Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension/Proof 2

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Theorem

Let $T$ be a finitely satisfiable $\LL$-theory.

There is a finitely satisfiable $\LL$-theory $T'$ which contains $T$ as a subset such that for all $\LL$-sentences $\phi$, either $\phi \in T'$ or $\neg\phi \in T'$.

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Lemma

Let $\LL$ be a language of predicate logic.

Let $T$ be a finitely satisfiable $\LL$-theory.

Let $\phi$ be an $\LL$-sentence.

Then either:

$T \cup \set \phi$

or:

$T \cup \set {\neg \phi}$

is finitely satisfiable.

Proof

Let $\AA$ be the set of finitely satisfiable extensions of $T$.

By the lemma, for each element $S$ of $\AA$ and each $\LL$-sentence $\phi$, either $S \cup \set \phi \in \AA$ or $S \cup \set {\neg \phi} \in \AA$.

$\AA$ has finite character, by the following argument:

Let $S \in \AA$.

Let $F$ be a finite subset of $S$.

Then $S$ is satisfiable and hence finitely satisfiable.

Thus in $\AA$.

Let $S$ be a theory on $\LL$.

Let every finite subset of $S$ be finitely satisfiable.

Then every finite subset of $S$ is satisfiable.

Therefore $S$ is finitely satisfiable.

Thus $\AA$ has finite character.

By the Restricted Tukey's Theorem, $\AA$ has an element $T'$ such that:

for each $\LL$-sentence $\phi$, either $\phi \in T'$ or $\neg \phi \in T'$.

$\blacksquare$

Boolean Prime Ideal Theorem

This theorem depends on the Boolean Prime Ideal Theorem (BPI), by way of Restricted Tukey's Theorem.

Although not as strong as the Axiom of Choice, the BPI is similarly independent of the Zermelo-Fraenkel axioms.

As such, mathematicians are generally convinced of its truth and believe that it should be generally accepted.