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 ∃x and ∀x 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. PRIME(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: ∀x∀y∀z((x · y) · z = x · (y · z)) - Identity: ∀x(x · e = x ∧ e · x = x) - Inverse: ∀x(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 ϕ(z) denote the first order formula ∀x(z · x = x ∧ x · z = x). The existence of an identity element is now expressed by the statement
∃zϕ(z) The existence of an inverse can be represented by
∀x∃y∃z(ϕ(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 ϕ(z).
2 Theorem G2 ∀z1∀z2(ϕ(z1) ∧ ϕ(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.
∀x(x 6= e → x · x 6= e) ∀x(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 = ∃x∃y(¬(x = y)) Similarly we have
GE3 = ∃x∃y∃z(x 6= y ∧ y 6= z ∧ x 6= z) where x 6= y denotes ¬(x = y). In general for k > 1, ^ GEk = ∃x1∃x2 · · · ∃xk( (xi 6= xj)) 1≤i