scythe33 wrote:lord_day wrote:Assertion:
The smallest positive real number is 1.
Proof:
Let r be the smallest positive real number. Either r > 1 or r = 1 or r > 1
If r > 1 then 0 < sqrt(r) < r and r is not the smallest positive real number.
If r < 1 then 0 < r^2 < r and r is not the smallest positive real number.
Therefore r = 1.
That is awesome.
No, it isn't!
What he gave was a proof by contradiction:
- If r > 1, there's a positive real smaller than r.
- If r < 1, then there's a positive real smaller than r.
- (he failed to mention) If r = 1, positive reals smaller than r exist.
The actual conclusion is that there's no such thing as the smallest positive real.
And this proof is even less interesting when you consider that there are many, many other ways of proving that there is no such thing as the smallest positive real, including such stupidly simple proofs like continuously multiplying your next guess by a number between 0 and 1, or by averaging your next best guess with 0.
His conclusion is completely invalid; it's as valid as a proof that says, "Suppose
r is both real and imaginary. For every real
r < 13,
r is not imaginary. For every real
r > 13,
r is not imaginary. Therefore
r = 13 is both real and imaginary." It's not a fucking proof. It's some kind of bizarre win-by-arbitrary-default condition that's implied and never considered for some reason. It's a non-sequitur.
