theory, theorems and axioms

1. According to Wikipedia

In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. Usually a deductive system is understood from context. An element $\phi\in T$ of a theory $T$ is then called an axiom of the theory, and any sentence that follows from the axioms ($T\vdash\phi$) is called a theorem of the theory.

a theory in a formal system is the set of axioms. But I don't know why I wrote it as the set of theorems in my old note. I am now sorting things out, so is the theory the set of axioms or the set of theorems?

2. Also Is the set of axioms required to be not deducible from each other under the set of inference rules? (i.e. to be minimal under the inference rules?)

I wonder if the set of theorems can be taken as a new set of axioms, which is equivalent to the original set of axioms?

Thanks.

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What is the first question? There is no question mark anywhere in there, and you reference the "question in the title," which is also not a question. – Thomas Andrews Jul 20 '14 at 17:41
And no, nothing requires the set of axioms to be independent. We often like independent axioms, but we certainly don't require them... – Thomas Andrews Jul 20 '14 at 17:42
@ThomasAndrews: My bad. the first question is: is the theory the set of axioms or the set of theorems? – Tim Jul 20 '14 at 17:44
It's a little weird that Wikipedia uses $\phi\in T$ as an axiom the first time, and then, in the same sentence, uses $\phi$ as a theorem. – Thomas Andrews Jul 20 '14 at 17:52

1 Answer

The problem with taking the theorems as axioms is that the set of theorems is not recursive, so you cant say for sure if a statement is a theorem before you have proved it. The set axioms should be recursive.

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(1) "taking the theorems as axioms" or "taking the theory as axioms" (2) Is the set of axioms always recursive? – Tim Jul 20 '14 at 17:46
It's certainly possible to have theories with axioms that are not recursive. We generally don't use them, but we can study them theoretically. The question wasn't whether theorems were axioms, but whether we define a theory by the theorems, of the axioms. – Thomas Andrews Jul 20 '14 at 17:49