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> a proof by contradiction is specifically a proof which shows that P is not false, and concludes that it is true. And this proof matches that description exac
by thaumasiotes 2y ago
> a proof by contradiction is specifically a proof which shows that P is not false, and concludes that it is true.
And this proof matches that description exactly, with P = "√2 ∉ ℚ". The only case where these would be different ideas is the case where ¬¬P ≠ P.
And of course, that can never happen.
- Smaug123 2y agoYour statement is, I think, being wilfully sloppy. To quote the outline of the proof: "First Euclid assumed √2 was a rational number.". To quote the proof itself: "Euclid's proof starts with the assumption that √2 is equal to a rational number p/q.". In two different places, the proof explicitly states that it is showing that "√2 in ℚ" is false. It is not showing that "√2 ∉ ℚ" is not false; such a proof would begin "Suppose that it were not the case that √2 ∉ ℚ", which is obviously not how the proof starts (and for good reason, because that would be much more confusing). By all means argue that "nobody cares about excluded middle"! You're probably right, and when I insist that "proof by contradiction" has a meaning that is correctly stated by Wikipedia and the nlab, I'm just like one of the old fogeys complaining about things like "could care less" or "irregardless"! But don't misquote arguments and say that they support your case when they don't.