But 1=0 is not a contradiction, it is simply false. All contradictions are false, but not all false statements are contradictions. But that is neither here no

But 1=0 is not a contradiction, it is simply false. All contradictions are false, but not all false statements are contradictions. But that is neither here nor there. What matters is that a provably false statement (in this case a contradiction, but that doesn't really matter) follows from some assumption, and therefore the assumption must be false. This is not circular reasoning, it is proof by reductio ad absurdum (https://en.wikipedia.org/wiki/Reductio_ad_absurdum https://en.wikipedia.org/wiki/Reductio_ad_absurdum).