Proof by contradiction not contradicting If the proof by contradiction never contradicts that means that the original statement is false right? Or is this way of thinking flawed?
 A: There can be three reasons for not being able to find a contradiction.


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*The original statement is indeed false, so when you assumed the contrary, you assumed something that is true, and so there is no contradiction, and hence you couldn't find a contradiction.

*The original statement is true, so when you assumed the contrary, you assume something that is false, but the theory within which you are working is not complete and in fact not powerful enough to generate a contradiction.

*The original statement is true, so when you assumed the contrary, you assumed something that is false, but you were unable to generate a contradiction even though the theory you are working with is powerful enough to generate that contradiction.
So, as you can see, there are several explanations for not being able to find a contradiction, and in some of those the statement is false, and in others the statement is true. Therefore, you cannot infer the statement is false just because you cannot derive a contradiction.
A: A contradiction implies that the original statement is false but not being able to obtain one doesn't say anything about the statement. 
