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What does the following mean, exactly?

When axiomatizing a theory, it is interesting to ask whether a set of axioms uniquely axiomatize the theory - $\textbf{that is, is it only this theory which makes this set of axoms true?}$

The first sentence, I understand, I think. We are asking whether the theory at hand can only be encaptured by a certain specific set of axioms. But then I am confused by the sentence in bold. What does it mean to say that the theory makes the axioms true? Did she (lecturer) mean to say it the other way around, i.e. asking "are these axioms the only ones which make the theory true?".

Peamo
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I think the question was correct as written. Usually --- maybe always --- a theory can be characterized by any of a number of different sets of statements within the theory chosen to be the axioms. However, say we look at Euclid's postulates without the parallel postulate. If you add the parallel postulate to it you get one theory; if you add one of the alternatives to the parallel postulate you get a different theory, but in both of them your original axioms are true. Therefore neither is the only theory in which those axioms hold.

Thus it is a mistake to think the question is "whether the theory at hand can only be encaptured by a certain specific set of axioms". Rather, the question is whether it is the only theory "encaptured" by the given set of axioms.

Michael Hardy
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