# Understanding definition of conservative extension of a theory

From Wikipedia

In mathematical logic, a logical theory $T_2$ is a (proof theoretic) conservative extension of a theory $T_1$ if the language of $T_2$ extends the language of $T_1$; every theorem of $T_1$ is a theorem of $T_2$; and any theorem of $T_2$ which is in the language of $T_1$ is already a theorem of $T_1$.

Is a theory, as noted by $T_i$, defined as the set of theorems and axioms of a formal system?

Does a language extending another language means the first language is a superset of the second?

Thanks and regards!

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Almost, and yes. A theory is simply a set of sentences in a formal language. Here, though, that set is taken to contain all of its logical consequences, so in effect your understanding is correct. – Brian M. Scott Feb 16 '12 at 2:19
@BrianM.Scott: Thanks! (1) Is a theory a set of all wff of a formal language, or a set of all axioms and theorems of a formal system? Is a theory a concept for a formal language or for a formal system? (2) Does a language extending another language means the first language is a superset of the second? – Tim Feb 16 '12 at 2:27
(2) Yes. (1) No, a theory in a language is simply a set of well-formed sentences (= formulas with no free variables) in that language. – Brian M. Scott Feb 16 '12 at 2:39
Hard to know what you mean by elements of a language. But for example, the logical and non-logical symbols are not sentences. Neither are the terms. A well-formed sentence has no free occurrences of variable symbols, a well-formed formula can and usually does. – André Nicolas Feb 16 '12 at 3:08
There are several kinds of relevant strings, for example terms, formulas, special kinds of formulas such as sentences, strings of sentences, particularly when they are proofs. Strings of sentences are not usually considered part of the language. – André Nicolas Feb 16 '12 at 3:18