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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 Gédel 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 Gédel’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 Godel
numbers.
Gédel’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 Gédel set up his paradox:
HOUSE_OVERSIGHT_015893