Reductio Ad Absurdum

From Conservapedia

Reductio ad absurdum, also called proof by contradiction, is a method of mathematical proof. It involves assuming the opposite of what one is trying to prove, and showing that this would lead to a contradiction. It works by the law of the excluded middle. The proof typically follows this structure:

1. Create an initial assumption
2. Follow a series of axiomatically valid steps
3. Reach a contradiction
4. Therefore the initial assumption is incorrect

An example of this is Euclid's proof of the infinitude of the primes:

1. Assume there are finitely many primes
2. Take the product of all primes and call it N. Since N+1 is not in our finite set of primes, it must be composite
3. By the fundamental theorem of arithmetic, N+1 has a prime factorization. But N+1 is not divisible by any of the previous primes
4. Since N+1 is composite, there must be a prime missing from our set of primes. But this set contains all primes
5. Therefore, our initial assumption ("there are finitely many primes") is invalid