# Consistency-related proof in predicate logic

1. Jun 23, 2006

### Mr.Cauliflower

Let L be language with n different constants and at least one predicate symbol. Prove that there exist 2 different maximal consistent sets formed of closed formulas of the language L.

2. Jun 23, 2006

### AKG

Do you know that if S is a consistent set of sentences (closed formulas), then there exists a maximal consistent extention S' of S (S' is an extention of S in the sense that $S' \subseteq S$)? To prove that there exist two different maximal consistent sets, find two consistent sets S and T such that any extention S' of S must be different from any extention T' of T.

3. Jun 29, 2006

### Mr.Cauliflower

You're right, I think I found it as Lindenbaum's lemma. Anyway, I don't know how to do this:

4. Jun 29, 2006

### AKG

What have you tried? As a hint, it's very very easy. S and T need not be complicated at all.