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human
Should mathematicians be worried about the implications of Gödel's incompleteness theorems?

From what I understand, Gödel proved that there does not exist any set of axioms, capable of describing arithmetic that are complete. That is, his incompleteness theorems showed that no matter the axioms we choose to base our arithmetic on, in that system, there will always exist truths that cannot be shown to be true.

My understanding of the matter isn't great, but nonetheless it leaves me disturbed. Should mathematicians be concerned about this result? Does this mean that, for example, something like the Riemann Hypothesis could ultimately be impossible to prove due to limitations in the definitions of our axioms? And since it seems we are limited by the way we define our axioms, would it ever be possible to get around this incompleteness by changing the way in which we define mathematics and it's axioms?
gpt
This is a subtle question, and there is some room for interpretation:

Goedel showed that for every axiom system that is strong enough to have
some basic reasoning about natural numbers, there exists a formula
that is not provable in this system, but which is true. 
This means that we actually know the truth of the formula, it is only the
axiom system that cannot prove it. If you do the obvious thing, which is 
adding the unprovable formula as a new axiom to the system, the strenghtened
system will have other formulas that are unprovable. 
So, there are always unprovable formulas, and if you add those as axiom,
there will be new unprovable formulas.

The goedel formulas are very specific in nature, very artificial, and
we know that they are true, so they are not a real problem.
If somebody would show that the Riemann hypothesis is equivalent
to a Goedel formula of some system, people would consider the Riemann
hypothesis solved.

It is, however, very unlikely that this will happen, because the Goedel
formulas all have something to do with computation. Natural numbers are strong enough to represent computation. The state of a computer can be expressed
by a natural number, and possible computations can be expressed by relations
between natural numbers. On the other hand, proving can be viewed as form of computation. 
Because of this, the system can reason about its own proofs represented 
within itself. It is this self referentiality which causes the incompleteness.

So, in order to be potential Goedel formula, the formula must have something
to do with computation. Since the Rieman hypothesis is about continues
functions on the complex numbers, it is very unlikely that it has.

We can still consider the possibility that  the Riemann hypothesis is independent
from current formalizations of set theory. There are formulas that
are independent of set theory (continuum hypothesis, axiom of choice),
but these formulas reason about ridiculously big sets. 
The Riemann hypothesis doesn't seem to do this. It is just about
complex numbers.

There is no reason to believe that the Riemann hypothesis is unprovable
in our common axiom systems. We are just not smart enough to find a proof.