CS486 Spring 1997

Lecture 15 Tuesday, March 11, 1997
Reading: Suppes p.20-23

Review

Peano Arithmetic
tactics and induction proof

Applied First-Order theories

adding equality
allowing functions and constants
using tactics

Definitions (lecture notes, Suppes)

theory extensions,
eliminating definitions
theorem on eliminating function definitions

Arithmetic theories

Q as a subsystem of
Q is finitely axiomatizable
Q is undecidable (Church's theorem again)
nonstandard elements
Presburger arithmetic - a decidable theory

Theories of integers,

object approach: groups, rings, integral domains as classes
ordered integral domains
Arith rule
induction (up and down)

On Definitions and Theory Extensions
Adding definitions of new predicates and operators to a theory constitutes an extension of the theory to . For example, we can add to

satisfies the formula when substituted for , that is

Sometimes new functions are added by means of the iota-operator (-operator) or an epsilon-operator (-operator). We can introduce whenever , the iota term denotes this unique object. So, for we can prove . Thus is well-defined, and we say

The -operator is used in some theories to pick out some element satisfying a predicate, so if then holds. This operator is more subtle to analyze since it implies the axiom of choice.

To simplify the notation, we can use in place of whenever holds.

We speak of these definitions as new abstractions. There are explicit rules for expanding an abstraction (the left hand side-) to its definitions (right hand side-).

Elimination of defined functions, constants and relations.

If extends theory , then we say that the new notations are eliminable iff there is an effective procedure which given any formula of produces a formula of such that

(i) If is in , then is .
(ii)
(iii) If then
When an elimination procedure exists, then we also know
(iv) iff and
(v) If are formulas of then iff .

Theorem

Let be a first-order theory with equality, =, and let
be a formula with only free, and suppose
, then the definition

is eliminable.
The effective elimination procedure replaces in any formula by a new variable to obtain and taking to be .

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