Is is possible to prove that two systems of axioms are equivalent

Question

The question is self explanatory. Is it possible to prove that two systems of axioms in two very different branches of math are equivalent? Is there a textbook to help me? Thanks!!

Answer 1

We answer a related question that removes the "two very different branches of math" part. Is there an algorithm which, given as input two recursive sets $A$ and $B$ of axioms, will determine whether or not $A$ and $B$ are equivalent?

The answer is no. For let $A$ be the first-order Peano axioms, and let $B$ be $A$ with the sentence $\varphi$ adjoined. Then such an algorithm would give a decision procedure for whether or not $\varphi$ is a theorem of first-order Peano arithmetic. But it is well-known that there is no such algorithm.

Peano arithmetic can be replaced by other undecidable theories, including finitely axiomatized ones. Most simply, we can look at the theory over the language that has a single binary predicate symbol $P$ and no axioms. So $A$ is the empty set. For it is known that there is no algorithm for determining whether or not a sentence is a theorem of the "empty" theory. (This theorem can be obtained from the undecidability of a certain version of arithmetic, or alternately from the theory of Turing machines.)

Answer 2

in principle it should be possible

first of all you need a dictionary between the two systems to relate the terms in the first system to the terms in the other.

then you need to prove that axioms of system 1 are theorems (provable) in system 2

and you need to prove that axioms of system 2 are theorems (provable) in system 1

good luck