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It seems to me that mathematicians or computer science theorists are still trying to come up a proof for conjecture they made in first or second order logic. By Godel's incompleteness theorems there are statements that could be true but unprovable.
My question is how do they know they are not wasting their time working on something unprovable? In general, are there any "work around" after Godel's incompleteness theorems?
Godel's result tells us that any sufficiently rich formal theory will permit propositions whose truth is not decidable within that theory.
In some cases it is possible to prove that a proposition is not decidable. A well known example of this is the continuum hypothesis of standard set theory.
In other cases it may not be possible to prove that a proposition is decidable because such a proof does not exist. In other words, decidability is not necessarily decidable.
So yes, it is entirely possible that people are completely wasting their time working on such problems as the Riemann Hypothesis, the Goldbach Conjecture, or any of the other well known conjectures.
EDIT
Considering the idea of a "work around". Firstly, it must be said that a mathematical theorem is a mathematical truth, so the idea of working around it seems opposed to the idea of mathematical truth. Having said that, if we are willing to bend the truth, then consider this :
Take the case of the continuum hypothesis of set theory. We can prove that this proposition is not decidable. We can also prove that both the continuum hypothesis and its negation are consistent with standard set theory in the sense that if we were to add either the hypothesis or its negation to our axioms, then the resulting theory would be consistent if set theory is consistent.
In this sense, we can work around the road block and explore a set theory where the continuum hypothesis is true ( or false ).
The axioms of any formal theory are generally taken to be self-evident truths. This is because we accept that some statements are true for no reason at all. Similarly, the continuum hypothesis, or any other conjecture which is not decidable, may be true for no reason whatsoever. However, in the case of the continuum hypothesis, its truth is clearly not self-evident.
This is more of a question of psychology than philosophy.
As you point out, a mathematical conjecture can be true, false or undecidable.
However, you're assuming that a mathematician will prefer one or other of the outcomes. This is by no means the case. In other words, if one was to show that, say, the Riemann Hypothesis was undecidable that would be as important a result as proving it true or false.
