Enderton 2.6.2

Let and be theories (in the same language) such that

(i) ,

(ii) is complete, and

(iii) is satisfiable. Show that .

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Before I get started, I need to understand some definitions related to this problem.

(*) AtheoryT is said to becompleteiff for every sentence , either or .

I am confused with an unsatisfiable theory, where a theory is defined to be a set of sentences closed under logical implication.

The textbook gives an example of an unsatisfiable theory, which is the theory consisting of all sentences of the language. Why is this unsatisfiable theory? Any description or example?

Thank you.