MATH 260A: Mathematical Logic | Fall 2019, UC San Diego Lecture 1 Instructor: Sam Buss Scribe: Sasank Mouli September 26, 2019 Mathematical logic consists of the following branches, which are used in various disciplines as shown below. This course will mainly cover Model theory and Proof theory. 1 First-order logic 1.1 Introduction First-order logic is a language used to make mathematical statements. It has the following elements as building blocks. Propositional/Boolean connectives are standard boolean functions used as a basis to express all boolean functions. The most commonly used ones are AND (^), OR (_), IMPLIES (!), IF AND ONLY IF ($) and NOT (:). Variables represented by x, y, z, ··· range over some nonempty universe of objects. 1 Quantifiers 9x and 8x are used to express the presence or absence of objects in the universe satisfying certain properties (specified by a boolean formula). Predicate symbols are used to express relations over the universe. For example, Equality (=), Less or equal to (≤) are a binary relations. P RIME(x) is a unary relation expressing whether or not x is prime. The arity of a predicate is the number of arguments to it. Function symbols represented by f, g, h ··· denote functions over the universe. For example, + is a binary function denoting addition in the universe. The Successor function S(x) gives the successor of x in the universe. In usual notation, S(x) = x + 1. The arity of a function is the number of arguments to it. Constants are denoted by c, d, ··· . They can be viewed as functions of arity zero. 0, 1 are examples of constants. A first-order language is a choice of predicate symbols, function symbols and constants. Equal- ity (=) is sometimes assumed to be present as a default predicate symbol. 1.2 Examples We will now show examples of first-order languages and statements that can be expressed using them. 1.2.1 Theory of Groups To express that a universe of objects forms a group, we use the following first-order language: Function symbols: The unary function −1 (Inverse), the constant symbol e (Identity), the binary function · (Group operation). Predicate symbols: Equality (=). We can now express the group axioms as First order statements: - Associativity: 8x8y8z((x · y) · z = x · (y · z)) - Identity: 8x(x · e = x ^ e · x = x) - Inverse: 8x(x · x−1 = e ^ x−1 · x = e) Closure under · (the group operation) follows automatically since the operation · is interpreted as a function over the universe. We can use the sub-language containing just the function symbol · and the predicate symbol = to make the same statements. Let '(z) denote the first order formula 8x(z · x = x ^ x · z = x). The existence of an identity element is now expressed by the statement 9z'(z) The existence of an inverse can be represented by 8x9y9z('(z) ^ x · y = z ^ y · x = z) If G2 denotes this new set of group axioms, then it implies the truth of statements such as the uniqueness of the inverse, i.e. any universe of elements satisfying the first-order formulae in G2 has a unique inverse element satisfying '(z). 2 Theorem G2 8z18z2('(z1) ^ '(z2) ! (z1 = z2)) Definition 1. A group is torsion free if it satisfies the following property: For integers n and all group elements x 6= e, xn 6= e. The statement that a set of elements is a torsion free group can be expressed as a set of infinitely many first-order formulae. 8x(x 6= e ! x · x 6= e) 8x(x 6= e ! x · (x · x) 6= e) . There are groups which have arbitrarily high torsion (e.g. the multiplicative group of nth roots of unity where n is prime) and thus a finite set of the above formulae will not suffice. In upcoming lectures we will prove the following theorem. Theorem There is no finite set of first-order formulae expressing that a set of elements is a torsion free group. 1.2.2 Statements about the size of the universe Let GE2 denote the statement that our universe has at least two distinct elements. GE2 = 9x9y(:(x = y)) Similarly we have GE3 = 9x9y9z(x 6= y ^ y 6= z ^ x 6= z) where x 6= y denotes :(x = y). In general for k > 1, ^ GEk = 9x19x2 · · · 9xk( (xi 6= xj)) 1≤i<j≤k V k where means a conjunction of 2 many inequalities. Note that the size of the above 1≤i<j≤k formula is O(k2) i.e. it uses O(k2) symbols. Homework Refomulate GEk with O(k) symbols. 1.2.3 Set theory The first order language for set theory consists of the predicate symbols 2 and =. For instance, the following first-order formula states that a set x is empty. 8y(y 62 x) 3 1.3 Things that are not in first-order logic V - Quantifying over infinitely many variables, e.g. 9x19x2 ··· ( i<j xi 6= xj) - Second order logic, which allows quantifying over sets of objects or equivalently has variable predicate symbols. - Modal quantifiers: ' is the statement \' is necessarily true". ' is the statement \' is possibly true." - Tense logics/CTL - Type theory 1.4 Formal definition of first-order formulae Below we formally define the syntax of first-order logic. Fix a language L of non logical symbols (i.e. predicate, function and constant symbols, = if necessary), logical symbols (^, _, !, $ , :), variables (x, y, z, ··· , a, b, c, ··· ) and other structural symbols like parentheses and comma. Definition 2. L-terms An L-term is inductively defined as follows. 1. Any variable x is a term 2. Any constant c is a term 3. If f is a function symbol of arity k and t1, ··· , tk are terms, then f(t1; ··· ; tk) is a term. Definition 3. L-formulae 1. If t1, t2 are terms t1 = t2 is a formula (an atomic formula) 2. If P is a predicate of arity k and t1, ··· , tk are terms then P (t1; ··· ; tk) is a formula (also an atomic formula) 3. If ' and are formulae then so are (' ^ ), (' _ ), (' ! ), (' $ ), :'. 4. If x is a variable, 8x', 9x' are formulae. We want L-formulae to have Unique Readability, i.e. there is a unique way to interpret every formula. This is ensured by the use of parentheses. We omit parentheses when reasonable and use infix notation. The order of precedence in the absence of parentheses is as follows :, 8, 9 ^, _ !, $ Operators of equal precedence are associated from right to left. For example, ' ^ ! _ χ is interpreted as (' ^ ) ! ( _ χ). ' ^ ^ χ is read as ' ^ ( ^ χ). ' ! ! χ means ' ! ( ! χ) which is the same as ' ^ ! χ. Expressions such as ' ^ _ χ should never be used since their meaning may be ambiguous. 4 MATH 260A: Mathematical Logic — Fall 2019, UC San Diego Lecture2 Instructor: Sam Buss Scribe: Ayoob Shahmoradi 1 Propositional Logic 1.1 Language • Connectives: ^, _, , Ñ, Ø • Variables (Var): x1, x2, ...p1, p2, ... 1.2 Syntax • Definition of a formula: – Any variable is an atomic formula. – If ϕ and ψ are formulas, so are p ϕq, pϕ ^ ψq, pϕ _ ψq, pϕ Ñ ψq, pϕ Ø ψq. 1.3 Semantics Variables range over T and F. Definition 1: A truth-assignment (t.a.) is a mapping τ: Var Ñ tT,F u. Then τ is the extension of τ to all formulas s.t.: • τppq=τppq if p is a variable. • τp ϕq= T iff τpϕq=F. • τpϕ ^ ψq= T iff τpϕq=T and τpψq=T. 1 • τpϕ _ ψq= T iff τpϕq=T or τpψq=T. • τpϕ Ñ ψq= T iff τpϕq=F or τpψq=T. • τpϕ Ø ψq= T iff τpϕq = τpψq. Definition 2. ϕ is a tautology{valid (written as ⊧ ϕ) if τpϕq=T for all truth assignments τ. Note that , ^, _, Ñ, Ø, respectively, stand for NEGATION, AND, OR, IF ... THEN, and IF AND ONLY IF. Definition 3. ϕ is satisfiable if for some truth assignment τ, τpφq=T. Theorem 1. ϕ is satisfiable iff ( ϕ) is not a tautology. Definition 4. ϕ tautologically implies ψ (ϕ ⊧ ψ) if every truth assignment that satisfies ϕ also satisfies ψ. Definition 5. ϕ is tautologically equivalent to ψ (ϕ )( ψq iff ϕ ( ψ and ψ ( ϕ. Theorem 2. ϕ ⊧ ψ iff ⊧ pϕ Ñ ψq. Let Γ be a set of formulas. Definition 6. τ satisfies Γ iff τ ⊧ ϕ for all ϕ P Γ.(τ ‰ Γ) Definition 7. Γ tautologically implies ϕ (Γ ⊧ ϕ) iff for all t.a. τ if τ ⊧ Γ then τ ⊧ ϕ. Theorem 3. If Γ is unsatisfiable and ϕ is any formula, then Γ ⊧ ϕ. Theorem 4 If for all ϕ, Γ ⊧ ϕ, then Γ is unsatisfiable. Theorem 5. Γ ⊧ ϕ iff Γ Y t ϕu is unsatisfiable. 1.4 CNF/DNF A conjunction of formulas is a formula pψ1 ^ ψ2q... ^ ψn ´ ¯ where each ψi is a formula. Similarly, a disjunction of formulas is a formula pϕ1 _ ϕ2q... _ ϕn ´ ¯ 2 where each ϕi is a formula. Now in order to define disjunctive normal form (DNF) and conjunctive normal form (CNF), we need the following definitions. • A literal is either a variable (p) or a negated variable ( p). • A clause is a disjunction of a finite set of formulas. • A CNF formula is a conjunction of a finite set of clauses. • A term is a conjunction of a finite set of literals. • A DNF formula is a disjunction of a finite set of terms. Example:pp Ø qq. • DNF: pp ^ qq _ p p ^ qq • CNF: p p _ qq ^ p q _ pq Theorem 6. Every formula ϕ is tautologically equivalent to a DNF formula and to a CNF formula.