Consistency of Logical Formulas has Finite Character

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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 1


This theorem requires a proof.
In particular: Use Compactness Theorem. And again, the "iff" in the definition of finite character is the interesting part
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Proof 2

Sufficient Condition

Let $\FF$ be a consistent set of formulas.

Aiming for a contradiction, suppose there exists $\GG \subseteq \FF$ where $\GG$ is finite such that $\GG$ is inconsistent.

But then from Set of Logical Formulas is Inconsistent iff it has Finite Inconsistent Subset, $\FF$ is itself inconsistent.

From this contradiction it follows that $\GG$ must be consistent.

As $\GG$ is arbitrary, it follows that every finite subset of $\FF$ is consistent.

$\Box$


Necessary Condition

Let every finite subset of $\FF$ is consistent.

Aiming for a contradiction, suppose $\FF$ is inconsistent.

Then by Set of Logical Formulas is Inconsistent iff it has Finite Inconsistent Subset there exists a finite subset $\GG \subseteq \FF$ such that $\GG$ is inconsistent.

But by hypothesis $\GG$ is consistent.

From this contradiction it follows that $\FF$ must also be consistent.

$\blacksquare$


Sources

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