I know that we can prove a contradiction in naive set theory.
Let D be a set of all sets that don't contain itself. Say D does not contain D. Then D contains D. That means D contains itself. A contradiction. Hence, D contains D. But that means D doesn't contain itself. That means D doesn't contain D again. But we presume D contains D. So another contradiction. Because either D contains itself and D does not contain itself is false, we're pretty much stuck.
So, set theory is pretty much upgraded to kingdom come to prevent this so called contradiction. I am still unsure how after so much upgrade the contradiction is gone.
How do we know that there will be no contradiction for group or other theory? How do I know if that from some axioms like commutative, associative, have identity, etc. suddenly I found out that A is a group and not a group or something like that.
Note: The problem with one contradiction is if you can proof one you can proof anything. Say I want to proof 1=0. Say 1 is not 0. Now, let Z be a set of all sets that doesn't contain it self. Yada yada yada... Contradiction. Hence, 1=0. The sun also rises in the west (at least for the last 300 years)