# How do you prove that any consistent set of sentences can be extended to a maximal consistent set?

Let S be a consistent statement set, and let s1 be an arbitrary statement not in S. If s1 is consistent with S, then the union of S and s1 is consistent; otherwise, the union of S and not s1 is consistent. Let S* be the union of the union of S with either si or not-si, depending on whether si is consistent with S. S* is consistent and maximal.

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