Logic: Lecture 7, 25 August 2015 -------------------------------- Free variables: Notation: phi(x1,...,xn) means that FV(phi) is contained in {x1,...,xn} Expressing properties in FOL Groups Axioms G1-G3 (Left) cancellation Abelian groups Element of order 2 ... element of order k All elements have a finite order "Infinite" formula Cannot express even with infinite set of finite formulas Equivalence relations Ax r(x,x) Ax Ay r(x,y) <=> r(y,x) Ax Ay Az (r(x,y) and r(y,z)) => r(x,z) Strict linear orders Ax ~(x < x) Ax Ay Az (x < y and y < z) => x < z Ax Ay (x < y or x = y or y < x) Peano's axioms for arithmetic R = {<}, F = {succ,+,*}, C = {0} Interpret over Nat = {0,1,2,...} ~Ex succ(x) = 0 Ax Ay (succ(x) = succ(y) => x = y) Ax Az (x < y <=> Ez (x + succ(z) = y) Ax (x + 0) = x Ax Ay (x + succ(y) = succ(x+y) Ax (x * 0) = 0 Ax Ay (x * succ(y)) = (x*y + x) These axioms on their own allow "non standard" models of arithmetic (which we will see later). To fully characterize Nat, we need an induction axiom. Any set X that contains 0 and is closed with respect to succ (i.e. x in X implies succ(x) in X) must contain all of Nat To formulate this in logic, need a second order quantifier AX, where X ranges over all subsets. To express this in first order logic: - A formula phi(x) with one free variable defines a subset: those values of x for which phi is true. - Induction axiom can be formulated as an infinite set of sentences, one for each formula phi(x) in the language with none free variable: (phi(0) and Ax (phi(x) => phi(succ(x))) => Ax phi(x) Cardinality At least two elements: Ex Ey ~(x = y) At least k elements: Ex1 ... Exn And_{i =/= j} ~(xi = xj) At most one element: Ax Ay (x = y) At most k-1 elements: Ax1 ... Axn Or_{i =/= j} (xi = xj) Structure is not finite: Infinite family At least two At least three ... At least k ... ======================================================================