A theory is unsatisfiable if no structure makes all sentences from the theory true. Any theory that contains a formula and its negation is unsatisfiable.
Let and be theories (in the same language) such that
(ii) is complete, and
(iii) is satisfiable. Show that .
Before I get started, I need to understand some definitions related to this problem.
(*) A theory T is said to be complete iff 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?
A theory is defined to be a set of sentences closed under logical implications.
As you said, if I am not mistaken, for an unsatisfiable theory T, there exists a sentence such that
and , where and .
I was having hard time finding an example of T and . Is there any concrete example of an unsatisfiable theory T and for the above?
Additionally, is there any example of an inconsistent theory T and a sentence such that
As you know, T is a theory iff T is a set of sentences such that for any sentence of the language,
If it logically implies anything, is it the theory consisting of all the sentences of the language? If the theory consists of all the sentences of the language, is the empty structure the only model for it?