CS 486: Spring 2001 Cornell University Prof. Robert Constable

April 19 Lecture Notes

Summary of past week

We have focused on translating various mathematical concepts into first-order logic, using examples of mathematics from the required textbook by Stewart and Tall, The Foundations of Mathematics. I’ll abbreviate this as S & T.

We have translated these concepts:

– equivalence relations and equality

– order relations

– algebraic structures The main translation combined these properties to show how to translate the definition of an

inductive ordered integral domain into FOL (first-order logic). This domain characterizes ℤ, the integers.

Today we will provide a more parsimonious account of numbers in FOL. These are the Peano axioms that define Peano Arithmetic (PA).

PA Axioms

Equality

E1:  x = x
E2:  x = y ⊃ (y = z ⊃ x = z)

Numbers

N1:  ¬ (s(x) = 0)
N2:  s(x) = s(y) ⇔ x = y
N3:  (A(o) ∧ ∀x.(A(x) ⊃ A(s(x)))) ⊃ ∀xA(x)
N4:  (m + 0 = m) ∧ (m + s(n) = s(m + n))
N5:  (m ∗ 0 = 0) ∧ (m ∗ s(n) = m ∗ n + m)

With these axioms we can prove the theorems in S & T, p. 151–158.

The above first-order axiomatization provides a bridge between the S & T book and the Smullyan book in the following precise sense. In S & T they establish facts about ℕ using informal set theory. If these facts are expressible in PA, then they are true in first-order logic and in PA.

Next week we will extend first-order logic to a new logic, called lambda-PRL. The RL stands for Refinement Logic and the P for Program. The first step is to add equality, =, as a primitive. We also add these axioms.

\begin{matrix}E1: & x = x \\ E2: & x = y ⊃ (A(x) ⇔ A(y)) \end{matrix}

Applying Compactness

Here is a remarkable result of compactness theorem applied to PA. Recall from Smullyan p. 65.

Compactness Theorem. If S is a denumerable set of first-order formulas such that every finite subset is satisfiable, then S is satisfiable.

Let S be a set of true first-order consequences of PA, choosing all particular instances of the axiom schema of induction. Suppose we also add the formulas for a new parameter a_{0}

a_{0} ≠ 0, a_{0} ≠ 1, a_{0} ≠ 2, ...

Also add consequences of using a_{0} in all theorems. Call this S′.

Every finite subset of S′ is satisfiable; we can take a_{0} to be a number bigger than any one mentioned. Hence S′ is satisfiable. This means that a_{0} behaves just like a natural number, but it is not any of the natural numbers, 0, 1, 2, 3, ...