LOGIC NOTES YIANNIS N. MOSCHOVAKIS

Transcription

1 LOGIC NOTES YIANNIS N. MOSCHOVAKIS July 16, 2012

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3 CONTENTS Chapter 1. First order logic A. Examples of structures B. The syntax of First Order Logic (FOL) C. Semantics of FOL D. First order definability E. Arithmetical functions and relations F. Quantifier elimination G. Theories and elementary classes H. The Hilbert-style proof system for FOL I. The Completeness Theorem J. The Compactness and Skolem-Löwenheim Theorems K. Some other languages L. Problems for Chapter Chapter 2. Some results from model theory A. More consequences of Completeness and Compactness B. The downward Skolem-Löwenheim Theorem C. Types D. Back-and-forth games E. 1 1 on countable structures F. Craig interpolation and Beth definability (via games) G. Problems for Chapter Chapter 3. Introduction to the theory of proofs A. The Gentzen Systems B. Cut-free proofs C. Cut Elimination D. The Extended Hauptsatz E. The propositional Gentzen systems F. Craig Interpolation and Beth definability (via proofs) G. The Hilbert program H. The finitistic consistency of Robinson s Q i

4 ii CONTENTS 3I. Primitive recursive functions J. Further consistency proofs K. Problems for Chapter Chapter 4. Incompleteness and undecidability A. Tarski and Gödel (First Incompleteness Theorem) B. Numeralwise representability in Q C. Rosser, more Gödel and Löb D. Computability and undecidability E. Computable partial functions F. The basic undecidability results G. Problems for Chapter Chapter 5. Introduction to computability theory A. Semirecursive relations B. Recursively enumerable sets C. Productive, creative and simple sets D. The Second Recursion Theorem E. The arithmetical hierarchy F. Relativization G. Effective operations H. Computability on Baire space I. The analytical hierarchy J. Problems for Chapter Chapter 6. Appendix to Chapters Chapter 7. Introduction to formal set theory A. The intended universe of sets B. ZFC and its subsystems C. Set theory without powersets, AC or foundation, ZF D. Set theory without AC or foundation, ZF E. Cardinal arithmetic and ultraproducts, ZFC F. Problems for Chapter Chapter 8. The constructible universe A. Preliminaries and the basic definition B. Absoluteness C. The basic facts about L D E. L and Σ F. Problems for Chapter Informal notes, full of errors, July 16, 2012, 11:13 ii

5 CHAPTER 1 FIRST ORDER LOGIC Our main aim in this fist chapter is to introduce the basic notions of logic and to prove Gödel s Completeness Theorem 1I.1, which is the first, fundamental result of the subject. Along the way to motivating, formulating precisely and proving this theorem, we will also establish some of the basic facts of Model Theory, Proof Theory and Recursion Theory, three of the main parts of logic. (The fourth is Set Theory.) 1A. Examples of structures The language of First Order Logic is interpreted in mathematical structures, like the following. Definition 1A.1. A graph is a pair G = (G, E) where G is a non-empty set (the nodes) and E G G is a binary relation on G, (the edges); G is symmetric or unordered if E(x, y) = E(y, x). A path in a symmetric graph G = (G, E) is a sequence of points (vertices) (x 0, x 1,..., x n ) such that there is an edge joining each x i with x i+1, i.e., E(x 0, x 1 ), E(x 1, x 2 ),..., E ( x n 1, x n ); a path joins its first vertex x 0 with its last x n. The distance between two vertices x, y which can be joined in G is the length (number of edges, n above) of the shortest path joining them, d(x, y) = min{n there exists a path x 0,..., x n with x 0 = x, x n = y}, and (by convention) it is 0 from a vertex to itself, d(x, x) = 0 and if x y and there is no path from x to y. The diameter of a symmetric 1

6 2 1. First order logic graph is the largest distance between two vertices, if there is a maximum distance, otherwise it is : diam(g) = sup{d(x, y) x, y G}. A symmetric graph is connected if any two distinct points in it are joined by a path, otherwise it is disconnected. Definition 1A.2. A partial ordering is a pair P = (P, ), where P is a non-empty set and is a binary relation on P satisfying the following conditions: 1. For all x P, x x (reflexivity). 2. For all x, y, z P, if x y and y z, then x z (transitivity). 3. For all x, y P, if x y and y x, then x = y (antisymmetry). A linear ordering is a partial ordering in which every two elements are comparable, i.e., such that 4. For all x, y P, either x y or y x. A wellordering is a linear ordering (U, ) in which every non-empty subset has a least element: i.e., for every X U, if X, then there exists some x 0 X such that for all x X, x 0 x. Definition 1A.3. The structure of arithmetic or the natural numbers is the tuple N = (N, 0, S, +, ) where N = {0, 1, 2,... } is the set of (non-negative) integers and S, +, are the operations of successor, addition and multiplication on N. The structure N has the following characteristic properties: (1) The successor function S is an injection, i.e., S(x) = S(y) = x = y, and 0 is not a successor, i.e., for all x, S(x) 0. (2) For all x, y, x + 0 = x and x + S(y) = S(x + y). (3) For all x, y, x 0 = 0 and x S(y) = x y + x. (4) The Induction Principle: for every set of numbers X N, if 0 X and for every x, x X = S(x) X, then X = N. These properties (or sometimes just (1) and (4)) are called the Peano Axioms for the natural numbers. Definition 1A.4. A field is a structure of the form K = (K, 0, 1, +, ) where 0, 1 K, + and are binary operations on K and the following field axioms are true. Informal notes, full of errors, July 16, 2012, 11:13 2

7 1A. Examples of structures 3 (1) (K, 0, +) is a commutative group, i.e., the following hold: 1. For all x, x + 0 = x. 2. For all x, y, z, x + (y + z) = (x + y) + z. 3. For all x, y, x + y = y + x. 4. For each x there exists some y such that x + y = 0. (2) 1 0 and for all x, x 0 = 0, x 1 = x. (3) The structure (K \{0}, 1, ) is a commutative group, and in particular x, y 0 = x y 0. Together with (2), this means that for all x, y in K, x y = 0 x = 0 or y = 0. (4) For all x, y, z, x (y + z) = x y + x z. Basic examples of fields are the rational numbers Q, the real numbers R and the complex numbers C, with universes Q, R, C respectively and the usual operations on these number sets. Definition 1A.5. The universe of sets is the structure V = (V, ) where V is the collection of all sets and is the binary relation of membership. We list here the most common set of axioms usually assumed about sets, not in the simplest way, but directly in terms of the basic membership relation, without introducing any auxiliary notions. (1) Extensionality: two sets are equal exactly when they have the same members, in symbols: x = y ( u)[u x u y]. (2) Emptyset and Pairing: there exists a set with no members, and for any two sets x, y, there is a set z whose members are exactly x and y, i.e., for all u, u z u = x or u = y. (3) Union: for each set x there exists a set z whose members are the members of members of x, i.e., for all u u z ( y x)[u y]. (4) Power: for each set x there exists a set z whose members are all the subsets of x, i.e., for all u, u z ( v u)[v x]. Informal notes, full of errors, July 16, 2012, 11:13 3

8 4 1. First order logic (5) Subsets: for each set x and each definite condition P (u) on sets, there exists a set z whose members are the members of x which satisfy P (u), i.e., for all u, u z u x and P (u). (6) Infinity: there exists a set z such that z and z is closed under the singleton operation, i.e., for every x, x z = {x} z. (7) Choice: for every set x whose members are all non-empty and pairwise disjoint, there exists a set z which intersects each member of x in exactly one point, i.e., if y x, then there exists exactly one u such that u y and also u z. (8) Replacement: for every set x and every definite operation F which assigns a set F (v) to every set v, the image F [x] of x by F is a set, i.e., there exists a set z such that for all u, u z ( v x)[u = F (v)]. (9) Foundation: every non-empty set x has a member z from which it is disjoint, i.e., there is no u X such that also u z. 1B. The syntax of First Order Logic (FOL) The name FOL abbreviates First Order Logic. It is actually a family of languages FOL(τ), one for each vocabulary τ, where τ provides names for the distinguished elements, relations and functions of the structures we want to talk about. FOL is also known as Lower Predicate Calculus (with Identity), or Elementary Logic with Identity or just Elementary Logic. Definition 1B.1. A vocabulary or signature is a quadruple τ = (Const, Rel, Funct, arity), where the sets of constant symbols Const, relation symbols Rel, and function symbols Funct have no common members and arity : Rel Funct {1, 2,... }. A relation or function symbol P is n-ary if arity(p ) = n. We will often assume that these sets of names are finite (as they are in the examples above), but it is convenient and useful to allow them to be arbitrary sets in the general case; and we should also keep in mind that any one or all of these sets may be empty. When they are all finite, we usually exhibit Informal notes, full of errors, July 16, 2012, 11:13 4

9 1B. The syntax of First Order Logic (FOL) 5 signatures by enumerating their symbols: for example, is a signature for graphs; τ g = (R) (with R binary) τ a = (0, S, +,.) is a signature for arithmetic (with 0 a constant, S a unary function symbol and +, binary function symbols); τ f = (0, 1, +, ) (with the appropriate arities) is a signature for fields; and τ = ( ) (with binary) is a signature for universe of sets. (We say a rather than the signature because the symbols R, S, +, etc. are arbitrary.) Definition 1B.2. The alphabet of the first order language with identity FOL(τ) comprises the symbols in the vocabulary τ and the following, additional symbols which are common to all FOL(τ). 1. The logical symbols & = 2. The punctuation symbols ( ), 3. The (individual) variables: v 0, v 1, v 2,... Here (not), & (and), (or) and (implies) are the propositional symbols, and (for all) and (there exists) are the quantifiers. Words are finite strings of symbols and lh(α) is the length of the word α. We use to denote identity of strings, We also set α β df α and β are the same string. α β df α is an initial segment of β, so that e.g., v 0 v 0 R(v 0 ). The concatenation of two strings αβ is the string produced by putting them together, with α first, so that α αβ. Definition 1B.3 (Terms and formulas). Terms are defined by the recursion: (a) Each variable is a term. (b) Each constant symbol is a term. (c) If t 1,..., t n are terms and f is an n-ary function symbol, then f(t 1,..., t n ) is a term. In abbreviated notation: where is read as or. t : v c f(t 1,..., t n ), Formulas are defined by the recursion: (a) If s, t are terms, then s = t is a formula. (b) If t 1,..., t n are terms and R is an n-ary relation symbol, Informal notes, full of errors, July 16, 2012, 11:13 5

10 6 1. First order logic then R(t 1,..., t n ) is a formula. (c) If φ, ψ are formulas and v is a variable, then the following are formulas: In abbreviated form, (φ) (φ) (ψ) (φ) & (ψ) (φ) (ψ) vφ vφ χ : s = t R(t 1,..., t n ) (the prime formulas) (φ) (φ) (ψ) (φ) & (ψ) (φ) (ψ) vφ vφ For the rigorous interpretations of these recursive definitions of sets see Problem x6.3. A formula is quantifier free if neither of the quantifier symbols, occurs in it. A formula is in prenex normal form (prenex) if it looks like φ Q 1 x 1 Q n x n ψ where each Q i or Q i and ψ is quantifier free. Terms and formulas are collectively called (well formed) expressions. Proposition 1B.4 (Parsing for terms). Each term t satisfies exactly one of the following three conditions. 1. t v for a uniquely determined variable v. 2. t c for a uniquely determined constant c. 3. t f(t 1,..., t n ) for a uniquely determined function symbol f and uniquely determined terms t 1,..., t n. Proposition 1B.5 (Parsing for formulas). Each formula χ satisfies exactly one of the following conditions. 1. χ s = t for uniquely determined terms s, t. 2. χ R(t 1,..., t n ) for a uniquely determined relation symbol R and uniquely determined terms t 1,..., t n. 3. χ (φ) for a uniquely determined formula φ. 4. χ (φ) & (ψ) for uniquely determined formulas φ, ψ. 5. χ (φ) (ψ) for uniquely determined formulas φ, ψ. 6. χ (φ) (ψ) for uniquely determined formulas φ, ψ. 7. χ vφ for a uniquely determined variable v and a uniquely determined formula φ. 8. χ vφ for a uniquely determined variable v and a uniquely determined formula φ. These propositions allow us to prove properties of expressions by structural induction, i.e., induction on the length of expressions; and we can also give definitions by structural recursion, i.e., recursion on the length of expressions, cf. Problem x6.4. Informal notes, full of errors, July 16, 2012, 11:13 6

11 1B. The syntax of First Order Logic (FOL) 7 Definition 1B.6 (Free and bound variables). Every occurrence of a variable in a term is free. The free occurrences of variables in formulas are defined by structural recursion as follows. 1. FO(s = t) = FO(s) FO(t). 2. FO( (φ)) = FO(φ), FO((φ) & (ψ)) = FO(φ) FO(ψ), and similarly for the other connectives. 3. FO( vφ) = FO( vφ) = FO(φ) \ {v}, meaning that we remove from the free occurrences of variables in φ all the occurrences of the variable v. An occurrence of a variable which is not free in an expression α is bound in α. The free variables of α are the variables which have at least one free occurrence in α; the bound variables of α are those which have at least one bound occurrence in α. To illustrate what these notions mean, consider the three formulas in the language of arithmetic φ : v 1 (+(v 2, v 1 ) = 0), ψ : v 5 (+(v 2, v 5 ) = 0), χ : v 1 (+(v 5, v 1 ) = 0) As we read these formulas in English (unabbreviating the formal symbols), the first two of them say exactly the same thing: that we can add some number to v 2 and get 0 which is true exactly when v 2 is a name of 0. The third formula says the same thing about whatever number v 5 names, which need not be the same as the number named by v 2. In short, the meaning (and truth value) of a formula does not change if we replace its bound variables by others, but it may change when we change its free variables. A customary example from calculus is the notation we use for integrals: for a b, a 0 x 2 dx = a 0 y 2 dy = a3 3 but a 0 x 2 dx b 0 x 2 dx = b3 3, which means that in the expression a 0 x2 dx the occurrences of x are bound, while a occurs freely. An expression is closed if it has no free occurrences of variables. A closed formula is a sentence. The universal closure of a formula φ is the sentence φ df v 0 v 1... v n φ, where n is least so that all the free variables of φ are among v 0,..., v n. Note that a variable may occur both free and bound within a formula. For example, the following are well formed by the rules: v 1 v 1 R(v 1, v 1 ), ( v 1 R(v 1 )) & ( v 2 S(v 1, v 2 )) Informal notes, full of errors, July 16, 2012, 11:13 7

12 8 1. First order logic (Think through what these formulas mean, and which variable occurrences are free or bound in them.) 1B.7. Abbreviations and misspellings. In practice we never write out terms and formulas in full: we use infix notation for terms, e.g., s + t for + (s, t) in arithmetic, we introduce and use abbreviations, we use metavariables (names) x, y, z, u, v,..., for the specific formal variables of the language, we skip (or add) parentheses or replace parentheses by brackets or other punctuation marks, and (in general) we are satisfied with giving instructions for writing out a formula rather than exhibiting the actual formula. For example, the following sentence says about arithmetic that there are infinitely many prime numbers: ( x)( y)[x y & ( u)( v)[(y = u v) (u = 1 v = 1)] where we have used the abbreviations x y : ( z)[x + z = y] 1 : S(0). The correctly spelled sentence which corresponds to this is quite long (and unreadable). Two useful logical abbreviations are for the iff (φ ψ) : ((φ ψ) & (ψ φ)) and the quantifier there exists exactly one (!x)φ : ( z)( x)[φ x = z], where z x. (Think this through.) We also set 0 i n φ i : φ 0 φ 1 φ n 0 i n φ i : φ 0 & φ 1 & & φ n (and analogously for more complex sets of indices). Definition 1B.8 (Substitution). For each expression α, each variable v and each term t, the expression α{v : t} is the result of replacing all free occurrences of v in α by the term t; we say that t is free for v in α if no occurrence of a variable in t is bound in the result of the substitution α{v : t}. The simultaneous substitution α{v 1 : t 1,..., v n : t n } is defined similarly: we replace simultaneously all the occurrences of each v i in α by t i. Note than in general α{v 1 : t 1 }{v 2 : t 2 } α{v 1 : t 1, v 2 : t 2 }. If α is an expression, then α{v 1 : t 1,..., v n : t n } is also an expression, and of the same kind term or formula. Informal notes, full of errors, July 16, 2012, 11:13 8

13 1C. Semantics of FOL 9 Definition 1B.9 (Extended expressions). An extended expression is a pair (α, (v 1,..., v n )) of a (well formed) term or formula α and a list of distinct variables, We set α(v 1,..., v n ) : (α, (v 1,..., v n )), and for any sequence of terms t 1,..., t n, we set α(t 1,..., t n ) : α{v 1 : t 1,..., v n : t n }. This is essentially a notational convention, to facilitate dealing with substitutions, and the pedantic distinction between expressions and extended expressions is not always explicitly noted: we may refer to a formula α( v), letting the notation indicate that we are really specifying both a formula α and a list v = (v 1,..., v n ). An extended expression α(v 1,..., v n ) is full if the list (v 1,..., v n ) includes all the variables which occur free in α. Definition 1B.10 (Sublanguages). If τ, τ are vocabularies and each symbol of τ is a symbol (of the same kind and with the same arity) in τ, we say that τ is a reduct of τ and we write τ τ. 1B.11. First order logic without identity. We will also work with the smaller language FOL, which is obtained by removing the symbol = and the clauses involving it in the definitions. There are no formulas in FOL (τ), unless the signature τ has at least one relation symbol, and so when we state results about FOL (τ), we will tacitly assume that the signature has at least one relation symbol. 1C. Semantics of FOL We interpret the terms and formulas of the language FOL(τ) in structures of signature τ, which include the examples in Section 1A and are defined in general as follows: Definition 1C.1 (Structures). A τ-structure is a pair A = (A, I ) where A is a non-empty collection of objects and I assigns to each constant symbol c a member I (c) of A; to each n-ary relation symbol an n-ary relation I (R) A n ; and to each n-ary function symbol f an n-ary function I (f) : A n A. The set A is the universe of the structure A, and the constants, relations and functions which interpret the symbols of the signature in A are its primitives. We set c A = I(c), R A = I(R), f A = I(f), so that the specification of a τ-structure can be given in the form A = (A, {c A } c Const, {R A } R Rel, {f A } f Funct ). Informal notes, full of errors, July 16, 2012, 11:13 9

14 10 1. First order logic In the typical case where there are only finitely many symbols in τ, we denote structures as tuples, as in Section 1A, so that a graph G = (G, R) is a (R)-structure and the structure N = (N, 0, S, +, ) of arithmetic is a (0, S, +, )-structure. Definition 1C.2 (Homomorphisms and isomorphisms). A homomorphism ρ : A B on one τ-structure A = (A, I) to another B = (B, J) is any mapping ρ : A B which satisfies the following three conditions: 1. For each constant c, ρ(c A ) = c B ; 2. for each n-ary function symbol f and all x 1,..., x n A, ρ(f A (x 1,..., x n )) = f B (ρ(x 1 ),..., ρ(x n )); 3. for each n-ary relation symbol R and all x 1,..., x n A, (1) R A (x 1,..., x n ) = R B (ρ(x 1, ),..., ρ(x n )); it is a strong homomorphism if it also satisfies (2) R A (x 1,..., x n ) R B (ρ(x 1, ),..., ρ(x n )), which is stronger than (1). An embedding ρ : A B is an injective strong homomorphism, and an isomorphism ρ : A B is an embedding which is bijective (one-to-one and onto). We set A B there exists an isomorphism ρ : A B, and when these conditions hold, we say that A and B are isomorphic. An isomorphism ρ : A A of a structure A onto itself is an automorphism of A, and a structure A is rigid if it has no automorphisms other than the (trivial) identity function id : A A, id(x) = x. The basic properties of homomorphisms in the next proposition are quite easy and we will leave the proof for Problem x1.4; here ρ π : V B is the composition of given π : V A and ρ : A B, (ρ π)(v) = ρ(π(v)). Proposition 1C.3. (a) If ρ : A B is a homomorphism, then for every term t and every assignment π into A, value B (t, ρ π) = ρ(value A (t, π)). Informal notes, full of errors, July 16, 2012, 11:13 10

15 1C. Semantics of FOL 11 (b) If ρ : A B is a surjective, strong homomorphism, then for every formula φ of FOL (τ) and every assignment π into A, (3) A, π = φ B, ρ π = φ, so that, in particular, for every FOL (τ)-sentence χ, (4) A = χ B = χ. (c) If ρ : A B is an isomorphism, then (3) holds for all FOL(τ)- formulas φ, and (4) holds for all FOL(τ)-sentences χ. Definition 1C.4 (Substructures). Suppose A = (A, I ), B = (B, J ) are τ-structures. We call A a substructure of B and B an extension of A and we write A B if the identity mapping id : A A an embedding of A to B, i.e., if the following conditions hold. 1. A B. 2. For each constant symbol c of τ, c B = c A A. 3. For each n-ary relation symbol R and all x 1..., x n A, R B (x 1..., x n ) R A (x 1..., x n ). 4. For each n-ary function symbol f and all x 1..., x n A, f B (x 1..., x n ) = f A (x 1..., x n ) A. For example, the field of rationals Q (the fractions) is a substructure of the field of real numbers R in the language of fields. Definition 1C.5 (Expansions and reducts). Suppose σ τ are signatures, A = (A, I ) is a σ-structure and B = (B, J ) is a τ-structure. We call A a reduct of B and B an expansion of A if A = B and for all symbols C σ, I (C) = J (C). If B is a given τ-structure and σ τ, we define the reduct of B to σ by deleting from B the objects assigned to the symbols not in σ, formally B σ = (B, J σ). Conversely, if τ σ, we can define expansions of B by assigning interpretations to the symbols in σ which are not in τ. The standard notation for this operation is (B, K ) = df the expansion of B by K, which is easier to understand in examples: (N, 0, S) and (N, 0, +) are reducts of the structure of arithmetic N obtained (in the first case) by deleting from the signature the symbols + and, so that (N, 0, S) = N {0, S}. Informal notes, full of errors, July 16, 2012, 11:13 11

16 12 1. First order logic A useful expansion of N is obtained by adding to the signature a symbol exp for exponentiation, and to N the corresponding operation, (N, exp) = the expansion of N by exp. It is important to keep clear the (trivial) distinction between substructuresextensions and reducts-expansions. Definition 1C.6 (Truth values). We will use the numbers 0 and 1 to denote the truth values, 0 for falsity and 1 for truth. Definition 1C.7 (Assignments). An assignment into a structure A is any function π : Variables A. If v is a variable and x A, then π{v := x} is the assignment which agrees with π on all variables except v, to which it assigns x: { x, if u v, π{v := x}(u) = π(u), otherwise. We call π{v := x} the update of π by (the reassignment) v := x. Definition 1C.8 (Denotations and satisfaction). The value or denotation of a term for an assignment π is defined by structural recursion on the terms as follows: 1. value(v, π) = df π(v). 2. value(c, π) = df c A. 3. value(f(t 1,..., t n ), π) = df f A (value(t 1, π),..., value(t n, π)). In the same way, by structural recursion on formulas, we define the truth value or denotation of a formula for an assignment π: { 1, if value(s, π) = value(t, π), 1. value(s = t, π) = df 0, otherwise. { 1, if R 2. value(r(t 1,..., t n ), π) = A (value(t 1, π),..., value(t n, π)), df 0, otherwise. 3. value( φ, π) = df 1 value(φ, π). 4. value((φ) & (ψ), π) = min(value(φ, π), value(ψ, π)). For we take the maximum and for implication we use value((φ) (ψ), π) = df value(( (φ)) (ψ)) = max(1 value(φ), value(ψ)). 5. value( vφ, π) = df max{value(φ, π{v := x}) x A}. 6. value( vφ, π) = df min{value(φ, π{v := x}) x A}. The denotation function depends on the structure, of course, although we suppressed this in the notation. When we need to exhibit the dependence we write value A (α, π) = value(α, π), Informal notes, full of errors, July 16, 2012, 11:13 12

17 and for formulas 1C. Semantics of FOL 13 A, π = φ df value A (φ, π) = 1. If A, π = φ, we say that the assignment π satisfies φ in A. Theorem 1C.9 (The Tarski truth conditions). The satisfaction relation on σ-structures, σ-formulas and assignments satisfies the following conditions: A, π = s = t value A (t, π) = value A (s, π) A, π = R(t 1,..., t n ) R A (value A (t 1, π),..., value A (t n, π)) A, π = φ A, π = φ A, π = φ & ψ A, π = φ and A, π = ψ A, π = φ ψ A, π = φ or A, π = ψ A, π = φ ψ either A, π = φ or A, π = ψ A, π = vφ there exists an x A such that A, π{v := x} = φ A, π = vφ for all x A, A, π{v := x} = φ Proof is by structural induction on formulas. Definition 1C.10 (Validity and semantic consequence).. For any τ- formulas φ, we set: = φ (for all A, π), A, π = φ; if = φ, we call φ valid or logically true, and if = φ ψ we say that φ logically implies ψ or ψ is a semantic (or logical) consequence of φ. Two sentences are semantically (or logically) equivalent if each is a semantic consequence of the other. Denotations of terms and formulas are defined in such a way that the value of an expression is a function of the values of its subexpressions. This is generally referred to as the Compositionality Principle for denotations, and it is the key to a mathematical analysis of denotations. The next theorem expresses it rigorously. Theorem 1C.11 (Compositionality). (1) If the σ-structure A is a reduct of the τ-structure B where σ τ, then for every σ-expression α and every assignment π, value A (α, π) = value B (α, π). (2) If π, ρ are two assignments into the same structure A and for every variable v which occurs free in an expression α, π(v) = ρ(v), then value A (α, π) = value A (α, ρ), Informal notes, full of errors, July 16, 2012, 11:13 13

18 14 1. First order logic so that, in particular, for any formula χ, A, π = χ A, ρ = χ. Proof of both claims is by structural induction on α. By appealing to compositionality, we set for each full extended term α(v 1,..., v n ) and each n-tuple (x 1,..., x n ) from A, α A [ x] = df value A (α, π{ v := x}) and similarly, for any full extended formula φ( v), (for any assignment π), A = φ[ x] df for some assignment π, A, π{ v := x} = φ for every assignment π, A, π{ v := x} = φ. These useful notations are even simpler for closed expressions: value A (α) = df value A (α, π) (α closed), A = φ df φ is true in A(φ a sentence) A, π = φ where π is any assignment. A τ-sentence is valid if it is true in all τ- structures. 1D. First order definability A proposition Φ of ordinary (mathematical) English, about a certain τ- structure A is expressed by a sentence φ of FOL(τ) if Φ and φ mean the same thing; similarly, a proposition Φ(x) about an arbitrary object x in a structure A is expressed by a formula φ(x) with one free variable x, if for each x A, Φ(x) and φ(x) mean the same thing. For example, ( x)[x + 0 = x] means every number added to 0 yields itself. We cannot make this notion of expressing precise unless we first define meaning rigorously for both natural language and FOL. On the other hand, we have a clear, intuitive understanding of it which is important for applications: roughly speaking, φ expresses Φ if we can construct the first from the second by straightforward translation, more-or-less word for word, and, but, also going to &, all, each, any going to, etc. For example, every number is either odd or even refers to the structure of arithmetic and translates to something of the form ( x)[φ(x) ψ(x)] where φ(x) and ψ(x) can be constructed to express the properties of being odd or even. Informal notes, full of errors, July 16, 2012, 11:13 14

19 1D. First order definability 15 As it turns out, all mathematical propositions and properties can be expressed by FOL(τ)-sentences or formulas on appropriate structures. This is one of the main discoveries of modern mathematical logic and the source of its applications to mathematics. We will explain how it works in the sequel, starting in this section with the theory of first order definability on a fixed structure. Definition 1D.1 (The basic local notions). Suppose A is a τ-structure. An n-ary relation R A n on A is first order definable or elementary on A, if there is a full extended formula χ(v 1,..., v n ) such that R( x) A = χ[ x] ( x A n ). A function f : A n A is A-explicit if for some full extended term α( v) f( x) = α A [ x] ( x A n ). A function f : A n A is first order definable or elementary on A if its graph G f ( x, w) f( x) = w is elementary on A, i.e., if there is a full extended formula χ( v, u) such that f( x) = w A = χ[ x, w]. The elementary functions and relations of the standard structure N of arithmetic are called arithmetical. The next theorem is useful, as it often frees us from needing to worry excessively about the formal syntax of FOL. Theorem 1D.2. The collection E(A) of A-elementary functions and relations on the universe of a structure A = (A, {c A } c Const, {R A } R Rel, {f A } f Funct ) has the following properties: (1) Each primitive relation R A is A-elementary; and the (binary) identity relation x = y is A-elementary. (2) For each constant symbol c and each n, the n-ary constant function g( x) = c A is A-elementary; each primitive function f A is A-elementary; and every projection function is A-elementary. P n i (x 1,..., x n ) = x i (1 i n) Informal notes, full of errors, July 16, 2012, 11:13 15

20 16 1. First order logic (3) E(A) is closed under substitutions of A-elementary functions: i.e., if h(u 1,..., u m ) is an m-ary A-elementary function and g 1 ( x),..., g m ( x) are n-ary, A-elementary, then the function f( x) = h(g 1 ( x),..., g m ( x)) is A-elementary; and if P (u 1,..., u m ) is an m-ary A-elementary relation, then the n-ary relation Q( x) P (g 1 ( x),..., g m ( x)) is A-elementary. (4) E(A) is closed under the propositional operations: i.e., if P 1 ( x) and P 2 ( x) are A-elementary, n-ary relations, then so are the following relations: Q 1 ( x) P 1 ( x), Q 2 ( x) P 1 ( x) & P 2 ( x), Q 3 ( x) P 1 ( x) P 2 ( x), Q 4 ( x) P 1 ( x) P 2 ( x). (5) E(A) is closed under quantification on A, i.e., if P ( x, y) is A-elementary, then so are the relations Q 1 ( x) ( y)p ( x, y), Q 2 ( x) ( y)p ( x, y). Moreover: E(A) is the smallest collection of functions and relations on A which satisfies (1) (5). Proof. To show that E(A) has these properties, we need to construct lots of formulas and appeal repeatedly to the definition of A-elementary functions and relations; this is tedious, but not difficult. For the second ( moreover ) claim, we first make it precise by replacing E(A) by F throughout (1) (5), and (temporarily) calling a class F of functions and relations good if it satisfies all these conditions so what has already been shown is that E(A) is good. The additional claim is that every good F contains all A-elementary functions and relations, and it is verified by structural induction on the formula χ such that some full extension of it χ( v) defines a given, A-elementary relation after showing, easily, that the graph of every A-explicit function is in F. The theorem suggests that E(A) is a very rich class of relations and functions. As it turns out, this is true for rich, standard structures like N, but not true for structures with simple primitives e.g., the plain (A) which has no primitives. We consider examples of these two kinds of structures in the next two sections. Informal notes, full of errors, July 16, 2012, 11:13 16

21 1E. Arithmetical functions and relations 17 1E. Arithmetical functions and relations Is the exponential function exp(t, x) = x t (x, t N) arithmetical? Not obviously but it is, as a corollary of a basic result about definition by recursion in N which we will prove in this section, and which has many important applications. Definition 1E.1 (Primitive recursion). A function f : N N is defined by primitive recursion from the number w 0 N and the binary function h(w, t) if it satisfies the following two equations, for all t: (5) f(0) = w 0, f(t + 1) = h(f(t), t); in greater generality, a function f : N n+1 N of n + 1 arguments on the natural numbers is defined by primitive recursion from the n-ary function g and the (n + 2)-ary function h if it satisfies the following two equations, for all t, x: (6) f(0, x) = g( x), For example, if we set f(0, x) = x, then (easily, by induction on t) f(t + 1, x) = h(f(t, x), t, x). f(t + 1, x) = S(f(t, x)), f(t, x) = t + x, and so addition is defined by primitive recursion from the two, simpler functions g(x) = x, h(w, t, x) = S(w), i.e., (essentially) the identity and the successor. Similarly, if we set f(0, x) = 0, f(t + 1, x) = f(t, x) + x, then, easily, f(t, x) = t x, and so multiplication is defined by primitive recursion from the functions g(x) = 0, h(w, t, x) = w + t, i.e., (essentially) the constant 0 and addition. More significantly (for our purposes here), exp(0, x) = x 0 = 1, exp(t + 1, x) = x t+1 = x t x = exp(t, x) x, so that exponentiation is defined by primitive recursion from the functions g(x) = 1, h(w, t, x) = w x, i.e., (essentially) the constant 1 and multiplication. Informal notes, full of errors, July 16, 2012, 11:13 17

22 18 1. First order logic Theorem 1E.2. If f : N n+1 N is defined by the primitive recursion (6) above and g, h are arithmetical, then so is f. To prove this we must reduce the recursive definition of f into an explicit one, and this is done using Dedekind s analysis of recursion: Proposition 1E.3. If f : N n+1 N is defined by the primitive recursion in (6), then for all t, x, w, (7) f(t, x) = w there exists a sequence (w 0,..., w t ) such that w 0 = g( x) & ( s < t)[w s+1 = h(w s, s, x)] & w = w t. Proof. If f(t, x) = w, set w s = f(s, x) for s t, and verify easily that the sequence (w 0,..., w t ) satisfies the conditions on the right. For the converse, suppose that (w 0,..., w t ) satisfies the conditions on the right and prove by (finite) induction on s t that w s = f(s, x). We can view the equivalence (7) as a theorem about recursive definitions which have already been justified in some other way; or we can see it as a definition of a function f which satisfies the recursive equations (6) and so justifies recursive definitions which is how Dedekind saw it. In any case, it reduces proving Theorem 1E.2 to justifying quantification over finite sequences within the class of arithmetical relations, and we will do this by an arithmetical coding of finite sequences whose construction requires a couple of basic facts from arithmetic. Proposition 1E.4 (The Division Theorem). For every natural number y > 0 and every x N, there exist exactly one q and one r such that (8) x = y q + r and 0 r < y. This is verified easily by induction on x. If (8) holds, we set quot(x, y) = q, rem(x, y) = r, and for completeness, we also let quot(x, 0) = 0, rem(x, 0) = x. Theorem 1E.5 (The Chinese Remainder Theorem). If d 0,..., d t are relatively prime numbers and w 0 < d 0,..., w t < d t, then there exists some number a such that w 0 = rem(a, d 0 ),..., w t = rem(a, d t ). Proof. Consider the set D of all (t + 1)-tuples bounded by the given numbers d 0,..., d t, D = {(w 0,..., w t ) w 0 < d 0,..., w t < d t }, which has D = d 0 d 1 d t members, and let A = {a a < D } Informal notes, full of errors, July 16, 2012, 11:13 18

23 1E. Arithmetical functions and relations 19 which is equinumerous with D. Define the function π : A D by π(a) = (rem(a, d 0 ), rem(a, d 1 ),..., rem(a, d t )). Now π is injective (one-to-one), because if f(a) = f(b) with a < b < D, then b a is divisible by each of d 0,..., d t and hence by their product D (which is what their being relatively prime implies); hence d b a, which is absurd since a < b < D. We now apply the Pigeonhole Principle: since A and D are equinumerous and π : A D is an injection, it must be a surjection, and hence whatever (w 0,..., w t ) may be, there is an a < d such that π(a) = (rem(a, d 0 ), rem(a, d 1 ),..., rem(a, d t )) = (w 0,..., w t ). The idea now is to code an arbitrary tuple (w 0,..., w t ) by a pair of numbers (d, a), where d can be used to produce uniformly t + 1 relatively prime numbers d 0,..., d t and then a comes from the Chinese Remainder Theorem. Lemma 1E.6 (Gödel s β-function). Set β(a, d, i) = rem(a, 1 + (i + 1)d). This is an arithmetical function, and for each sequence of numbers w 0,..., w t there exist numbers a and d such that β(a, d, 0) = w 0,..., β(a, d, t) = w t. Proof. The β-function is arithmetical because it is defined by substitutions from addition, multiplication and the remainder function, which is arithmetical since rem(x, y) = r ( q)[x = yq + r & r < y]. To find the required a, d which code the tuple (w 0,..., w t ), set and verify that the t + 1 numbers s = max(t + 1, w 0,..., w t ), d = s! d 0 = 1 + (0 + 1)d, d 1 = 1 + (1 + 1)d,..., d t = 1 + (t + 1)d are relatively prime. (If a prime p divides 1 + (1 + i)s! and also 1 + (1 + j)s! with i < j, then it must divide their difference (j i)s!, and hence it must divide one of (j i) or s!; in either case, it divides s!, since (j i) s, and then it must divide 1, since it is assumed to divide 1 + (1 + i)s!, which is absurd.) It is also immediate that w i < d = s!, by the definition of s, and so the Chinese Remainder Theorem supplies some a such that as required. (w 0,..., w t ) = (rem(a, d 0 ),..., d t ) = (β(a, d, 0),..., β(a, d, t)) Informal notes, full of errors, July 16, 2012, 11:13 19

24 20 1. First order logic Proof of Theorem 1E.2. By the Dedekind analysis and using the β- function to code tuples, we have [ f(t, x) = w ( a)( d) β(a, d, 0) = g( x) ] & ( s < t)[β(a, d, s + 1) = h(β(a, d, s), s, x)] & β(a, d, t) = w Thus the graph of f is arithmetical, by the closure properties of the arithmetical functions and relations in Theorem 1D.2. There is no standard definition of rich structure, but the following notion covers the most important examples: Definition 1E.7 (Structures with tuple coding). A copy of N in a structure A is a structure N = (N, 0, S, +, ) such that: 1. N is isomorphic with the structure of arithmetic N. 2. N A. 3. The set N, the object 0 and the functions S, + and are all A- elementary. A structure A admits tuple coding if it has a copy of N and there is an A-elementary function γ : A n+1 A such that for every tuple w 0,..., w t A, there is some a A n such that γ( a, 0) = w 0, γ( a, 1) = w 1,..., γ( a, t) = w t, where 0, 1,..., t are the A-numbers 0, 1,..., t (i.e., the copies of these numbers into A by the given isomorphism of N with N ). In this definition, γ plays the role of the β-function in N, and we have allowed for the possibility that triples (n = 3) or quadruples (n = 4) are needed to code tuples of arbitrary length in A using γ. We might have also allowed the natural numbers to be coded by pairs of elements of A or tweak the definition in various other ways, but this version captures all the interesting examples already. The key result is: Proposition 1E.8. Suppose A admits coding of tuples, g : A n A, h : A n+2 A; are A-elementary, f : A n+1 A, and for t N, y / N, (9) f(0, x) = g( x), f(t + 1, x) = h(f(t, x), t, x)), f(y, x) = y; it follows that f is A-elementary. It can be used to show that structures which admit tuple coding have a rich class of elementary functions and relations. Informal notes, full of errors, July 16, 2012, 11:13 20

25 1F. Quantifier elimination 21 Example 1E.9 (The integers). The ring of (rational) integers (10) Z = (Z, 0, 1, +, ) (Z = {..., 2, 1, 0, 1, 2,... }) admits tuple coding. To see this, we use the fact that N Z, and it is a Z-elementary set because of Lagrange s Theorem, by which every natural number is the sum of four squares: x N ( u, v, s, t)[x = u 2 + v 2 + s 2 + t 2 ] (x Z). We can then use the β-function (with some tweaking) to code tuples of integers. Example 1E.10 (The fractions). The field of rational numbers (fractions) Q = (Q, 0, 1, +, ) admits tuple coding. This is a classical theorem of Julia Robinson which depends on a nontrivial, Q-elementary definition of N within Q. Example 1E.11 (The real numbers, with Z). The structure of analysis (R, Z) = (R, 0, 1, Z, +, ) admits tuple coding. This requires some work and it is not a luxury that we have included the integers as a distinguished subset: the field of real numbers R = (R, 0, 1, +, ) does not admit tuple coding. We will discuss this very interesting, classical structure later. 1F. Quantifier elimination At the other end of the class of structures which admit tuple coding are some important, classical structures which are, in some sense, very simple : the elementary functions and relations on them are quite trivial. We will consider some examples of such structures in this section, and we will isolate the property of quantifier elimination which makes them simple much as tuple coding makes the structures in the preceding section complex. Informal notes, full of errors, July 16, 2012, 11:13 21

26 22 1. First order logic We list first, for reference, some simple logical equivalences which we will be using, and to simplify notation, we set for arbitrary τ-formulas φ, ψ and any τ-structure A: φ A ψ A = φ ψ, φ ψ = φ ψ. Proposition 1F.1 (Basic logical equivalences). (1) The distributive laws: (2) De Morgan s laws: φ & (ψ χ) (φ & ψ) (φ & χ) φ (ψ & χ) (φ ψ) & (φ χ) (φ & ψ) φ ψ (φ ψ) φ & ψ (3) Double negation, implication and the universal quantifier: φ φ, φ ψ φ ψ, xφ ( x) φ (4) Renaming of bound variables: if y is a variable which does not occur in φ and φ{x : y} is the result of replacing x by y in all its free occurrences, then (5) Distribution law for over : xφ yφ{x : y} xφ yφ{x : y} x(φ 1 φ n ) xφ 1 xφ n (6) Pulling the quantifiers to the front: if x does not occur free in ψ, then xφ & ψ x(φ & ψ) xφ ψ x(φ ψ) xφ & ψ x(φ & ψ) xφ ψ x(φ ψ) xφ ψ x[φ ψ] xφ ψ x[φ ψ] (7) The general distributive laws: for all natural numbers n, k and every doubly-indexed sequence of formulas φ i,j with i n, j k, i n j k φ i,j (11) f:{0,...,n} {0,...,k} i n φ i,f(i). i n j k φ i,j (12) f:{0,...,n} {0,...,k} i n φ i,f(i). Informal notes, full of errors, July 16, 2012, 11:13 22

27 1F. Quantifier elimination 23 Proof. To see (11), fix a structure A and an assignment π and compute: A, π = i n j k φ i,j for each i n, there is some j k such that A, π = φ i,j there is a function f : {0,..., n} {0,..., k} such that for all i n, A, π = φ i,f(i), where (in the implication from left to right) the function f in the last equivalence assigns to each i n the least j = f(i) k such that A, π = φ i,j. The dual (12) is established by taking the negation of both sides of (11) applied to φ i,j, pushing the negation through the conjunctions and disjunctions using De Morgan s laws, and finally using the obvious φ i,j φ i,j. 1F.2 (Literals). For the constructions in the remainder of this section, it is useful to enrich the language FOL(τ) with propositional constants T, F for truth and falsity. We may think of these as abbreviations, T : x(x = x), F : x(x x), considered (by convention) as prime formulas. A literal is either a prime formula R(t 1,..., t n ), s = t, T, F, or the negation of a prime formula. Proposition 1F.3 (Disjunctive normal form). Every quantifier-free formula χ is (effectively) logically equivalent to a disjunction of conjunctions of literals which has no more variables than χ: i.e., for suitable n, n i, and literals l ij (i = 1,..., n, j = 1,..., n i ) whose variables all occur in χ, χ χ φ 1 φ n, where for i = 1,..., n, φ i l i1 & & l ini. By the definition in the proposition, x = y (z = z) is not a disjunctive normal form of x = y (if all three variables are distinct), even though x = y x = y (z = z) Proof. We show by structural induction that for every quantifier-free formula χ, both χ and its negation χ are logically equivalent to a disjunction of conjunctions of literals, among which we count T (truth) and F (falsity). The result is trivial in the Basis, when χ is prime, since χ and χ are in disjunctive normal form with n = 1, n 1 = 1, and l 11 χ or l 11 χ. In the Induction Step, the proposition is immediate for χ χ 1, since the Induction Hypothesis gives us disjunctive normal form for χ 1 χ and χ 1 χ. Informal notes, full of errors, July 16, 2012, 11:13 23

28 24 1. First order logic If χ is a disjunction or conjunction of χ 1 and χ 2, we may assume that the disjunctive normal forms for χ 1 and χ 2 χ 1 i<n j<k χ 1,i,j, χ 2 i<n j<k χ 2,i,j given by the induction hypothesis have the same number of disjuncts and conjuncts, by padding adding harmless insertions of T and F. We get immediately a disjunctive normal form for the disjunction: ( ) ( χ 1 χ 2 i<n j<k χ ) 1,i,j i<n j<k χ 2,i,j [either i < n and j<k χ 1,i,j or n i and ] where i<2n χ i,j { χ 1,i,j, if i < n, χ 2,i n,j, otherwise. i<2n j<k χ 1,i n,j j<k χ 2,i,j To get a disjunctive normal form for the conjunction χ 1 & χ 2, we use the distributive laws (11), (12) which give us equivalent conjunctive normal forms χ 1 i< n j< k χ 1,i,j, χ 2 i< n j< k χ 2,i,j for the conjuncts, and using these we compute as above: ( ) ( χ 1 & χ 2 i< n j< k χ 1,i,j & i< n j< k χ 2,i,j where χ i,j { χ 1,i,j, χ 2,i n,j, if i < n, otherwise. ) i<2 n j< k χ i,j We now use (11) again to get a disjunctive normal form for χ 1 & χ 2. These two computations also give us disjunctive normal forms for the negations of disjunction and conjunctions by appealing to the De Morgan Laws, and also for implication, using χ 1 χ 2 χ 1 χ 2. Proposition 1F.4 (Prenex normal forms). Every formula χ is (effectively) logically equivalent to a formula χ Q 1 x 1 Q n x n ψ (ψ quantifier-free) in prenex form, whose free variables are among the free variables of χ. Definition 1F.5 (Quantifier elimination for structures). A quantifierfree normal form for a formula χ in a structure A, is any quantifier-free formula χ (in which T or F may appear) whose variables are among the free variables of χ and such that χ A χ. Informal notes, full of errors, July 16, 2012, 11:13 24

29 1F. Quantifier elimination 25 A structure A admits elimination of quantifiers, if every formula χ has a quantifier-free normal form in A; and it admits effective elimination of quantifiers, if there is an effective procedure which will compute for each χ a quantifier-free normal form for χ in A. 1F.6. Quanitifier elimination and decidability. To see the importance of this notion, suppose the vocabulary τ is purely relational, i.e., it has no constant or function symbols. Now the only quantifier-free sentences are T and F ; and so if a τ-structure A admits effective quantifier elimination, then we can effectively decide for each sentence χ whether it is logically equivalent in A to T or F in other words, we have a decision procedure for truth in A. More generally, suppose τ may have constants and function symbols and A admits effective quantifier elimination: if we have a decision procedure for quantifier-free sentences (with no variables), then we have a decision procedure for truth in A. The hypothesis is, in fact, satisfied in most structures that occur naturally in mathematics, including (trivially) the structure of arithmetic N = (N, 0, 1, +, ); so we cannot expect that N admits effective quantifier elimination, because we don t expect it to be decidable and in time we will prove that it is not decidable. Lemma 1F.7 (Quantifier elimination test). If every formula of the form χ x[χ 1 & & χ n ] (where χ 1,..., χ n are literals) is (effectively) equivalent in a structure A to a quantifier-free formula whose variables are all among the free variables of χ, then A admits (effective) quantifier elimination. Proof. Let F be the set of formulas which (effectively) have quantifier free forms in A. By (3) of Proposition 1F.1, it is enough to show that F contains all literals, which it clearly does; that it is closed under, & and, which it clearly is; and that it is closed under existential quantification. For the latter, if χ xφ with φ quantifier-free, we bring φ to disjunctive normal form, so that χ x[φ 1 φ n ] xφ 1 xφ n where each φ i is a conjunction of literals and then we use the hypothesis of the Lemma. Proposition 1F.8. For each infinite set A, the structure A = (A) in the language with empty vocabulary admits effective quantifier elimination. Informal notes, full of errors, July 16, 2012, 11:13 25