Lecture 10

Automating Unsatisfiability Proofs for First-Order Logic

First-order logic can be used to aximatize essentially all of math, through set theory.

Goal: a procedure such that if $S$ is a first-order formula (or an enumerable sequence of formulas i.e. sequence where we can compute $k$ -the element for each $k$ ), then we can detect in finite amount of time the case when $S$ is unsatisfiable

Observations:

to prove $F$ is valid, we show $\lnot F$ is unsatisfiable
to show that $F$ is true in an axiomatization given as an enumerable sequence $S$ , we check that the sequence $\lnot F$ ; $S$ is unsatisfiable
given a well-defined mathematical statement, we can detect in finite amount of time if this statement follows from a well-defined set of axioms

Countable Models for Quantified First-Order Logic

Normal Forms for First-Order Logic

Preliminary Discussion on Models

Ground Terms

Herbrand's Expansion Theorem

Continued in Exercise 10