Using the compactness and completeness theorems

Godel’s completeness theorem (not to be confused with his incompleteness theorems) says that a theory has a model if and only if it is consistent, i.e. no contradiction is proved by the theory. This is the heart of model theory as it lets us answer questions about theories by looking at models and vice-versa.

One should not confuse the completeness theorem with the notion of a complete theory. A complete theory is a theory that contains every sentence or its negation.

— Wikipedia on Model theory

2012.09.20 Thursday ACHK

In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms.

The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.

The syntactic definition states that a theory is consistent if and only if there is no formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.

If these semantic and syntactic definitions are equivalent for a particular logic, the logic is complete.[clarification needed][citation needed]

— Wikipedia on Consistency

2012.09.17 Monday ACHK

Any proof of the Completeness Theorem consists always of two parts.

First we have show that all formulas that have a proof are tautologies. This implication is also called a Soundness Theorem, or soundness part of the Completeness Theorem.

The second implication says: if a formula is a tautology then it has a proof. This alone is often called a Completeness Theorem. In our case, we call it a completeness part of the Completeness Theorem.

— Cse371, Math371, LOGIC, Fall 2011

— Professor Anita Wasilewska

2012.09.14 Friday ACHK