more from Herbert B. Enderton

### Single Idea 9995

#### [catalogued under 5. Theory of Logic / K. Features of Logics / 6. Compactness]

Full Idea

If a wff is tautologically implied by a set of wff's, it is implied by a finite subset of them; and if every finite subset is satisfiable, then so is the whole set of wff's.

Clarification

a 'wff' is a well-formed formula

Gist of Idea

Proof in finite subsets is sufficient for proof in an infinite set

Source

Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5)

Book Reference

Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.142

A Reaction

[Enderton's account is more symbolic] He adds that this also applies to models. It is a 'theorem' because it can be proved. It is a major theorem in logic, because it brings the infinite under control, and who doesn't want that?