in light of theorem 29the entire issue is moot, because although as accessibility relations the notions aredifferent, nevertheless it turns out that the corresponding modal operators coincide.In this sense, we can equivalently think of possibility with respect to either notion,and the result will be the same. Theorem 29. In Mod( T ) for any first-order theory, direct extension possibilityand embedded extension possibility are equivalent for assertions ϕ in the full modallanguage L with parameters from M . M | = ϕ [ a ] ⇐⇒ M | = ϕ [ a ] Similarly, direct extension necessity and embedded extension necessity are also equiv-alent. M | = ϕ [ a ] ⇐⇒ M | = ϕ [ a ] Proof.
We aim to prove that M | = ϕ [ a ] if and only if M | = ϕ [ a ]. The forwarddirection is immediate, since direct extensions M ⊆ N are instances of embeddedextensions j : M → N using the inclusion embedding j ( x ) = x . Conversely, if M | = ϕ [ a ], then there is some embedded extension j : M → N with N | = ϕ [ j ( a )].The model M is isomorphic via j with the range of the embedding M ′ = j " M , andthis model directly extends to N . So M ′ | = ϕ [ j ( a )], and therefore M | = ϕ [ a ]by the renaming lemma, as desired. The necessity case now follows by duality. (cid:3) It follows inductively that complex formulas involving and have the samemeaning as the corresponding assertions using and .Going beyond this, Sam Adam-Day has proved (in forthcoming work) that thetwo potentialist systems are actually bisimilar, which completely explains why theyexhibit the same modal truths.
Theorem 30 (Adam-Day) . For any first-order theory T , the potentialist systemsconsisting of Mod( T ) under the direct extension relation or under the embeddedextension relation, are bisimilar. Therefore, they exhibit exactly the same modaltruths. Modal validities
One of the central research tasks of modal model theory is to discover the modalprinciples and validities that hold in a given potentialist system and the models in it.This has been the focus of prior work on set-theoretic potentialism [Ham03], [HL08,HL13], [HLL15], [HL19], [HW19], [HW17], and arithmetic potentialism [Ham18].In modal model theory generally, we should like to move beyond set theory andarithmetic to the models of any given theory, to graphs, groups, fields, orders, andwhat have you. We aim to discover which modal principles are valid in Mod( T ) fora first-order theory T , considering this as a potentialist system. A key definition isthe following: Definition 31.
A modal assertion ϕ ( p , . . . , p n ), with propositional variables p i ,is valid at a world M in a potential system W for an allowed language of instances,if all substitution instances ϕ ( ψ , . . . , ψ n ) arising for ψ i in that language are trueat M in W .Thus, a modal validity ϕ ( p , . . . , p n ) stands as a template scheme for all themodal truths ϕ ( ψ , . . . , ψ n ) that arise by substituting the propositional variables p i with allowed substitution assertions ψ i . Whether a modal assertion is valid or notis often highly sensitive to the particular language of allowed substitutions ψ , forinstance, whether we are allowing ψ only from the language L , or from L or L ,or whether parameters from the model M are allowed. When speaking of a modalvalidity, therefore, one must take care to clarify precisely the class of substitutioninstances for which validity is asserted.For this reason, we should particularly like to emphasize that when we say amodal theory such as S4.2 is valid in a model M of a potentialist system, we aretreating S4.2 as a propositional modal theory, that is, as a set of assertions inpropositional modal logic P , rather than as a logic, as a proof system with rulesof inference. In general, because the validities are sensitive to the precise classof substitution instances and the allowed parameters, they do not always interactwell with the inference rules typically used when defining a modal logic, if thoserules would be taken to be applicable generally in the broader predicate-logic modalcontext with variables and quantifiers. For example, a modal principle ϕ ( a ) canbe valid at a model M using any parameter a from that model, but not valid insome extension M ⊆ N using a parameter from the extension N (see remarks aftertheorem 39 for a specific example). In this case, ∀ x ϕ ( x ) will be valid at M , butnot ∀ x ϕ ( x ), and therefore the validities at M are not closed under necessitation,even when the modal theory S5 is valid there in the sense of definition 31.Let us begin by reviewing some easy lower bounds. Consider the potentialistsystem consisting of the class Mod( T ) of all models of a fixed first-order theory T .Before stating the theorem, we recall some useful terminology, particularly on thedistinction between convergence and amalgamation. The class of models Mod( T )exhibits embedding convergence , if for any model M and embeddings M ⊂ ∼ N and M ⊂ ∼ N , then there is a model N with N ⊂ ∼ N and N ⊂ ∼ N . Convergenceis thus a form of local directedness, since it asserts that models N and N havea common embedding extension, provided that these models are themselves em-bedding extensions of a common model M . Note that for convergence there is norequirement that the diagram of embeddings commutes. This is the key difference ODAL MODEL THEORY 17 between convergence and amalgamation and is why convergence leads only to va-lidities for sentences rather than also for assertions with parameters. Namely, theclass of models has amalgamation , if whenever j : M → N and j : M → N are embeddings, there there are embeddings h : N → N and h : N → N to acommon model N , such that the diagram commutes, meaning h ◦ j = h ◦ j . Theorem 32. (1) The modal theory S4 is valid at every model in Mod( T ) with respect to allsubstitution instances, as is every instance of the converse Barcan formula: ∀ x ϕ ( x ) → ∀ x ϕ ( x ) (2) If the class of models of T is convergent under embeddability ⊂ ∼ , then S4.2 is valid at every model in
Mod( T ) for L -sentences.(3) If the class of models of T exhibits amalgamation, then S4.2 is valid at everymodel M in Mod( T ) for L -assertions with parameters from M .(4) If the models of T are linearly pre-ordered by embeddability ⊂ ∼ , then S4.3 isvalid at every model M in Mod( T ) for L -assertions with parameters from M . Note that the theorem is concerned with the convergence and amalgamationproperties using the embedding extension relation ⊂ ∼ , rather than direct extension ⊆ , and this is important, since ⊆ is almost never convergent, while in importantcases the embedding extension relation ⊂ ∼ exhibits both convergence and amalga-mation. Proof hints.
We leave the proof details as an exercise in modal reasoning for thereader. For the first part of statement (1), use that the direct extension relation ⊆ is reflexive and transitive; for the converse Barcan scheme, use that the domains areinflationary with respect to the accessibility relation. For statement (2), the mainfact is that convergence is sufficient to establish the validity of ϕ → ϕ forany sentence ϕ , using theorem 29 to transfer from ⊂ ∼ to ⊆ . If there are parametersinvolved, however, as in statement (3), then one should use amalgamation to verify ϕ ( a ) → ϕ ( a ); the commutativity of the diagram enables one to know thatit is the same a being referred to around both sides of the diagram. For statement(4), linearity is sufficient to verify the validity of the S4.3 axiom ( ϕ ∧ ψ ) → [ ( ϕ ∧ ψ ) ∨ ( ψ ∧ ϕ )]. (cid:3) Let us turn now to the more difficult issue of providing upper bounds on thevalidities of a potentialist system. A general method has emerged in a series ofpapers [HL08], [HLL15], [HL19], [HW19], [HW17], [Ham18], developing the controlstatement technique of establishing upper bounds on the modal validities of a po-tentialist system. With this method, as in theorem 33, one uses the existence ofvarious kinds of control statements in the potentialist system—buttons, switches,dials, ratchets, or railyards—to establish upper bounds on the class of modal va-lidities of the system.The main advantage of the control statement method is that it enables one toanalyze and discover the modal validities of a potentialist system by using princi-pally only expertise in the subject matter of the object theory, rather than tech-nical expertise in the foundations of modal logic. To construct control statementsfor potentialist conceptions in set theory, group theory, or graph theory generallyrequires only set-theoretic, group-theoretic, or graph-theoretic ideas, respectively.
With those ideas and with theorem 33, one can often establish important factsabout the modal validities of one’s potentialist conception.Let us quickly review some of the key kinds of control statements. A button is astatement ϕ that is possibly necessary, that is, for which ϕ ; it is pushed when ϕ holds, otherwise unpushed. A switch is a statement ψ for which ( ψ ∧ ¬ ψ ).A dial is a sequence of statements d , . . . , d n , such that necessarily, exactly oneof the statements is true, and any of them is possible. A ratchet is a sequence ofbutton statements r , . . . , r n such that each is possibly necessary, each necessarilyimplies the previous, and V k Assume W is a potentialist system.(1) If a world M in W has arbitrarily large finite families of independentswitches, then the modal validities at M are contained within S5.(2) If a world M in W has arbitrarily large finite families of independent but-tons and switches, or independent buttons and dials, then the modal validi-ties are contained within S4.2 .(3) If a world M in W has arbitrarily long ratchets, independent of switches anddials (as many as desired), then the modal validities are contained within S4.3 .(4) If a world M in W admits railyard labelings of every finite pre-tree, thenthe modal validities at M are exactly S4 .In each case, the relevant language of substitution instances would be any languagecontaining the control statements and closed under Boolean connectives. For proof of theorem 33, we refer the reader to [HLL15] for statements (1), (2),and (3), and to [Ham18] for statement (4).Let us illustrate these ideas in the case of modal graph theory. Theorem 34. In graphs, S4 . ⊆ modal validities of graphs ⊆ S5 . Indeed, for validities with respect to substitution instances in the language of graphtheory with parameters, every graph validates either exactly S4.2 or exactly S5 , andboth cases are realized. Specifically:(1) The modal theory S4.2 is valid in any graph for any assertion at all, withparameters.(2) The validities of any graph, with respect even just to substitution instancesfor sentences in the language of graph theory, is contained within S5 . ODAL MODEL THEORY 19 (3) Some graphs have their validities exactly S4.2 , with respect even just tosentences in the language of graph theory.(4) Some graphs have S5 as valid, even for assertions in the full modal languageof graph theory L ∼ , with parameters.(5) For validities with respect to substitution instances in the language of graphtheory with parameters, every graph validates either exactly S4.2 or exactly S5 .Proof. For the S4.2 lower bound, it suffices by theorem 32 to observe that the classof all graphs exhibits the amalgamation property. And indeed it does, since if graph G embeds into graphs H and H , then we may take copies of H and H extending G and for which the sets of respective new vertices are disjoint. We may then forma graph H with embedded copies of H and H , which therefore amalgamate over G as desired.For the S5 upper bound, it suffices to show that every graph admits arbitrarilylarge finite dials. Let d n be the graph-theoretic assertion that there are exactly n isolated points, and let d ≥ N assert that there are at least N isolated points. Theseare assertable by sentences in the first-order language of graph theory. For anyfinite number N , we may consider the statements d n for n < N and d ≥ N , whichtogether form a dial, since every graph either has some specific number of isolatedpoints n < N , or else it has at least N isolated points; and every graph can beextended to a graph making any one of these dial statements true.To show that the lower bound of S4.2 is realized, it suffices to find a graphadmitting arbitrarily large finite families of independent buttons and dials. We canuse the dials d n mentioned just previously, and for buttons, let b k (for k ≥ 3) assertthat there is a cycle of length k in the graph. This is a button, since every graphcan be extended to a graph with a k -cycle, and once there is a k -cycle, then itremains in all larger extension graphs. Furthermore, these buttons and dials areindependent, since we may add any number of isolated points without changing theexistence of k -cycles, and we may add any k -cycle without changing the number ofisolated points. If a graph lacks k -cycles for infinitely many k , then these buttonswill be unpushed, and consequently the validities of the graph will be containedwithin S4.2 and hence identical to S4.2, as claimed for statement (3).Statement (4) will follow from theorem 35 in the next section from the observa-tion that the class of graphs is closed under unions of chains. Theorems 39 and 41identify exactly the graphs that validate S5.Statement (5) will similarly be proved in the next section with corollary 40. (cid:3) Maximality principle A model M in a potentialist system satisfies the maximality principle with re-spect to a language when it satisfies all instances of the S5 axiom ϕ → ϕ for ϕ in that language. Our goal in this section is to determine necessary and sufficientconditions for a model M to satisfy the maximality principle with respect to variouslanguages.Let us begin by establishing in very general circumstances that there will besome models validating S5. Theorem 35. Suppose that W is a potentialist system.(1) If every countable chain of extensions in W has an upper bound in W , thenevery model can be extended to one that validates S5 with respect to anyfixed countable family of substitution instances.(2) If every (set-sized) chain of extensions in W has an upper bound in W ,then every model can be extended to one that validates S5 with respect tosentences in any fixed set-sized language.(3) If every (set-sized) chain of extensions in W has its union in W , thenevery model can be extended to one that validates S5 with respect to anyfixed set-sized language, allowing parameters.Proof. For statement (1), assume that the potentialist system has every countablechain bounded. Fix the assertions ϕ , ϕ ,. . . that will be used for substitution, andconsider any model M . If this model thinks ϕ , then there is an extension M ⊆ M with M | = ϕ . Thus, we have pushed this button. Now, push eachnext button in turn: if M n | = ϕ n , then extend to M n ⊆ M n +1 | = ϕ n . Thus,we build a chain of models. M ⊆ M ⊆ M ⊆ M ⊆ · · · By assumption, there is a model N extending every model M n in the chain. Thekey observation is that N | = ϕ n → ϕ n , since if ϕ n is possibly necessary over N ,then it was possibly necessary at stage n and therefore pushed earlier, making ϕ n necessary in N , as desired.For statement (2), it is a similar idea, but with a longer chain. Enumerate thesentences as ϕ α for α < δ , and assume that every chain in W of size at most δ has an upper bound. Start in any model M , and then push button ϕ , if possible,by finding an extension M | = ϕ . Continue to M , M , pushing each button inturn, if possible. At limit stages ξ , we use the chain hypothesis to find a model M ξ extending all earlier M α for α < ξ . In this way, we build a chain of models M α , for α < δ , such that ϕ α is pushed in M α +1 , if it is possibly necesssary over M α . M ⊆ M ⊆ M ⊆ · · · ⊆ M α ⊆ · · · Using the chain hypothesis a final time, there is a model M δ extending the chain.Any statement ϕ α that is possibly necessary over M δ was possibly necessary over M α , and therefore made necessary in M α +1 and hence is still true in M δ . So M δ validates all instances of the maximality principle for these sentences.Statement (3) is a bit more subtle, since we are allowing parameters. Becausethe models are growing in size, we cannot seem to predict in advance how manybuttons we will need to push. Nevertheless, we will be able to catch our tail using thecontinuity assumption and an iteration of iterations. Assume that the potentialistsystem is closed under unions of chains. Validating the S5 axiom p → p amountsessentially to a closure operation, and so there will be a model validating everyrequired instance. Start in any model M , and then using the method of statement(2), we may push all the buttons ϕ ( a ) that arise over M using parameters a ∈ M for statements ϕ in any set-sized language. Thus, we’ll find a model M satisfying allinstances of the S5 axiom using parameters in M . We can similarly find a furtherextension M satisfying the instances of S5 using parameters in M . Continuing inthis way, consider the sequence of extensions M ⊆ M ⊆ M ⊆ · · · ⊆ M n ⊆ · · · ODAL MODEL THEORY 21 whose limit M ω = S n M n is a model in W by our closure assumption, and it willsatisfy all instances of S5 using any parameters in M ω , as desired. (cid:3) In the case of Mod( T ) for a first-order theory T , this result plays out as follows. Corollary 36. (1) For any first-order theory T , every model of T can be extended to one inwhich S5 is valid in Mod( T ) for all sentences in L or indeed any set-sizedlanguage.(2) If the theory is ∀∃ axiomatizable, then every model of it can be extended toone in which S5 is valid in Mod( T ) for all assertions in L or indeed anyset-sized language, with parameters.Proof. A simple compactness argument shows that every chain h M α | α < θ i ofmodels of T has an upper bound in Mod( T ). Simply consider the theory consistingof T together with the atomic diagrams S α<θ ∆ ( M α ). This theory is finitelyconsistent, and hence consistent, and any model of it provides an upper bound forthe chain. Therefore, by theorem 35, every model of T can be extended to a modelin which S5 for any set of sentences in any set-sized language.For statement (2), if the theory is ∀∃ axiomatizable, then unions of chains ofmodels of T will be models of T , placing us into the continuity case of theorem 35,statement (3). In this case, therefore, we can find a limit model that will validateS5 for any fixed set-sized language, allowing parameters from the model. (cid:3) One cannot in general omit the requirement of the ∀∃ axiomatization, in lightof the following observation, which provides an ∃∀ theory T having no models thatvalidate S5 for L with parameters. Observation 37. There is an ∃∀ axiomatizable first-order theory T for which Mod( T ) admits no models with the maximality principle with parameters.Proof. In the language of graph theory, let T assert that “there is a node adjacentto all other nodes,” and consider the resulting potentialist system Mod( T ) of graphshaving this property. No graph like this can satisfy the maximality principle withparameters, since it is possibly necessary that any given node a lacks an edge tosome other node, since we can extend the graph by adding a node to which a is not adjacent, even while adding a new universally adjacent node in order tosatisfy T . (cid:3) A similar example would be the theory of dense linear orders with a leastelement—it is a button to make any particular element non-least. Both of thesetheories have ∃∀ axiomatizations, and it seems that this argument will work withany truly ∃∀ theory. Such theories are not closed under unions of chains, and thisis how they escape the conclusions of statement (3) in theorem 35 and statement(2) of corollary 36.We would now like to understand more precisely which models validate S5, andto do so, let us begin with modal graph theory, which illustrates the central ideasin a concrete manner. The countable random graph is (up to isomorphism) theunique countable saturated graph. This graph is characterized by the finite patternproperty , which asserts that for any two disjoint finite sets of vertices A and B ,there is a vertex v adjacent to every vertex in A and to no vertex in B . Thefinite pattern property allows you to find a new node satisfying exactly any desired pattern of connectivity to finitely many previous nodes, and by iterating this in aback-and-forth manner, one can easily show both that any two countable graphswith this property are isomorphic and also that any graph with this property isuniversal for all countable graphs. Theorem 38. A countable graph G satisfies the maximality principle ϕ (¯ a ) → ϕ (¯ a ) for all ϕ in the language of graph theory with parameters ¯ a ∈ G if and only if G isthe countable random graph.Proof. ( → ) Suppose that G satisfies the maximality principle assertion with pa-rameters. Every instance of the finite pattern property has the form ∃ x ( ^ i x ∼ a i ) ∧ ( ^ j x ≁ b j ) , with finitely many parameters a i , b j ∈ G . Every such statement is possibly neces-sary, since we can extend the graph to add such a vertex, if necessary, and oncethere is one, it remains in all further extensions. So by the maximality principle,this statement must already be true in G . And so G fulfills the finite patternproperty, and since it is countable, it is therefore the countable random graph.( ← ) Suppose that G is the countable random graph. We aim to show that G satisfies the maximality principle. To show this, suppose that G | = ϕ (¯ a ) forsome ¯ a = ( a , . . . , a n ) from G . So there is an extension graph G ⊆ H such that H | = ϕ (¯ a ). By theorem 20, we may assume that H is countable. Therefore H embeds into G by the universality of the countable random graph, and furthermore,we can find such an embedding that fixes the parameters in ¯ a . So G extends theimage of H under that embedding, and so G | = ϕ (¯ a ), as desired. (cid:3) We can mount several refinements of this theorem. The argument actually showsthat the countable random graph exhibits the maximality principle for assertionsin the modal language L ∼ , not just L ∼ . And conversely, having the maximalityprinciple merely for existential assertions in the language of graph theory is sufficientto ensure the finite pattern property; so these all will be equivalent. In addition,we didn’t use countability as such, but rather merely the finite pattern property(which together with countability characterizes the countable random graph), andso a more erudite version of the theorem is the following: Theorem 39. The following are equivalent for any graph G .(1) The graph G satisfies the maximality principle ϕ (¯ a ) → ϕ (¯ a ) for everyassertion ϕ in the modal language of graph theory L ∼ , allowing parame-ters ¯ a = ( a , . . . , a n ) from G .(2) The graph G satisfies the maximality principle ϕ (¯ a ) → ϕ (¯ a ) for everyassertion ϕ in the language of graph theory L ∼ , allowing parameters ¯ a =( a , . . . , a n ) from G .(3) The graph G satisfies the maximality principle ϕ (¯ a ) → ϕ (¯ a ) for everyexistential assertion ϕ = ∃ x ϕ ( x, ¯ a ) in the language of graph theory, with ϕ quantifier free, allowing parameters ¯ a = ( a , . . . , a n ) from G .(4) The graph G fulfills the finite pattern property.Proof. (1 → → 3) These implications are immediate. ODAL MODEL THEORY 23 (3 → 4) Every instance of the finite pattern property is a possibly necessaryexistential statement, which must therefore already be true in the graph.(4 → 1) This is the main part of the argument; we argue as in theorem 38.Suppose that G fulfills the finite pattern property and satisfies ϕ (¯ a ) for some ϕ ∈ L ∼ , with parameters ¯ a from G . So there is a graph extension G ⊆ H | = ϕ (¯ a ). The theory ∆( G ) ∪ ∆ ( H ) is consistent, since any finite part of ∆ ( H ) canbe realized inside G over the parameters ¯ a by successive uses of the finite patternproperty. Any model N of this theory will have G ≺ N and H ⊆ N . Because ofthe latter, we will have N | = ϕ (¯ a ), and therefore by the key lemma, we will deduce G | = ϕ (¯ a ), since ϕ is in L ∼ . (cid:3) Theorem 39 shows that a model can validate S5 for assertions in L with pa-rameters, while this fails in an extension (using parameters of the extension). Forexample, the countable random graph certainly fulfills the finite pattern property,but if we were to add a new isolated vertex, we would of course no longer fulfill thefinite pattern property. Corollary 40. For any graph G , the validities of G with respect to the language ofgraph theory with parameters is either exactly S4.2 or exactly S5 .Proof. This corollary amounts to theorem 34 statement (5). If G fulfills the finitepattern property, then the validities of G are exactly S5 by theorem 34. So supposethat G does not fulfill the finite pattern property. So there are finitely many a , . . . , a n and b , . . . , b m for which there is no vertex x connected to every a i andto no b j . This is an unpushed button. For any other vertex u not amongst the a i sand b j s, we may consider the statement asserting that there is a vertex adjacentto every a i and also to u , and to none of the b j . This also is a button, which isunpushed in G , and these buttons are independent, since one can push any oneof them without pushing any of the others. And furthermore, these buttons areindependent of the dials asserting that there are exactly k isolated points or at least N isolated points, as in theorem 34. So we’ve found arbitrarily large independentfamilies of buttons and dials, and so by theorem 33 the validities of G are containedwithin S4.2. But since S4.2 is valid in any graph, the validities of G are exactlyS4.2, as desired. (cid:3) Let us now also prove a parameter-free version of theorem 39. Theorem 41. The following are equivalent for any graph G .(1) The graph G satisfies the maximality principle ϕ → ϕ for all sentences ϕ in the modal language of graph theory L ∼ .(2) The graph G satisfies the maximality principle ϕ → ϕ for all sentences ϕ in the language of graph theory L ∼ .(3) The graph G satisfies the maximality principle ϕ → ϕ for all existentialsentences ϕ = ∃ x , . . . , x n ϕ ( x , . . . , x n ) , with ϕ quantifier free in thelanguage of graph theory L ∼ .(4) The graph G is universal for finite graphs.Proof. (1 → → 3) These implications are immediate.(3 → 4) The assertion that G contains an isomorphic copy of a given finite graphis a possibly necessary existential statement, since we can add such a copy to anygiven graph, and so it must already be true in G , if G satisfies the maximality principle for such assertions. So any such graph G as in statement (3) is universalfor finite graphs.(4 → 1) Suppose that G is universal for finite graphs. We want to prove thatit fulfills every instance of the maximality principle ϕ → ϕ for sentences ϕ inthe modal language of graph theory. Assume G | = ϕ . This means that thereis an extension G ⊆ H for which H | = ϕ . Consider the theory Th( G ) ∪ ∆ ( H ),taking the elementary theory of G in the language of graph theory (sentences only,no parameters) together with the atomic diagram of H (using names for everyelement of H ). This theory is finitely consistent, since any finite part of H can berealized inside G by finite universality. So the theory is satisfied in some graph,which because of Th( G ) will satisfy the full theory of G and because of ∆ ( H ) willcontain a copy of H . So there is a model N of this theory, which we may take as adirect extension H ⊆ N . Since N extends H , it follows that N | = ϕ . And since N satisfies the theory of G , it follows from theorem 10 that G | = ϕ . So G satisfies allinstances of the sentential maximality principle, as desired. (cid:3) We should like now to generalize this analysis to arbitrary theories, not justgraph theory. What is really going on? The model-theoretic feature in play here is existential closure . A model M is existentially closed with respect to a theory T , ifevery existential statement ∃ x , . . . , x n ϕ ( x , . . . , x n , ¯ a ), with ϕ quantifier-free, thatis true in some extension M ⊆ N | = T + ∃ ¯ x ϕ (¯ x, ¯ a ), with parameters ¯ a from M , istrue already in M . Theorem 42. In the potentialist system Mod( T ) for any theory T in a first-orderlanguage L , the following are equivalent:(1) M | = ϕ (¯ a ) → ϕ (¯ a ) for formulas ϕ ∈ L with parameters ¯ a from M .(2) M | = ϕ (¯ a ) → ϕ (¯ a ) for formulas ϕ ∈ L with parameters ¯ a from M .(3) M | = ϕ (¯ a ) → ϕ (¯ a ) for existential formulas ϕ = ∃ ¯ xϕ (¯ x, ¯ a ) ∈ L , with ϕ quantifier free and parameters ¯ a from M .(4) M is existentially closed in Mod( T ) .Proof. (1 → → 3) These implications are immediate.(3 → 4) Any instance of existential closure is a possibly necessary existentialstatement, since once an existential statement becomes true, it remains true in allfurther extensions.(4 → 1) Suppose that M is existentially closed in Mod( T ). We aim to prove themaximality principle in M for assertions in the modal language L with parame-ters. Suppose that M | = ϕ (¯ a ) for such an assertion ϕ . So there is an extension M ⊆ H | = T with H | = ϕ (¯ a ). Consider the theory ∆( M ) ∪ ∆ ( H ), consistingof the full elementary diagram of M together with the atomic diagram of H . Thistheory is finitely consistent, since any finite combination of atomic assertions truein H must already be realizable in M , since it is existentially closed. So there isa model N of this theory, which we may take to have M ≺ N and H ⊆ N . Since H ⊆ N , we know that N | = ϕ (¯ a ), and since M ≺ N it follows by the key lemmathat M | = ϕ (¯ a ), as desired. (cid:3) This theorem has a consequence when every model in Mod( T ) validates S5. Corollary 43. For any first-order theory T , the following are equivalent:(1) S5 is valid in Mod( T ) at every model, with parameters, for assertions in L . ODAL MODEL THEORY 25 (2) S5 is valid in Mod( T ) at every model, with parameters, for assertionsin L .(3) S5 is valid in Mod( T ) at every model, with parameters, for assertions in L .(4) T is model complete.(5) The modally trivializing logic p ↔ p (and so also p ↔ p ) is valid in Mod( T ) at every model, with parameters, for assertions in L .Proof. (1 → → 3) Immediate.(3 → 4) If S5 is valid in Mod( T ) at every model, with parameters, for assertionsin L , then by theorem 42, it follows that every model of T is existentially closed.But [Hod93, theorem 8.3.1] shows that the models of T are all existentially closedif and only if the theory is model complete.(4 → 5) If the theory T is model complete, then by theorem 17, we have modalitytrivialization in the full modal language L .(5 → 1) If modalities trivialize in L , then p ↔ p is valid at every world, forassertions with parameters. In particular, ϕ ( a ) → ϕ ( a ) will hold at every worldof Mod( T ) for assertions ϕ in L , with parameters. (cid:3) Corollary 44. For any first-order theory T , the modal validities of Mod( T ) forassertions in L , L , or L , with parameters, are never exactly S5 .Proof. Corollary 43 shows that if S5 is valid in Mod( T ) at every world for anyof those languages, with parameters, then p ↔ p and p ↔ p are also valid.But these latter validities are not part of S5, and so the modal validities are not exactly S5. (cid:3) We can similarly establish a parameter-free version, defining that a model M is sententially existentially closed in Mod( T ) if whenever M ⊆ N | = T and N satisfiesan existential sentence ∃ x , . . . , x n ϕ ( x , . . . , x n ), with ϕ quantifier free, then thesentence is already true in M . In graph theory, this is equivalent to being universalfor finite graphs. Theorem 45. In the potentialist system Mod( T ) for any theory T in a first-orderlanguage L , the following are equivalent:(1) M | = ϕ → ϕ for sentences ϕ ∈ L .(2) M | = ϕ → ϕ for sentences ϕ ∈ L .(3) M | = ϕ → ϕ for existential sentences ϕ = ∃ x , . . . , x n ϕ ( x , . . . , x n ) with ϕ quantifier free in L .(4) M is sententially existentially closed in Mod( T ) .Proof. (1 → → 3) These implications are immediate.(3 → 4) Any instance of sentential existential closure is a possibly necessaryexistential sentence, which by statement (3) must be true in M .(4 → 1) Suppose that M is sententially existentially closed in Mod( T ). To verifythe sentential maximality principle, suppose that M | = ϕ for some sentence ϕ ∈ L . So there is an extension M ⊆ H | = T with H | = ϕ . Consider the theoryTh( M ) ∪ ∆ ( H ) consisting of the theory of M (no parameters) together with theatomic diagram of H , using names for every element of H . This theory is finitelyconsistent, since any finite part of the atomic diagram of H is realized inside M by sentential existential closure. So there is a model N of the theory, which has M ≡ N and H ⊆ N . Since H ⊆ N , we know that N | = ϕ , and since M ≡ N , itfollows by theorem 10 that M | = ϕ , as desired. (cid:3) Note that if a first-order theory T is complete, then corollary 11 shows that themodally trivializing logic σ ↔ σ is valid in Mod( T ) for sentences σ in L . Inparticular, the maximality principle σ → σ will also be valid for such sentences. Observation 46. The maximality principle for assertions in the intermediatemodal language L is not necessarily equivalent to the maximality principle inthe full modal language L . In the class of graphs, the countable random graphsatisfies the maximality principle for L ∼ with parameters, but it does not satisfythe maximality principle for L ∼ , even merely for sentences.Proof. We have already proved that the countable random graph satisfies the max-imality principle in the language L ∼ with parameters. But this does not extendto the full modal language L ∼ because the countable random graph is countable,whereas being uncountable is a possibly necessary statement of L ∼ , since everygraph can be extended to an uncountable graph, and once a graph is uncountable,this remains true in all further extensions. (cid:3) In light of the results of sections 2 and 7, any graph satisfying the maximalityprinciple in the full modal language L ∼ must be enormous, because it is a buttonthat the graph exceeds in size any of the cardinalities that we proved to be express-ible in this language, such as the first i -hyper-fixed-point cardinal. Indeed, becauseof the uncertain status of question 54, it is not clear whether we can even definein ZFC a truth predicate for modal graph-theoretic truth in the full language; andwithout such a truth predicate, it is not sensible to assert the maximality princi-ple in that full modal language. For this reason, the maximality principle in thefull modal language may be subject to the subtle metamathematical difficulties wediscuss in section 9.7. The interpretative power of modal graph theory We should like now to demonstrate the strength of modal graph theory by prov-ing that it can interpret various other mathematical theories and structures.Let us begin with finite modal graph theory, where we consider the potentialistsystem consisting of all finite graphs. This is a proper class, but if desired one canreduce to a set by considering the finite graphs using a vertex set contained in theset of natural numbers N . By the renaming lemma, any modal truth in the classof all finite graphs will be realized in these kinds of graphs, and so modal truth isnot affected by this restriction. Theorem 47. True arithmetic is interpretable in finite modal graph theory. Thereis a translation ϕ ϕ ∗ from arithmetic sentences ϕ to sentences of modal graphtheory ϕ ∗ , such that h N , + , · , , , < i | = ϕ if and only if ϕ ∗ holds in every finitegraph in the class of finite graphs.Proof. The strategy will be to represent numbers within graph theory and thento show that the arithmetic operations and structure are definable in modal graphtheory. To begin, we shall represent the numbers with graphs of the following form: ODAL MODEL THEORY 27 We represent each number n with a vertex (in red) whose neighbors form a cycle oflength n + 3. Our use of this +3 offset enables a convenient absoluteness property:if a vertex represents a number in a graph, then in any extension of this graphwhere it still represents a number, it must represent the same number as originally.(This wouldn’t have been true, for example, had we represented 0 with an isolatednode or 2 with a node adjacent to two adjacent nodes.) So we represent n withan ( n + 3)-cycle. A node encodes a number exactly when it’s neighbors form anonempty connected set with all vertices of degree 2, and this is expressible in thelanguage of finite modal graph theory.The arithmetic order relation n ≤ m on these representations is expressible inthe language of finite modal graph theory. Namely, if we have a representation of n and a representation of m , then n ≤ m if and only if possibly, there is a nodewhose neighbors are each adjacent to distinct neighbors of n and m , in such a waythat these associations form an injection from all but three neighbors of n with atmost all but three neighbors of m , like this: n m This particular figure illustrates that 4 ≤ 5. We can similarly express that n and m represent the same number, by asserting that the association uses all but threeneighbors of each vertex.We can also express that n + m = r for number representatives using the languageof finite modal graph theory. We simply say that possibly, they all still representnumbers and there is such a one-to-one correspondence from all but three neighborsof n and all but three neighbors of m with all but three neighbors of r . And we canexpress that nm = r , by asserting that possibly, they all still represent numbersand there is a node adjacent to nodes that are each adjacent to a distinct node of r , and to neighbors of n and m , in such a way that except for three neighbors eachof n , m and r , every pair of nodes from n and m arises exactly once in associationwith a neighbor of r .Finally, let us explain how to translate arithmetic assertions to finite modal graphtheory. Every arithmetic assertion is equivalent to an arithmetic assertion in whichthere are no compound terms, so that all atomic formulas have the form x + y = z , x · y = z , x = y , x < y , x = 0, or x = 1, where x , y and z are variable symbols.We can translate these into the language of finite modal graph theory by assertingthat x , y and z code numbers obeying the relevant identity, as we described above.Next, we extend the translation recursively:( ¬ ϕ ) ∗ = ¬ ϕ ∗ ( ϕ ∧ ψ ) ∗ = ϕ ∗ ∧ ψ ∗ ( ∃ x ϕ ) ∗ = ∃ x (cid:0) ^ v ∈ FV( ϕ ) v represents a number ∧ ϕ ∗ (cid:1) In the exists case, we include the assertion that all the variables of the formulastill represent numbers, since we only want to consider extensions in the which theother parts of the graph still represent numbers. This is where it is important thatour number representation is absolute under extensions.It now follows by induction that the truth of an arithmetic sentence ϕ in N isequivalent to the truth of the modal graph translation ϕ ∗ in any particular finitegraph. (cid:3) Since one can encode finite graphs into arithmetic, it follows that we can alsomake a converse translation of finite modal graph theory into arithmetic, and sothese theories are mutually interpretable. They are actually bi-interpretable, sinceeach can see how it is that it is encoded into the translated copy of the other,although we shall not formulate a precise notion of bi-interpretation of models andpotentialist systems here. Corollary 48. Arithmetic is interpretable in modal graph theory, using the classof all graphs (not just finite graphs).Proof. The point is that since theorem 3 shows that finiteness is expressible inmodal graph theory, we can define the class of finite graphs within the class of allgraphs, and therefore we can define the finiteness modal operators Fin and Fin of true in some finite extension and true in all finite extensions , respectively. (cid:3) Let us consider next the case of countable modal graph theory. Let H ω be theset of all hereditarily countable sets, the sets having a countable transitive closure.This is a model of ZFC − , meaning Zermelo-Frankael set theory without the powerset axiom (but see [GHJ16] concerning a subtlety about the axiomatization). Weshall show that truth in the hereditarily countable sets is interpretable in countablemodal graph theory. Let us denote by Γ ⊕ Λ the disjoint sum of graphs Γ and Λ. Theorem 49. Hereditarily countable set theory is interpretable in countable modalgraph theory. We shall represent hereditarily countable sets with countable graphsand vertices and define a translation ϕ ϕ ∗ of set-theoretic assertions ϕ to modalgraph-theoretic assertions ϕ ∗ , such that h H ω , ∈i | = ϕ ( a , . . . , a n ) ⇐⇒ Γ ⊕ · · · ⊕ Γ n | = ϕ ∗ (ˆ a , . . . , ˆ a n ) , where (Γ i , ˆ a i ) is a countable graph and vertex representing the set a i .Proof. Every hereditarily countable set a is an element of a countable transitiveset t , such as the transitive closure of its singleton TC( { a } ). The structure h t, ∈i is a countable set with a well-founded extensional relation ∈ , and every such well-founded extensional relation on a countable set is isomorphic to a countable tran-sitive set via the Mostowski collapse.We define that a graph code for a hereditarily countable set is a graph consistingof a node t , whose neighbors are related by a well-founded and extensional relation x y , defined to hold when there are nodes between x and y as depicted here: x y In addition, there shall be a copy of ω in the sense of theorem 5 together with abijection of the copy of ω with the neighbors of t and the nodes used in the ODAL MODEL THEORY 29 relation. The property of being a graph code is expressible in modal graph theory,because the extensionality of is expressible directly without modal operatorsby the assertion that distinct neighbors of t have distinct sets of -predecessorsamongst the neighbors of t , and the well-foundedness of is expressible by theassertion that necessarily, in any extension in which the copy of ω is still a copy of ω and still provides a bijection with the neighbors of t and the supplemental nodesused in the relation, then there is no node whose neighbors are a set of neighborsof t having no -minimal element. Note that we use the bijective copy with ω toensure that in the graph extension, there are no new neighbors of t and no newinstances of the relation; this is a form of absoluteness in our representationsimilar to what we had used in theorem 47 in the finite case. In addition, there willbe a vertex ˆ x pointing at t and at one of the neighbors x of t . The graph, togetherwith the vertex ˆ x code the set that would result from the image of x under theMostowski collapse of the relation on the neighbors of t .We can express that two graph codes (Γ , x ) and (Γ ′ , x ′ ) code the same set byasserting that possibly, there is a node pointing at nodes that are each adjacent toa node in Γ and to a node in Γ ′ , as in the proof of theorem 47, in such a way thatthey make a isomorphic correspondence between -downward closed subsets of Γand Γ ′ , which furthermore associates x with x ′ . It will follows that the set codedby x via Γ will be the same as the set coded by x ′ via Γ ′ . Similarly, we can expressthat one code (Γ , x ) codes an element of another (Λ , y ), if (Γ , x ) is isomorphic tothe code formed by a -predecessor of y in Λ. This tells us how to interpret theatomic formulas x = y and x ∈ y in terms of the codes. And we can simply extendthe interpretation recursively through Boolean connectives and quantifiers as intheorem 47, establishing the equivalence stated in the theorem by induction. (cid:3) Corollary 50. Second-order arithmetic is mutually interpretable with countablemodal graph theory.Proof. Second-order arithmetic is bi-interpretable with the structure h H ω , ∈i , andthis latter structure can define representations of any countable graph up to iso-morphism. By the renaming lemma, the modal truths of any countable graph arethe same as for the copies available in H ω , and so this structure can interpretcountable modal graph theory. (cid:3) We could have used this set-theoretic representation even in the case of finitemodal graph theory, representing every hereditarily finite set with a finite exten-sional relation relation on a set together with a bijection to a finite cycle (therebyproviding the absoluteness). This would show that h H ω , ∈i is interpretable in finitemodal graph theory.But similarly, let us use the method on larger cardinals. The key requirement isa notion of stability for the cardinality in question. Let us define that a cardinal κ is stably representable in modal graph theory if there is a property φ expressible inmodal graph theory with the following properties:(1) There is a graph G with a vertex v satisfying φ ( v ).(2) If φ ( v ) holds in a graph G of a vertex v , then v has exactly κ many neighborsin G .(3) The truth of φ ( v ) in G depends only on the induced subgraph consisting of v and its neighbors and the neighbors of the parameters (suppressed). (4) If φ ( v ) holds in G and also in an extension graph H , then v has the sameneighbors in G as in H .This was the key property of our representation of ω that enabled the absolutenessfeature we used in theorem 49. Theorem 51. If a cardinal κ is stably representable in modal graph theory, then h H κ + , ∈i is interpretable in modal graph theory.Proof. We can use the same method as in theorem 49, except using a bijectionwith a stable representation of κ , rather than a copy of ω . In this way, we rep-resent sets in H κ + using well-founded extensional relations on sets of size κ , witha distinguished vertex pointing at the set being represented. Once again, we canexpress the equivalence of codes and the set membership relation on the codes inthe language of modal graph theory. (cid:3) Since H κ is definable inside H κ + , it follows that we can also interpret h H κ , ∈i inmodal graph theory, when κ is stably representable. Theorem 52. (1) If κ is stably representable in modal graph theory, then so is κ + .(2) If κ is stably representable in modal graph theory, then so is κ .(3) If κ is stably representable in modal graph theory, then so are ℵ κ and i κ .(4) If κ is stably representable in modal graph theory, then so is the next i -fixed-point above κ .(5) If κ is stably representable in modal graph theory, then so is the first i -hyper-fixed-point above κ —the first cardinal λ that is the λ th i -fixed-pointabove κ .Proof. Suppose that κ is stably representable. We shall represent κ + by a vertexpointing at κ + neighbors that are well-ordered by the relation, with the first κ many of them fulfilling the stable representation of κ , and such that every node u amongst the κ + beyond κ is adjacent to a node that points at nodes forming abijection between the -predecessors of u and the κ initial segment of the order.That is, the whole order does not have size at most κ and not only do all theseinitial segments have size κ , but they come equipped already with the bijectionswitnessing that they have size κ . We can assert easily that is a linear order;to assert that it is a well-order, we just have to claim that in every extension inwhich the copy of κ remains stable and the -relation is still a linear order and thebijections of initial segments are still bijections with κ , there is no node pointing ata nonempty set with no -minimal element. If this is true, then it really must be awell-order, and if it is a well-order, then because of the stability with κ , none of thenodes in the order can gain new predecessors; and the κ + size order itself cannotgain new elements on top of all the previous nodes, because then those nodes wouldhave κ + many predecessors and therefore could not exhibit a bijection with the setof size κ . So we have a stable copy of κ + and so it is stably representable.To stably represent 2 κ , we begin with a stable representation of κ , together witha node P whose neighbors are each adjacent to distinct subsets of the set κ manynodes, and such that furthermore, necessarily, in any extension in which the size κ set fulfills its definition, if a node is adjacent to a subset of the set of size κ ,then some neighbor of P already realizes that same subset. The point is that if theneighbors of P did not already represent the full power set, then we could extend ODAL MODEL THEORY 31 by adding a node pointing out the missing the subset, and it would violate thisnecessity requirement. But if P does represent the full power set, then any newpattern will already be represented. So we have a stable representation of P ( κ ),which has size 2 κ .If κ is stably representable, then we can stably represent a copy of κ that iswell-ordered by ; this is simply a set of size κ that is linearly ordered by ,such that in every extension in which the copy of κ is stable (so that the order hasgained no new elements), there is no node pointing at a set having no -minimalelement; and such that every proper initial segment of the order has size less than κ , which is to say, that in no extension in which κ is preserved is there a bijectionof κ with an initial segment of the order. So we have now a stable representationof a well-order of length κ . To represent i κ , we associate the least element of the κ -sequence with a copy of ω , and at each successor node, we associate the powerset of the previous node; and at limit nodes, we associate a set that is exhibited inbijection with the union of the prior sets. Since our copy of κ is stable, we will havetherefore iteratively and stably represented the iteration of the power set. Finally,we assert that there is a vertex whose neighbors are placed into bijection with allthe associated sets at each stage; this will be a stable representation of i κ . Wecan stably represent ℵ κ in a similar manner, simply by representing the successorcardinal at each step instead of the power set.We now iterate this to find a stable representation of the next i -fixed point.Consider the least i -fixed point, which is the cardinal θ = sup θ n , where θ = ω and θ n +1 = i θ n . To stably represent this, we have a copy of ω pointing out nodesto represent each θ n , each with a well-order, and then at each θ n +1 we assert thatthe previous construction has been already arranged, so that the node representing θ n +1 is pointing out a set of size i θ n , along with a well-order of it. The point isthat this whole iteration is sufficiently uniform that we can describe that a graphhas carried out the whole construction to reach the limit node, which is the i -fixedpoint. To find the next i -fixed point after a given stably representable cardinal,we simply start with θ representing κ + 1, rather than ω .Similarly, since the process of representing the next i -fixed point is stably rep-resentable, we can represent the process of iterating it sufficiently to find the firsthyper-fixed-point. (cid:3) Theorem 52 has consequences for the possibility or impossibility of L¨owenheim-Skolem conclusions in modal graph theory. Because these various sizable cardinalsare expressible in modal graph theory, they form bounds on the sizes of possibleelementary substructures and extensions. A comparatively small graph, after all,cannot be elementary in a large one in modal graph theory, if the size of the smallgraph is expressible in the language of modal graph theory.The theorem shows that the set-theoretic structures H ω , H ω , H ω , V ω ωω +5 andso on are all interpretable in modal graph theory. And indeed much more. Corollary 53. Set-theoretic truth in V θ , where θ is the first i -hyper-fixed point, isinterpretable in modal graph theory.Proof. The theorem shows that the first i -hyper-fixed point is stably representable,and so theorem 51 shows that V θ , which is the same as H θ , is interpretable in modalgraph theory. (cid:3) And we may proceed similarly to the next i -hyper-fixed point, the next hyper-hyper-fixed point, and so on. These will all also be stably representable, and so wecan interpret set-theoretic truth for a quite a long way into the cumulative hierarchy.In this sense, modal graph theory can serve as a foundation of mathematics.Does it ever stop? Question 54. Is set-theoretic truth, that is, truth in the full set-theoretic universe V , interpretable in modal graph theory? The key obstacle concerns the question of whether we can find sufficient stablerepresentations of any given set. It is not clear how to translate definable cardinalsin ZFC, such as the first Σ -correct cardinal, into stable representations in modalgraph theory.There is a related question that we have been unable to resolve. In any graph G , let us denote by G x the set of neighbors adjacent to a given node x . • (Equinumerosity problem) Can we express in the language of modal graphtheory that the neighbor set G x of vertex x is equinumerous with the neigh-bor set G y of vertex y ? • (Cardinal comparability problem) Can we express in the language of modalgraph theory that the neighbor set G x of vertex x has cardinality less thanor equal to the neighbor set G y of vertex y ?We conjecture that the answers are both negative. A positive answer to the car-dinal comparability problem, of course, will imply a positive answer to the equinu-merosity problem.If it should turn out that contrary to our conjecture, the cardinal comparabilityproblem has a positive solution, then we would like to mention that we will beable to express in modal graph theory that a relation such as is well-ordered.That is, if we have a node whose neighbors are linearly ordered by the relation,then the relation is well-ordered if and only if it is possible that every node in theorder is adjacent to a set of neighbors, such that the cardinality comparisons ofthose neighbor sets agrees with the order. The point is that in ZFC, any set ofcardinal sizes is well-ordered by comparability (this fact is equivalent to the axiomof choice).This observation seems important, since the ability to express the well-orderconcept in modal graph theory seems likely to be highly relevant for the capacityof modal graph to interpret set-theoretic truth.8. Actuality operator @ Let us now discuss the fruitful extension of the modal language by means ofthe actuality operator @, which allows one to refer to the actual world and moregenerally to the various worlds that are in effect referenced during the course ofinterpreting a modal statement. Let us explain the usage and semantics by meansof several examples. The statement ( ϕ → ( ψ ↔ @ ψ )), expressing the assertionthat “necessarily, if ϕ is true, then ψ holds if and only if ψ is actually true.” Thisis true in a world w , when in every world u that w can access, if ϕ is true in u ,then ψ is true in u if and only if ψ is true in w . Thus, the @ operator in effectreferences truth in the original world w . Let us introduce the predication @ x tomean that individual x is actual, that is, that @ ∃ u u = x , which asserts that x is an individual of the actual world. Similarly, we introduce the quantifier ∀ @ x ϕ ODAL MODEL THEORY 33 to mean that ∀ x (@ x → ϕ ), or more simply, that ϕ holds (in the current possibleworld of evaluation) for all actual x , that is, for all x in the actual world. And thequantifier ∃ @ xϕ means ∃ x (@ x ∧ ϕ ), asserting that there is some actual x fulfilling ϕ in the current possible world of evaluation.For example, in the modal language of graph theory with actuality, the assertion ∃ x ∀ y ( x ∼ y ↔ (@ y ∧ @ ∀ z ¬ y ∼ z ))expresses that it is possible that there is a node adjacent to all and only the isolatednodes of the actual world.When the modal depth of the assertion increases, there are several worlds at playin the semantics of the assertion, and a more complex usage and semantics for @allows us to refer to them directly. The unadorned @ refers as above to the world inwhich the overall statement is being evaluated, and otherwise the subscripts @ , @ and so on refer to the worlds referenced by corresponding subscripts on the modaloperators. For example, the assertion( ∃ x ∀ @ y T xy ) ∧ ∃ x ∀ @ y T xy asserts that possibly someone is taller than every actual person, and furthermore,this is necessarily true. The second clause asserts that in every accessible world,it is possible that someone is taller than everyone in that world, not just in theoriginal world. The operator here is not a different modal operator than , forthe two necessities themselves have the same meaning; rather, the subscript on tells us to which world @ will now refer inside the scope of .To illustrate further, consider the assertion that necessarily there are two possibleworlds, such that every individual who is tall in one of them is short in the other.One might be tempted at first to formulate it like this: (cid:2) ∀ x ( T x → @ Sx ) ∧ ∀ x ( T x → @ Sx ) (cid:3) . This expression, however, is not well formed; the semantics of it are not composi-tional, since the @ does not occur under the scope of and @ does not occurunder the scope of . One can, however, properly express exactly the desiredstatement like this: @ (cid:2) @ ∀ x ( T x → @ Sx ) ∧ @ ∀ x ( T x → @ Sx ) (cid:3) . In this way, the actuality operator allows for a rich expression of truth and possi-bility relations between worlds.In the potentialist system of all graphs, the assertion (cid:2) ∃ x ∀ y ( x ∼ y ↔ @ y ) ∧ ∃ z ∀ y ( z ∼ y ↔ (@ y ∧ ¬ @ y )) (cid:3) is true at a graph G exactly when there is a larger graph G with a node x thatis connected to all and only the nodes of G and a further graph G with a node z connected to all and only the nodes of G that are in G but not in G . Theorem 55. Both the equinumerosity problem and the cardinal comparabilityproblem are expressible in the language of modal graph theory with actuality L , @ ∼ .Proof. It suffices to express only the comparability problem. For any graph G , let G v denote the nodes in G adjacent to v . The relation | G v | ≤ | G w | is expressible bysaying: there is some larger graph H with a node that points at nodes that indexan injective correspondence from the actual neighbors of v to the actual neighborsof w . This is the same idea as used with ≤ in the proof of theorem 47. (cid:3) The main point we should like to make is that modal graph theory with theactuality operator can interpret full set-theoretic truth. Theorem 56. The modal logic of graph theory with actuality can interpret set-theoretic truth. There is a translation of set-theoretic assertions to modal graphassertions ϕ ϕ ∗ , such that a set-theoretic sentence ϕ is true in the set-theoreticuniverse ( V, ∈ ) if and only if ϕ ∗ is true in the empty graph, or in any particulargraph.Proof. This theorem follows from the methods used in proving the theorems ofsection 7. The main point is that the actuality operator @ will enable us avoid theneed for stable representations, since with @ we can directly refer to the desiredcoded structure.So we may code any set with a node whose neighbors form a set that under therelation is well-founded and extensional. And so any set is represented by point-ing to such a well-founded extensional relation and a point in it. As in theorem 51,we can define equivalence of codes and the set membership relation. And we cantranslate set-theoretic assertions into modal graph theory by extending this inter-pretation of the atomic truths through the Boolean connectives, and interpretingthe set-theoretic ∃ x as “ ∃ x such that x codes a set,” just as in section 7. (cid:3) There are some subtleties about the metamathematical status of this theorem,since we cannot refer to set-theoretic truth inside set theory. One can interpret thetheorem in ZFC as a theorem scheme, a separate claim about interpreting Σ n set-theoretic truth in modal graph theory, for each metatheoretic n , but always usingthe same interpretation method. Alternatively, we can interpret the theorem as atrue theorem in an extension of ZFC to a theory such as KM or just GBC + ETR ω ,which proves the existence of first-order set-theoretic truth predicates. Question 57. Is actuality @ expressible in modal graph theory? We conjecture not, but we do not know how to prove this. One idea may beto show that the equinumerosity problem is not expressible in the language ofmodal graph theory—perhaps this can be proved by means of modal pebble games.If successful, this would show actuality is not expressible in modal graph theory,because with actuality, we can express equinumerosity.9. Set-theoretic and meta-mathematical issues Although we have claimed to define the semantics for modal truth in Mod( T ),the meta-mathematics of this definition involves some set-theoretic subtleties. Thebasic problem is that Mod( T ) is a proper class and the recursive definition of truthis not a set-like recursion. Therefore, we can’t seem to undertake the definitionlegitimately in ZFC. But in Kelley-Morse set theory, for example, or even merely inG¨odel-Bernays set theory with the axiom of elementary transfinite recursion ETR,or even just ETR ω , then we can prove that there is a solution of the recursivedefinition of modal truth. We refer the readers to [GHH + 20] for further discussionof the role of ETR in defining truth predicates. Question 58. In ZFC can one define the satisfaction relation for modal graphtheory for the class of all graphs? ODAL MODEL THEORY 35 This question is related to the question whether modal graph theory interpretsset-theoretic truth. If it does, then the answer to this question will be negative, sinceby Tarski’s theorem on the non-definability of truth one cannot define first-orderset-theoretic truth within first-order set theory. Theorem 59. No ZFC -definable class defines the satisfaction relation for modalgraph theory with actuality, that is, a truth predicate for the class of graphs in L , @ ∼ .Proof. By theorem 56, set-theoretic truth is interpretable in modal graph theorywith actuality. So by Tarski’s theorem on the non-definability of truth, there canbe no definable truth predicate. (cid:3) Meanwhile, in the stronger class-based theories such as KM or GBC + ETR,there is a satisfaction class for first-order set-theoretic truth, and with this we candefine the satisfaction relation for modal graph theory with actuality. To our way ofthinking, the necessary set-theoretic difficulty of defining the modal semantics fora potentialist system such as Mod( T ) poses a philosophical difficulty for advocatesof potentialism seeking a simplified, reduced ontology, for it is as difficult to definethe modal semantics for modal graph theory with actuality, for example, even whenone’s ontology is reduced to having only a set-sized graph at a time, as it is to definethe semantics for full-blown set-theoretic truth. Even ZFC cannot do it. In thissense, it seems impossible for the potentialists to have the parsimonious ontologythey seek, if they also wish to have a coherent potentialist semantics. References [GHH + 20] Victoria Gitman, Joel David Hamkins, Peter Holy, Philipp Schlicht, and KamerynWilliams. The exact strength of the class forcing theorem. Journal of Symbolic Logic ,2020, 1707.03700.[GHJ16] Victoria Gitman, Joel David Hamkins, and Thomas A. Johnstone. What is the theoryZFC without Powerset? Math. Logic Q. , 62(4–5):391–406, 2016, 1110.2430.[Ham03] Joel David Hamkins. A simple maximality principle. Journal of Symbolic Logic ,68(2):527–550, 2003, math/0009240.[Ham18] Joel David Hamkins. The modal logic of arithmetic potentialism and the universalalgorithm. ArXiv e-prints , pages 1–35, 2018, 1801.04599. Under review.[HL08] Joel David Hamkins and Benedikt L¨owe. The modal logic of forcing. Trans. AMS ,360(4):1793–1817, 2008, math/0509616.[HL13] Joel David Hamkins and Benedikt L¨owe. Moving up and down in the generic multi-verse. Logic and its Applications, ICLA 2013 LNCS , 7750:139–147, 2013, 1208.5061.[HL19] Joel David Hamkins and Øystein Linnebo. The modal logic of set-theoretic poten-tialism and the potentialist maximality principles. Review of Symbolic Logic , October2019, 1708.01644.[HLL15] Joel David Hamkins, George Leibman, and Benedikt L¨owe. Structural connectionsbetween a forcing class and its modal logic. Israel Journal of Mathematics , 207(2):617–651, 2015, 1207.5841.[Hod93] Wilfrid Hodges. Model Theory , volume 42 of Encyclopedia of Mathematics and itsApplications . Cambridge University Press, 1993.[HW17] Joel David Hamkins and W. Hugh Woodin. The universal finite set. ArXiv e-prints ,pages 1–16, 2017, 1711.07952. Manuscript under review.[HW19] Joel David Hamkins and Kameryn J. Williams. The σ -definable universal finite se-quence. ArXiv e-prints , 2019, 1909.09100. Under review. (Joel David Hamkins) Professor of Logic, University of Oxford & Sir Peter StrawsonFellow, University College, High Street, Oxford OX1 4BH, United Kingdom E-mail address : [email protected] URL : http://jdh.hamkins.org (Wojciech Aleksander Wo loszyn) Mathematical Institute, University of Oxford, An-drew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX26GG, United Kingdom & St Hilda’s College, Cowley Place, Oxford, OX4 1DY, UnitedKingdom E-mail address : [email protected] URL ::