Consistency of Logical Formulas has Finite Character/Proof 1

Theorem

Let $P$ be the property of collections of logical formulas defined as:

$\forall \FF: \map P \FF$ denotes that $\FF$ is consistent.

Then $P$ is of finite character.

That is:

$\FF$ is a consistent set of formulas if and only if every finite subset of $\FF$ is also a consistent set of formulas.

Proof

This theorem requires a proof. In particular: Use Compactness Theorem. And again, the "iff" in the definition of finite character is the interesting part You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.