# Lecture 2: Syntax. January 24, 2018

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1 Lecture 2: Syntax January 24, 2018 We now review the basic definitions of first-order logic in more detail. Recall that a language consists of a collection of symbols {P i }, each of which has some specified arity n i (which is a nonnegative integer). Fix, once and for all, an infinite set V of variables. Given subset V 0 V, a formula of L with free variables in V 0 consists of one of the following five things: (1) An expression x = y, for some pair of elements x, y V 0 (which might be the same). (2) An expression P i (x 1,..., x ni ), where P i is a predicate of the language L having arity n i, and each x i is an element of V 0 (the x i need not be distinct). (3) An expression ϕ ϕ, where ϕ and ϕ are previously constructed formulae with free variables in V 0. (4) An expression ϕ, where ϕ is a previously constructed formula with free variables in V 0. (5) An expression ( x)[ϕ], where x V \V 0, where ϕ is a previously constructed formula with free variables in V 0 {x}. Definition 1. Let L be a language. An L-structure is a set M together with a subset M[P i ] M ni for each predicate symbol P i of L. Let M be an L-structure, let ϕ be a formula of L with free variables {x 1,..., x n }, and suppose we re given an n-tuple of elements c = (c 1,..., c n ) in M. The relation M ϕ( c) is defined by recursion as follows: (1) If ϕ is the formula x i = x j, then M ϕ( c) if and only if c i = c j. (2) If ϕ is the formula P i (x j1,..., x jni ), then M ϕ( c) if and only if and only if the tuple (x j1,..., x jnj ) belongs to M[P i ]. (3) M (ϕ ϕ )( c) if and only if either M ϕ( c) or M ϕ ( c). (4) M ( ϕ)( c) if and only if it is not true that M ϕ( c). (5) If ϕ has the form ( y)ψ for some formula ψ with free variable {x 1,..., x n, y}, the M ϕ( c) if and only if there exists some element a M such that M ψ( c, a). We let M[ϕ] denote the set {(c 1,..., c n ) M n : M ϕ(c 1,..., c n )}. Definition 2. A sentence in a language L is a formula ϕ with no free variables. A theory T consists of a language L, together with a collection of sentences in the language L (which we refer to as the axioms of T. A model of a theory T is an L-structure M such that M ϕ for each axiom ϕ of T. In this case, we write M T. 1

2 Notation 3 (Substitutions of Variables). We will typically denote a formula by ϕ( x), where x = (x 1,..., x n ) is an enumeration of the set of free variables in ϕ. However, we will constantly engage in a certain abuse of this notation. Suppose that ϕ( x) is a formula with free variables in a set of variables V 0 = {x 1,..., x n }, and that we are given another finite set of free variables V 1, together with a map V 0 V 1 (x i V 0 ) (y i V 1 ). We can then form a new formula with free variables in V 1 by replacing each occurrence of the variable x i by y i ; we will denote this formula by ϕ(y 1,..., y n ). Note that we need not assume that the map V 0 V 1 is bijective. For example, if ϕ(x, y) is a formula with free variables x and y, then we can consider ϕ(z, z) as a formula with a single free variable z. We can also consider the case where V 0 is a subset of V 1 and each y i is equal to x i ; in this case, we can regard ϕ( x) as a formula with free variables in V 1, where it just happens that some of the variables have never been mentioned. The preceding discussion comes with a caveat: for the substitution procedure ϕ(x 1,..., x n ) ϕ(y 1,..., y n ) to be sensible, we must assume that the variables of V 1 do not appear as bound variables in the formula ϕ. For example, if ϕ(x) is a formula of the form ( y)r(y, x), then it is sensible to write ϕ(z) for the formula ( y)r(y, z) for any variable z distinct from y, but we don t want to think of the sentence ( y)r(y, y) as an instance of the formula ϕ. Remark 4. We have opted for some economy here in the interest of keeping our definition compact. Using the preceding operations we can define several others. For example: We define ϕ( x) ϕ ( x) to be (( ϕ( x)) ( ϕ ( x))), We define ϕ( x) ϕ ( x) to be ( ϕ( x)) ϕ ( x). We define ϕ( x) ϕ ( x) to be (ϕ( x) ϕ ( x)) (ϕ ( x) ϕ( x)). Given a sequence of variables y = (y 1,..., y m ) and a formula ϕ( x, y), we define ( y)[ϕ( x, y)] to be ( y)( y 2 ) ( y m )[ϕ( x, y)]. Given a sequence of variables y = (y 1,..., y m ) and a formula ϕ( x, y), we define ( y)[ϕ( x, y)] to be ( y)[ ϕ( x, y)]. Given a sequence of variables y = (y 1,..., y m ) and a formula ϕ( x, y), we define (! y)[ϕ( x, y)] to be ( y)[ϕ( x, y)] ( y)( z)[(ϕ( x, y) ϕ( x, z)) (y 1 = z 1 ) (y m = z m )] where z = (z 1,..., z m ) is some list of auxiliary variables which are not bound in ϕ. Let T be a first order theory. Our next goal is to introduce a category Syn 0 (T ), which we will refer to as the weak syntactic category of T. The objects of Syn 0 (T ) are just formulas in the language of T. To avoid confusion, if ϕ( x) if a formula in the language of T, we will write [ϕ( x)] for the corresponding object of Syn 0 (T ). Roughly speaking, we want to think of [ϕ( x)] as the collection of all tuples x satisfying ϕ. However, we view this as an abstract entity, existing without reference to any fixed model of T. Definition 5 (Morphisms in Syn 0 (T )). Let X = [ϕ( x] and Y = [ψ( y)] be objects of Syn 0 (T ). A morphism from X to Y in Syn 0 (T ) consists of a collection of maps f M : M[ϕ] M[ψ], defined for every model M of T, which are definable in the following sense: ( ) After renaming the variables to arrange that the variables x = (x 1,..., x m ) to not appear in ψ and the variables y = (y 1,..., y n ) do not appear in ϕ, there exists a formula θ( x, y) such that, for each model M T, we have M[θ] = Γ(f M ) := {(c 1,..., c m, d 1,..., d n ) : M ϕ(c 1,..., c m ) and f M (c 1,..., c m ) = (d 1,..., d n )} 2

3 Proposition 6. Let T be a first-order theory. Then: (a) Let X = [ϕ( x)] be an object of Syn 0 (T ). Then the collection of identity maps is a morphism id X : X X of Syn 0 (T ). {id M[ϕ] : M[ϕ] M[ϕ]} MT (b) Let X = [ϕ( x)], Y = [ϕ ( y)], and Z = [ϕ ( z)] be objects of Syn 0 (T ), and suppose we are given morphisms X f Y g Z given by maps f M : M[ϕ] M[ϕ ] and g M : M[ϕ ] M[ϕ ] for each model M T. Then the collection of composite maps g M f M determines a morphism from X to Z in Syn 0 (T ), which we will denote by g f. Proof. To prove (a), we first choose some auxiliary variables (y 1,..., y n ) not appearing in ϕ( x). We then observe that graph of the identity map id : M[ϕ] M[ϕ] is given by M[θ], where θ( x, y) is the formula ϕ( x) (x 1 = y 1 ) (x n = y n ). To prove (b), we note that after suitable renaming of variables we may assume that there are formulas θ( x, y) and θ ( y, z) such that M[θ] = Γ(f M ) and M[θ ] = Γ(g M ) for each model M T. Then we have Γ(g M f M ) = M[ρ], where ρ( x, z) is the formula ( y)[θ( x, y) θ ( y, z)]. Corollary 7. For every theory T, Syn 0 (T ) is a category (with the composition law described in Proposition 6). We will refer to Syn 0 (T ) as the weak syntactic category of T. Remark 8. Let T be a (typed) first-order theory and let ϕ be a sentence in the language of T. We will write T ϕ if M ϕ for every model M of T. Let X = [ϕ( x)] and Y = [ψ( y)] be objects of Syn 0 (T ). Assume that the variables of x to not appear in ψ and the variables of y do not appear in ϕ (which we can always arrange by renaming variables). Unwinding the definitions, we see that the morphisms from X to Y in Syn 0 (T ) are given by equivalence classes of formulas θ( x, y) such that T ( x, y)[θ( x, y) ϕ( x) ψ( y)] ( x)[ϕ( x) (! y)θ( x, y)] (More informally, this statement can be read as T implies that θ defines the graph of a function from tuples x satisfying ϕ to tuples y satisfying ψ. ) Here we consider two formulas θ( x, y) and θ ( x, y) to be equivalent if T ( x, y)[θ( x, y) θ ( x, y)]. More informally: θ and θ are equivalent if T implies that they define the graph of the same function. Warning 9. At this point, the reader might reasonably object that Syn 0 (T ) is defined by reference to the models of T, and is therefore not really syntactic in nature. To obtain a fully syntactic definition, we should define the objects of Syn 0 (T ) to be formulas in the language of T, with morphisms from X = [ϕ( x)] and Y = [ψ( y)] given by equivalence classes of formulas θ( x, y) satisfying T ( x, y)[θ( x, y) ϕ( x) ψ( y)] ( x)[ϕ( x) (! y)θ( x, y)], where we consider θ( x, y) and θ ( x, y) to be equivalent if T ( x, y)[θ( x, y) θ ( x, y)]. To make this definition precise, we need to say what it means for a sentence ϕ to be provable in a theory T. After doing so, we ultimately discover that the result is equivalent to the definition we have given above, by virtue of Gödel s completeness theorem. However, this is in some sense cheating : for example, later in this course we would like to apply these ideas to recover a proof of Gödel s theorem. We will return to this point later. 3

4 Construction 10. Let T be a (typed) first-order theory and let M be a model of T. For every object X = [ϕ( x)] in the weak syntactic category Syn 0 (T ), we let M[X] denote the set M[ϕ] = {(c 1,..., c n ) : M ϕ(c 1,..., c n )}. It follows immediately from the definition of morphism in Syn 0 (T ) that we can regard the construction X M[X] as a functor from Syn 0 (T ) to the category Set of sets. Proposition 11. Let T be a first-order theory and let M and N be models of T. The following data are equivalent: (1) The datum of an elementary embedding f 0 : M N (2) The datum of a natural transformation f from M[ ] to N[ ] (in the category of functors from Syn 0 (T ) to Set). Proof. Assume first that f 0 : M N is an elementary embedding. For every formula ϕ(x 1,..., x n ) in the language of T and every c 1,..., c n M, we have (M ϕ(c 1,..., c n )) (N ϕ(f 0 (c 1 ),..., f 0 (c n ))). It follows that the construction (c 1,..., c n ) (f 0 (c 1 ),..., f 0 (c n )) determines a map of sets f ϕ : M[ϕ] N[ϕ]. We claim that these maps determine a natural transformation of functors. In other words, for every morphism g : [ϕ(x 1,..., x n )] [ψ(y 1,..., y m )] in the weak syntactic category, the diagram M[ϕ] f ϕ N[ϕ] g M M[ψ] f ψ g N N[ψ] commutes. Equivalently, we wish to show that the map f n+m 0 : M n+m N n+m carries the graph Γ(g M ) into the graph Γ(g N ). Since g is a morphism in the syntactic category, we can choose a formula θ( x, y) such that Γ(g m ) = M[θ] and Γ(g N ) = N[θ]. We are therefore reduced to showing the implication M θ(c 1,..., c n, d 1,..., d m ) N θ(f 0 (c 1 ),... f 0 (c n ), f 0 (d 1 ),... f 0 (d m )), which follows from our assumption that f 0 is an elementary embedding. Now suppose that we are given a natural transformation of functors f : M[ ] N[ ], given by a collection of maps f ϕ : M[ϕ] N[ϕ]. Taking ϕ(x) to be the formula x = x, we obtain a map f 0 : M N. We claim that f 0 is an elementary embedding. Let ϕ(x 1,..., x n ) be any formula in the language of T. It follows from the naturality of f that we have a commutative diagram M[ϕ] f ϕ N[ϕ] M n f n 0 N n, where the vertical maps are inclusions. From the existence of such a diagram, we deduce the implication (M ϕ(c 1,..., c n )) (N ϕ(f 0 (c 1 ),..., f 0 (c n ))). Applying the same argument to the formula ϕ( x), we deduce the reverse implication, so that f 0 is an elementary embedding. We leave it to the reader to verify that these constructions are inverse to one another. 4

5 The conclusion of Proposition 11 is that it is not really necessary to distinguish between a model M of T and the associated functor M[ ] : Syn 0 (T ) Set; they are essentially the same data. One can think of the functor M[ ] as an unbiased presentation of the datum of a model of T : one does not think of a model as something that has a single underlying set: instead, one considers all the sets which can be constructed from the model by means of a first-order formula. Of course, if we want to think about models as a special kind of functor, we will need to decide what kind of functors that we want to allow: Question 12. Let T be a first-order theory and let F : Syn 0 (T ) Set be a functor. When does there exist a model M of T and an isomorphism of functors F M[ ]? We will take up Question 12 in the next lecture. We close by describing a mild generalization of the formalism described above. Recall that in the previous lecture, we introduced the notion of a projective plane. A projective plane is naturally described as a pair of sets P and L (the points and lines, respectively), together with a binary relation between elements of P and elements of L. This is most naturally described by a slight variation on the preceding definitions. Variant 13 (Typed First-Order Logic). A typed language L consists of a set T of types together with a set {P i } of predicate symbols. Each predicate symbol has an arity which is no longer a nonnegative integer, but a finite sequence t = (t 1,..., t n ) of types. We assume that each variable name is assigned a type, in such a way that there are infinitely many variable names of each type. The notion of formula is defined as before, with two caveats: (0) We add a new formula (for any number of free variables), to be understood simply as false. This is necessary only in the case where there are no types (hence no variables); otherwise, we could just pick a variable x and replace by a contradictory assertion like ( x)[ (x = x)]. (1 ) We allow formulas of the form x = y only when x and y are variables names of the same type. (2 ) If P i is a predicate of arity (t 1,..., t n ), then we allow formulas of the form P i (x 1, x 2,..., x n ) only when each x j has type t j. An L-structure consists of a set M[t] for every type t T, together with a subset M[P i ] M[t 1 ] M[t n ] for each predicate symbol P i of arity (t 1,..., t n ). More generally, for any formula ϕ(x 1,..., x n ) with variables x i of types t i, we define the notion M ϕ(c 1,..., c n ) for (c 1,..., c n ) M[t 1 ] M[t n ] by induction as before, with the added clause that M (c 1,..., c n ) is always false. We set M[ϕ] = {(c 1,..., c n ) M[t 1 ] M[t n ] : M ϕ(c 1,..., c n )}. Example 14. In the special case where there is only one type, Variant 13 reduces to the definitions given above it. Remark 15. In the case where there are only finitely many types, Variant 13 represents no real gain in generality. For example, instead of axiomatizing the theory of projective planes in terms of two sets P and L (together with the binary incidence relation P L), one could consider the structure given by the disjoint union P L, augmented by two new unary relations U(x) = x is a point V (x) = x is a line ( x)(u(x) V (x)) ( x) [U(x) V (x)]. However, this sort of trick does not work if we want to allow infinitely many types. 5

6 Example 16 (Propositional Logic). In the situation of Variant 13, suppose that the set T of types is empty. Then the language L consists of a collection of symbols {P i }, all of which are forced to have arity 0. The set V of variable names is empty (note that we still have infinitely many variables of each type, vacuously). All formulas are sentences, which can be built by combining the symbols P i and by means of the negation operation and the disjunction operation. An L-structure is equivalent to the datum of a map {P i } {True, False} which assigns a truth value to each P i. This determines, more generally, a truth value for each sentence ϕ built from the P i. Example 17 (The Syntactic Category of a Propositional Theory). Let T be a propositional theory (that is, a typed first order theory with no types). Then every formula in the language of T is a sentence. Moreover, for every pair of sentences ϕ and ψ, we have { if T (ϕ ψ) Hom Syn0 (T )([ϕ], [ψ]) = otherwise. In other words, the category Syn 0 (T ) is equivalent to the partially ordered set of equivalence classes of sentences. This partially ordered set is a Boolean algebra, with join given by (ϕ, ψ) ϕ ψ and complements given by ϕ ϕ. 6

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