HOUSE_OVERSIGHT_015893.jpg

Known Unknowns
203

You have to step back and think about the problem in the round and then devise some additional rules to handle the circumstances. Mathematics is like this also.

Here is how Gödel proved his result.

It is easy to turn logic or any text into numbers. That’s how this book is stored on my laptop. All we need do is translate sentences into ASCII or Unicode. In this way, any theory can be reduced to a string of numbers.

Since Gödel’s proof predates the invention of the computer, he had to come up with a novel way to store information. He deployed an old Roman invention; a substitution code. The number one was represented by 1, two by 2 and the symbols by larger numbers, for example, ‘=’ was coded as 15 and so on. He then raised a sequence of prime numbers to the power of each of these codes and multiplied all the results together. This generated a single enormous but unique number that he could later factor back into its constituent parts to recover the information. This is a truly complicated solution to a very simple problem. Today we would solve it by storing each number in the memory of a computer as an array.

Let’s use the easier table method to store things and code as follows: 000 will stand for ‘start of proof’. Each step in the proof will start with 00 and each symbol in the proof starts and ends with a zero. This way we can code one plus one equals two as follows.

0000001110454011101210222000000

I think this is simple enough for you to guess the coding scheme. Hint: 111 stands for 1. The scheme is on my website if you can’t work it out. Using this technique, any series of mathematical statements can be turned into a number. As a series of mathematical statements is a proof, we can generate proof numbers. They are just the sequential list of all the instructions. These numbers are sometimes referred to as Gödel numbers.

Gödel’s next step was to say one number demonstrates the proof of another number. For example, the number 000820962 might demonstrate the proof of another number 000398... This is the mathematical equivalent of my saying a Word file demonstrates the truth of your mathematical theorem. Any statement can be represented by numbers, provided you have a consistent coding scheme that allows you to get back to the meaning.

Now Gödel set up his paradox:

HOUSE_OVERSIGHT_015893