Lately I lose sleep at night worrying about the consistency of various theories. Linear algebra, number theory, set theory etc.
Question 1: How do we know things like Russell's paradox (which ultimately lead to reformulation of axiomatic set theory) won't repeat themselves in these theories?
Many times you take a problem and look at it from a different direction, reaching a conclusion that is consistent with what you already know. Sometimes it seems like magic.
A consistent theory is one that you cannot prove a claim and its opposite using its axioms. (reaching a contradiction)
Question 2: It is said that when a model is found for a certain theory, that theory is consistent. Why? Why can't it be that our model is inconsistent?
Case in point, non euclidian geometries - not agreeing with the 5th postulate, how do we know these are consistent. Perhaps I can construct something using such a geometry that would contradict itself, maybe this doesn't prove the 5th postulate doens't follow from the first 4.