Ethics
(I)God is Morality
Evolution
Dembski Dismissed
Godel
(I)Godel
(II)The Basic Idea
Postmodernism
(I)Five Points
Skeptics Boot Camp
(I)Abusing Fallacies
(II)Catch-Phrase Philosophy
(III)All or Nothing
(IV)Faith is Bad
Man Of The World
Tuesday, 10 January 2006
Index
Ethics (I)God is Morality
Evolution
Godel
Postmodernism
Skeptics Boot Camp
Posted by gadianton2
at 12:01 AM
Updated: Tuesday, 31 January 2006 6:32 PM
Sunday, 8 January 2006
Gödel (II) The Outline Topic: Godel Here is an example of a simple axiomatic system:
A v B = B v A (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. |