Proving unicorns exist! (using maths of course)
- Saanvi Karanjalkar
- Oct 31, 2018
- 2 min read
Maths is supposed to be consistent. It’s a subject where there are supposed to be no contradictions, where all true statements could be proved. However, Gödel showed that even maths has limitations where there are true statements which cannot be proved.
A consistent system in maths is one where logically, once we prove a statement, we should then not be able to prove the opposite of it, as the original statement is then proven to be false. But if maths was inconsistent we would be able to prove ANYTHING, even the opposite. Essentially, we would be able to prove false statements since one must be true and other false. It’d lead to us questioning ALL our mathematical proofs as they can lead to conclusions that just aren’t true.
Let’s make this easier by applying it to two simpler statements. These take place in an inconsistent system where opposites can be proved (this is important to remember).Here, the statement ‘there are flying pigs and no pigs can fly’ is true. Even though they are mutually exclusive, they are true.
Claim: ‘There are flying pigs OR unicorns exist’
What this means is that since we can prove there are flying pigs, unicorns must NOT exist to make the claim true (remember we've used the word OR not and). However, we know the statement 'there are flying pigs' must be false as we can ALSO prove pigs can't fly. Since we just proved the claim to be true previously, it must still be true. For that to happen when pigs can't fly, unicorns must have to exist.
We would be able to replace unicorns exist with literally anything we wanted to prove, such as 'God exists' or 'everyone has smallpox'. This is one of the reasons why we definitely do NOT want an inconsistent system.
Gödel's first incompleteness theorem states that any math system must be incomplete OR inconsistent. We just have to hope ours is incomplete NOT inconsistent as even a consistent math system cannot prove its consistency (Gödel's second incompleteness theorem). This could be good news as it's proof of incompleteness but it looks like we just have to work in one with the lingering possibility that it could just be inconsistent.
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