>>80
It's easy to get contradictions or paradoxes in natural language - just look at the liar paradox, or various self-refernece sentences.
Contradictions and paradoxes are not always obvious, even when talking about formal systems. For example, naive set theory is inconsistent. We don't even know if modern set theories are consistent. By Godel's incompleteness theorms, we cannot even know if arithmetic is consistent, and that any consistent system containing (or being stronger than) arithmetic cannot ever claim its own consistency as a theorem (if you do, the system is inconsistent). Claiming consistency as an axiom is okay(not that you'd really know if the new system is consistent), but any belief about arithmetic's consistency is a matter of "religion" (since it cannot be known, despite that it would be utterly strange for it to not be consistent) - you cannot know that in a countable infinitity of inferences (in well-defined first order logic) there is no contradiction. Still, such belief is common in almost all scientists, along with belief such as any finite sentence written in the language of First Order Logic + Peano Arithmetic has a truth value (true or false). In more complex systems, such a belief may not be warranted - some statements can be undecidable or independent in the system, this is especially common with infinitary set theories, examples of independent axioms: Continuum hypothesis, Axiom of Choice, various "high" non-constructive cardinal axioms.
You define something, but you don't know if it's sound or consistent(free of contradictions), or if it's true or false.
Let me give a more complex example, consider Goldbach's conjecture "Every even integer greater than 2 can be expressed as the sum of two primes". It can be defined in arithmetic. It can be computably verified (given unbounded resources) by a Turing Machine, yet if it's true, the process will never terminate (it will never find an integer which isn't the sum of 2 primes). Which means that to prove or disprove Goldbach's conjecture (and many other mathematical hypotheses) one must show if one particular process halts or not.
Some set theories makes bets about the behavior of the infinite, or in this case, infinite processes, and if those bets are correct, you will be able to say if some process terminates or not, even if you would not know of that given only the "beliefs" of Peano Arithmetic (definitions of successor, addition, multiplication and an induction axiom; if you don't like the induction axiom, you can avoid it, and you'll get Robinson Arithmetic instead, but in RA, even if you can still compute anything, you can prove much less, not even simple sentences like: "x + y = y + x").
Consider Goodstein's theorem (
http://en.wikipedia.org/wiki/Goodstein's_theorem ), it states that some particular (computable) sequence terminates, yet it cannot be proven in Peano Arithmetic, but can be proven in some stronger systems (such as some set theories). Of course, we do know that the Halting problem isn't generally solvable, but the idea here that with stronger systems, more things can be proven and more (particular) cases of the problem can be solved. The downside of such betting is that the stronger you make the theory (by adding axioms) the more you risk adding contradictions and making it inconsistent (can prove anything, thus it no longer talks about truth).
Mathematicians cannot always know if a theory has contradictions or not ("always" knowing will require you to have a solution to the halting problem), despite knowing its axioms, but they can hope to discover such contradictions if they exist (otherwise they will work on a false theory).