Topic: Godel
Here is an example of a simple axiomatic system:
A v B = B v A
A = C
The first statements reads, "A or B equals B or A." From these two sentences, we
can create other sentences, like A v B = B v C. If every statement that can be
expressed within this system can be constructed by various combinations of the
two statements, then the system is said to be complete. This trivial system is
certainly complete. The system logicians were especially interested in is one
which could reduce mathematics (arithmetic) generally to basic propositions of
which the above are examples. But the system must be complete in order to be
successful. The implication of the system not being complete would be that there
are mathematical truths which can't be proven. Plainly speaking, the goal here
is to be able to "know everything" by drawing deductions from trivial
assumptions. Gödel demonstrated a statement that can be expressed within the
logical foundations of mathematics which can't be derived from the system,
rendering any logic powerful enough to do math generally, incomplete. Let's fast
forward through the rules of the game and all the key plays necessary for the
setup and concentrate on the structure of the final slam dunk which brought down the
glass.
(Loosely following Braithwaite's Introduction)
Important Definitions
v Gen q (v,w): The q (v,w) part means the variables "v" and "w" hold a relation to each other "q". The "v Gen" part means to substitute other variables for "v" in this relation.
Gödel number: An actual number which he has decided represents a formula - kind of reversing the roles of numbers and formulas. Why a number opposed to just another variable will make sense in the next section.
Start with v Gen q (v,w). We will define q as "not a proof." So this reads, q is a relation between "v" and "w" such that for all Gödel numbers (representing formulas) that could be substituted for v, v is NOT a proof of w. Now let's substitute the Gödel number for the formula "v Gen q (v, w)" into w, let's say that's 23. So we get, v Gen q (v, 23) which means, "For all Gödel numbers of formulas that we could substitute for v (all formulas), none of these formulas are proof of the formula with the Gödel number 23. Well, if no formula is a proof of the formula known trivially as 23, then there is no proof for formula 23. Of course this formula that can't be proven is "v Gen q (v, w)" - itself. So this is the formulae which can be rightly constructed, but declares of itself, that it can't be proven. He then shows that if the formula could be proven, the negation of the formula could also be proven, making the foundations of mathematics inconsistent. So we can choose between incomplete yet consistent and complete but inconsistent. Finally, he shows that if we get tricky and try to assume this formally undecidable proposition away, another can be constructed and this ad infinitum.
In The Matrix Reloaded, it's revealed that Neo is, obviously, the Gödel sentence
of the Matrix. There had been prior versions of the Matrix and every one had an
anomaly that would crop up. The architect had never been able to fix the problem by
integrating prior anomalies (Gödel statements) into the structure of the system.
A new one would always appear.