Gambling All Of Mathematics

"A chess-master may gamble a piece, or even the entire game; but a mathematician writing an ad absurdum argument is gambling all of mathematics." - Couldn't find the source.

- This post can also be considered a hint to Solving Conundrums.

What is an ad absurdum argument? As always, you can see wikipedia, but let me take one of the most famous examples, and then give an explanation. Essentially the argument goes like this: Assume the opposite of what you want to show, arrive at a contradiction, conclude the opposite of what you assumed.

Example of an ad absurdum ("to the absurd") argument:

Are there a finite or infinite number of primes? (Primes are numbers p that cannot be factored into any other factors but p and 1. Primes per definition are > 1.)

Well, assume the statement Q to be true:

Q: 'There is a finite number of primes'

Then there are exactly n primes for some number n, and we can number these primes p_1, p_2, p_3 ... to p_n. Then we make a new number s by multiplying all these primes, and then adding 1 (s = p_1*p_2*...*p_n +1). This new number s will not be divisible by any of our primes, so the only possible factors of s are itself and 1. Hence there is at least n+1 primes. But this is impossible as n is the total number of primes.

We assumed Q and ended up with a contradiction, so Q must be wrong. Hence there is an infinite number of primes.

Example end.

So how does this work? I assume that we agree on what Logic is (the axioms and rules), as it would be far too cumbersome to write it out. Last time we talked about consistency of logic. This can be represented by the following axiom:

((P) and (not P)) equivalent to false

What we got in the proof was that the number of primes was exactly n and n+1, that is, exactly n and not n. What we got was equivalent to false by the above axiom. We showed that:

Q implies false

Now we want to use something called the rule of transposition:

(A implies B) is equivalent to ((not B) implies (not A))

In total we get

(not false) implies (not Q), which gives

true implies (not Q), which gives

not Q, which is

'There is an infinite number of primes'

“Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.” - Sherlock Holmes