User:Boris Tsirelson/Sandbox1
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Consistent or inconsistent
If a theory states that 2+2=5, it is a paradox but not yet a contradiction. By "paradox" people may mean
- a contradiction;
- an apparent contradiction;
- something counterintuitive;
- something surprising;
- something ironic;
etc. In contrast, a contradiction (in a mathematical theory) is, by definition, a pair of theorems (of the given theory) such that one is the negation of the other. Thus, two theorems
are still not a contradiction. Two theorems
are a contradiction.
If a contradiction exists in a given theory, this theory is called inconsistent. Otherwise, if no contradiction exist (rather than merely not found for now), the theory is called consistent.