A Little Conundrum

Consider the sentence :

“Thsi sentence hsa three errors.”

The puzzle is to find the errors. Two of them are easy to find :

• “Thsi” should read “This”,
• “hsa” should read “has”,

but where’s the third error?

See if you can find it! Search and search… eventually it will dawn on you that

• the sentence, in fact, has only two errors, not three.

So that’s the third error! Found it!

But wait…. that means the sentence really does have three errors. So we were wrong about the third error.

I guess it has only two errors after all.

Only two errors! But the sentence says it has three! Aha! There’s the third error!

Oh. That means it only has two errors. Hmm….

Now if this is confusing you, let’s cut to the chase, and ask the basic question –

• Is the sentence in error about the number of errors it contains? Does it contain two? or three?

I showed my son this little conundrum this afternoon. He recognized it quickly as a variant of similar paradoxes I’ve shown him in the past – all related to the Epimenides Paradox “This sentence is not true“.

Although this seems, in ways, like a silly logical trick, it keeps turning up in different disguises as you dig deeper into mathematics. In fact, it changed the course of mathematical thinking in the 1930’s, when a young mathematician named Kurt Gödel constructed the mathematical equivalent of the sentence “This statement can’t be proven“

This meant that the statement had to be true – or mathematics riddled with deep contradictions.

• If the statement could be proven, it would be wrong. The proof would also be an example proving the opposite of the statement, so that there would be mathematically provable statements whose opposites could also be proven. This would be the equivalent of someone finding a proof that 2+2 was 5.
• If the statement could not be proven, however, that would mean it was true – but unprovable. This would mean that there are mathematical statements that could never be proven one way or the other.

This, and similar ideas, had a big impact on how mathematician thought about their topic over the following decades, and affected other fields also, including computer sciences and the design of modern computers.