Re:All Horses Are the Same Color
by ajuc on Sunday December 05, 2021 @02:09PM
(#29451981)
> But you're not trying to prove X = 3 and starting out by assuming X = 3.
No, you're trying to prove
for all N>=1: A(N) is true
and start by proving that
IF A(N) is true THEN A(N+1) is true
This implication can be true even if A(N) is false, and your final proof will only use the implication, not the assumption used when proving that implication.
Assumptions in math proofs can have scopes. For example in many proofs you split the domain into subsets and prove that something is true assuming X>=0 and X<0 separately. Naively you would say that we cannot assume X>=0 and then assume X<0 in the same proof because that's a contradicion.
But these assumptions were used in different scopes so there's no contradiction.
No, you're trying to prove
and start by proving that This implication can be true even if A(N) is false, and your final proof will only use the implication, not the assumption used when proving that implication.Assumptions in math proofs can have scopes. For example in many proofs you split the domain into subsets and prove that something is true assuming X>=0 and X<0 separately. Naively you would say that we cannot assume X>=0 and then assume X<0 in the same proof because that's a contradicion.
But these assumptions were used in different scopes so there's no contradiction.