Most of us have heard about Gödel via Douglass Hofstadter, someone who has been utterly mystified by Hofstadter, or someone who has been inspired by Hofstadter to invent their own Gödel mystery cult.  Being susceptible to mysticism myself I knew I'd need to be strong when I set out to get to the bottom of Gödel a few years back.  It really is a temptation to extrapolate too far. I mean, take possibly the most important result in logic - ever - and combine it with the one subject the general public would have an interest in where Gödel's theorem might have an application, the mind - which just in fact might conceal the greatest mystery ever - and how could the average person not be seduced? Hell, how could the average person not want to be seduced? While I've personally only scratched the surface of mathematical logic, I think I learned enough in my little excursion to benefit others who have become interested in Gödel through popular literature understand his theorem and how to skeptically consider the applications.  I think the best way to present Gödel given the walls I ran into is, after the remainder of this brief introduction, forgo all the context and just spell it out in grossly oversimplified terms. Then I'll add some context, along with considerations to the philosophy of mind.  I'll have some final notes that step through his actual proof and translate the main ideas.

To over-oversimplify, Gödel will show that from no formal system, or rather, list of assumptions and rules for making deductions from those assumptions, can we produce all true statements of the formal system.  In other words, Gödel will discover a statement that is true within a formal system, but this statement can't be derived from the formal system.  Why that's important to math and logic I'll save for later. But a brief note on the importance of the result to the philosophy of mind, since that's what's the interesting part for most people.  It would seem the only analog we have for describing how the brain works, or how the mind works, is a machine. And probably a fast one, like a computer.  But a machine is just the physical embodiment of rules.  So if Kurt Gödel's brain is a machine, then how could he have ever come up with his "Gödel sentence?"

Posted by gadianton2 at 9:22 PM

Updated: Tuesday, 7 February 2006 7:42 PM

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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.
 

Posted by gadianton2 at 12:01 AM

Updated: Tuesday, 7 February 2006 7:43 PM

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