• Simplified proofs

Here are some demonstrations (simplified proofs) of 'edges of logic' where we can 'know that we cannot know'.

Demonstrating that √2 is irrational

Suppose p / q = √2 , where p and q are integers with no common factors (e.g. not like 14 & 10 which both can be divided by the common factor 2).

Squaring gives

p² / q² = 2

So p² = 2 q²

Which means p must be even (being twice some other number).

If p² is even, then so is p.

If p is even, then p² must be divisible by 4.

Then, q² must be even, so q must be even which contradicts the supposition that p and q do not share the common factor 2.

Hence √2 cannot be written as a ratio of known numbers. It can only be written as an infinite sequence of decimal digits in a non-repeating pattern:

1.414213562373095 …

(" … " indicates non-repeating sequence of digits)

Demonstrating that an infinite list of numbers cannot be complete and Gödel's incompleteness theorem

A very important part of Kurt Gödel's incompleteness theorem is the demonstration that a supposedly exhaustive infinite list of numbers cannot include all possible numbers. It is known as Cantor diagonalisation after George Cantor while he was considering infinite sets of numbers (such as the list of all integers and the list of all irrational numbers). Cantor diagonalisation has the following argument:

Consider a infinite list of numbers computed in order by some computation where each computation, n, produces the n th number listed in order, e.g.:

1 4 1 4 2 6 9 5 4 8 4 …

2 5 9 9 0 2 5 0 8 3 5 …

3 1 1 2 6 2 1 4 7 9 0 …

4 1 5 4 2 9 5 3 2 6 2 …

5 0 0 1 0 5 5 3 4 5 0 …

6 3 6 2 3 5 1 7 4 7 1 …

7 6 4 5 6 5 7 7 9 9 0 …

8 8 3 6 6 1 0 5 0 6 0 …

9 0 1 1 0 0 0 0 0 1 3 …

. …

The bold digits down the diagonal are now used to compute a new number constructed by taking each bold digit and changing it.

E.g. the number from the diagonal:

4 9 2 2 5 1 7 0 1 …

can be changed (by adding 1, for example):

5 0 3 3 6 2 8 1 2 …

This new number, though computed, cannot be a member of the computed list, since it is different from every number in the list (since the n th digit in the new number differs from the n th digit of the n th number in the list, for all n).

The list of computed numbers, even though infinite in length, is incomplete, where numbers exist which cannot be in that list.

Gödel's incompleteness theorem then extends this principle to any theory made up of consistent rules (axioms) as a list of statements that can be encoded to refer to statements within that theory. Gödel produced a relationship between statements of a theory and numbers. The result, relating to Cantor diagonalisation, shows that a suitably structured theorem has at least one statement that translates to the following form:

"It is true that this statement has no proof"

Gödel showed that this is generally true, where for example Euclidian geometry has the axiom that states:

"Two parallel lines will never meet"

It was known from an early stage that this statement had a perplexing uncertainty to it and when curved spaces and geometries were developed clearly came to be true only when space was defined to be flat - with the set condition that parallel lines cannot meet.

Gödel's proof was found at a time when the exact opposite was also being attempted to be proved, i.e. that mathematical theorems could be complete and self evidently & absolutely true. In principle Gödel's theorem is very important.