Who Shaves the Barber?

I just finished listening to a podcast called Radio Lab.  It's a great "science nerd" podcast and this one was about paradox.

Bertrand Russell took mathematics and turned it on its ear around 1901.  His paradox can be written as follows:

What does this mean?

Any definable collection is a set.  Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition.  This contradiction is Russell's paradox.

Another way to look at this is the Barber's Paradox.  Although it has been attributed to Russel, he denied it:

Suppose there is a town with just one male barber. In this town the barber shaves only those men in town who do not shave themselves.  The question arises: Who shaves the barber? This question results in a paradox because, according to the statement above, he can either be shaven by: himself,  or the barber (which happens to be himself). However,  neither of these possibilities are valid! This is because: if the barber does shave himself, then the barber (himself) must not shave himself. If the barber does not shave himself, then the barber (himself) must shave himself.

Of course: this statement is a lie.... is another example.

Kurt Friedrich Gödel developed incompleteness theorems. They are two theorems of mathematics that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic.  The theorems are important both in mathematical logic and in the philosophy of mathematics.

The actual theorems are someone complicated...but think if the summary as:  one can set a sceario in which there is no answer that is right......and even math (something that seems logical and "provable") may not be logical or provable.

The application of the incompleteness theorems helped me understand them:

You need to understand Goldbach's conjecture first:

Goldbach's conjecture:

One can always write ANY even number  (greater than 4) as a sum as two prime numbers.  For example 12 (even number) can be written as 7 + 5.  Another example 24 (11+13) or 36 (19+17).

However when we apply incompleteness theorems we run into a problem:

This (Goldbach's conjecture) has been checked out to trillions.  To this day there is not one single counter example to Goldbox conjecture. You might think it is true.  In reality IT IS NOT TRUE it is undecidable.  The reason for this is that it is NOT probable.  All we can do is check each even number...but no matter how many numbers we check we can never PROVE IT, because we never run out of even numbers.

This of course is where the podcast ended.

So how many things do you know that feel true...are really just undecidable?