Set of Logical Formulas is Inconsistent iff it has Finite Inconsistent Subset

Theorem

Let $\FF$ be a collection of logical formulas.

Then:

$\FF$ be inconsistent

if and only if: there exists a finite subset of $\FF$ which it itself inconsistent.

Proof

Sufficient Condition

Let $\FF$ be inconsistent.

Then it is possible to assemble a proof in a finite set of statements of a contradiction.

This finite set of statements uses within it a finite subset $\GG \subseteq \FF$ of the logical formulas of $\FF$.

Hence $\GG$ is that inconsistent finite subset of $\FF$ whose existence is proposed.

$\Box$

Necessary Condition

Let $\FF$ have a finite subset $\GG$ which is inconsistent.

Then it is possible to assemble a proof of a contradiction using logical formulas of $\GG$.

But those logical formulas are also logical formulas of $\FF$.

Then by definition $\FF$ is likewise inconsistent.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles