Consistent Set of Logical Formulas is Subset of Maximally Consistent Set

Theorem

Let $\LL$ be a formal language used in the field of symbolic logic.

Let $\FF$ be the set of logical formulas of $\LL$.

Let $\FF$ be countable.

Let $S$ be a consistent subset of $\FF$.

Then $S$ is a subset of some maximal consistent set of formulas.

Proof

This theorem requires a proof. In particular: This may be the same as Consistent Set of Formulas can be Extended to Maximal Consistent Set. Smullyan and Fitting are vague here. They may be trying to state Lindenbaum's Lemma, but if this and that are equivalent is not clear. 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.

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: $\S 6$ Another approach to maximal principles