# Automating Satisfiability 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 is a first-order formula (or an *enumerable sequence* of formulas i.e. sequence where we can compute -the element for each ), then we can detect in finite amount of time the case when is **unsatisfiable**

Observations:

- to prove is valid, we show is unsatisfiable
- to show that is true in an axiomatization given as an enumerable sequence , we check that the sequence ; 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

Compactness Theorem for Propositional Logic

Normal Forms for First-Order Logic

Preliminary Discussion on Models

Semidecision Procedure for First-Order Logic (without Equality)

## Resolution in First-Order Logic

Instantiation plus Ground Resolution

Non-Ground Instantiation and Resolution