I have an fundamental question about what Mathematica can and cannot do.

I have a book which presents a certain physical theory in an axiomatic manner. The axioms make heavy use of mathematics.

Some representative examples of the axioms it contains are as follows (this is certainly not the complete set of axioms provided):

1. Let $M^3$ be a 3-dimensional metric space with metric tensor $g$.
2. Every $U_{\gamma }\in \left\{U_{\gamma }\right\}$ and it's first order derivatives are smooth on $M^3$.
3. Every $\rho \in \{\rho \}$ is a function from $B M^3 T$ to a set of nonnegative reals, Lebasgue integrable over any finite region of $M^3$.
4. For every $\gamma \in \Gamma$, every ..., there exists at least one $k \in K$ such that $\rho \ddot{X}=\text{divT}-\rho \nabla U$.

Where the symbols have their usual meanings in mathematical physics. Then, after many more axioms and definitions in this vein, several 'theorems' are stated and proved from the axioms and definitions.

My questions are as follows:

1. It is somehow possible to represent these mathematical statements in Mathematica? The focus here is not on the specific axiom system I chose, but axiom systems of this general nature. For example, this particular axiom system uses tensors, which may or may not be representable in Mathematica, but that is not the point. Even if tensors are not representable, axioms systems of this nature which happen not to use tensors may be representable (or not). Whether or not they (likely) are, is the question I'm asking.
2. If representable, is it then possible to use FullSimplify (or any other feature of Mathematica) to check if other statements follow from using these as assumptions, which is my main goal, or otherwise manipulate these statements in any manner? (For example, I know how to use FullSimplify to determine whether or not the square root of 2 is rational, but of course, that is trivial compared to anything involving these axioms.)
3. If not representable, is there any third-party package for Mathematica, free or commercial, which would make the goal possible?
4. If still not possible, is there any other computer algebra system currently existing, free or proprietary, which would make the goal possible (Magma, maybe?)?
5. If still not possible, why do you think that is the case? Is it somehow fundamentally impossible (which I doubt) or difficult to do this using a computer? Is this technology we (as in the human race) are on the verge of achieving, or something which might happen decades later, or not at all?

In asking this question, am I fundamentally misunderstanding the nature and purpose of Mathematica and what it can do? If so, why? I had the impression that Mathematica specializes in symbolic manipulation, as opposed to something like MATLAB, which specializes in numerical computation. My example certainly seems to be of symbolic manipulation, though of an abstract nature.

Note:

I am aware that the main goal is, in principle, achievable using an automated theorem prover like Prover9, but in practice, impractically difficult, requiring axioms of all the mathematics involved - sets, numbers, analysis, geometry, etc, all built on top of each other and ultimately on first order logic - possibly thousands of axioms, causing a combinatorial explosion in any proof that is attempted.