Which condition characterizes a maximally consistent set S of formulas in a deductive system?
Hard
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Get StartedWhich condition characterizes a maximally consistent set S of formulas in a deductive system?
Options:
- S is consistent and for every formula φ either φ ∈ S or ¬φ ∈ S
- S is finite and contains all theorems of the system
- S contains only provable formulas and no nonprovable ones
- S is inconsistent and remains inconsistent under any extension
Correct answer: S is consistent and for every formula φ either φ ∈ S or ¬φ ∈ S
Explanation: A maximally consistent set is consistent and for each formula φ either φ or ¬φ belongs to S. Lindenbaum's lemma guarantees such extensions of consistent sets in countable languages.
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