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
...
======================================================================