*Slate* runs a good debunking of romantic popular misinterpetations of Godel’s theorem. Key quote

The precise mathematical formulation that is Gödel’s theorem doesn’t really say “there are true things which cannot be proved” any more than Einstein’s theory means “everything is relative, dude, it just depends on your point of view.”

I’ve lost count of the number of times I’ve seen dubious appeals to intuition or claims about chaos theory and the like supported with reference to Godel’s theorem, but I have derived the following proposition:

Quiggin’s metatheorem: Any interesting conclusion derived with reference to Godel’s theorem is unfounded.

Feel free to evaluate with reference to the post title.

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and I’d just kinda of worked out one part of Hofstadter. Bah and anyway if all bloggers are liars then ahhhhhhhhh infinite loooooooooooooop

With the resurgence of creationism the Law of Thermodynamics is a bit whiffy these days two

Actually, JQ, Godel is responsible for two incompleteness theorems. The following explanations and proofs are from http://www.math.hawaii.edu/~dale/godel/godel.html

Godel’s First Incompleteness Theorem. Any adequate axiomatizable theory is incomplete. In particular the sentence “This sentence is not provable” is true but not provable in the theory.

Proof. Given a computably generated set of axioms, let PROVABLE be the set of numbers which encode sentences which are provable from the given axioms.

Thus for any sentence s,

(1) is in PROVABLE iff s is provable.

Since the set of axioms is computably generable,

so is the set of proofs which use these axioms and

so is the set of provable theorems and hence

so is PROVABLE, the set of encodings of provable theorems.

Since computable implies definable in adequate theories, PROVABLE is definable.

Let s be the sentence “This sentence is unprovable”.

By Tarski, s exists since it is the solution of:

(2) s iff is not in PROVABLE.

Thus

(3) s iff is not in PROVABLE iff s is not provable.

Now (excluded middle again) s is either true or false.

If s is false, then by (3), s is provable.

This is impossible since provable sentences are true.

Thus s is true.

Thus by (3), s is not provable.

Hence s is true but unprovable.

Note 1. An analysis of the proof shows that the axioms don’t have to be true. It suffices that (a) the system is consistent and (b) it can prove the basic facts needed to do arithmetical computations, e.g., prove that 2+2=4. The latter is needed to encode sequences of numbers and insure that computable sets are definable.

Note 2. Godel discovered that the sentence “This sentence is unprovable” was provably equivalent to the sentence CON:

“There is no with both and in PROVABLE”.

CON is the formal statement that the system is consistent.

Since s was not provable, and since s and CON are equivalent,

CON is not provable. Thus —

Godel’s Second Incompleteness Theorem. In any consistent axiomatizable theory (axiomatizable means the axioms can be computably generated) which can encode sequences of numbers (and thus the syntactic notions of “formula”, “sentence”, “proof”) the consistency of the system is not provable in the system.

The theories of real numbers, of complex numbers, and of Euclidean geometry do have complete axiomatizations. Hence these theories have no true but unprovable sentences. The reason they escape the conclusion of the first incompleteness theorem is their inadequacy, they can’t encode and computably deal with finite sequences.

Quiggin’s metatheorem is uninteresting. If it were a self-referential paradox it might have minor interest as yet another example of such, but it isn’t even that since – being uninteresting – it falls outside its own scope.

Now if you want really mind bendingly interesting, try Newcomb’s Paradox – and reflect on the fact that it can actually be constructed and observed experimentally (I’ll tell you how in detail later – it involves the Telephone Indian or Inverse Pyramid swindle).

Any interesting conclusion derived with reference to Godelâ€™s theorem is unfounded, but for that still not necessarily untrue, particularly when it comes to bloggers.

There’s a collection of them. Kuhn’s “paradigm shift” and Shannon’s communications theory are two more examples of theories that have a totally different meaning in the Science faculty and the Arts faculty.

One of the points to emerge from Godel’s work is the distinction between truth and proof. If people demand formal proof of true statemements they are liable to get upset when the proof cannot be provided and become prone to relativism, or corrosive skepticism, or some kinds of pomo stances or dadaism (anything goes).

However it is logically possible to operate in the “critical preference” mode, using the truth as a regulative standard for descriptive statements without claiming proof, or certainty or even inductive probability. In this perspective, Godel’s work can be seeen as a support for Popper’s non-authoritarian epistemolology and his theory of conjectural objective knowledge.

This has major political and social implications, as Popper described in the Introduction to “Conjectures and Refutations” and as I have sketched in the introduction to a collection of essays on these matters.

http://www.the-rathouse.com/introrandi.html

â€œthere are true things which cannot be provedâ€? is about as close to a plain-english language statement of Godel’s theorem as you’re gonna get.

So the theorem “doesn’t _really_ say ‘there are true things which cannot be proved'” only in the very uninteresting sense that no mathematical theorem ever says exactly what its plain-english counterpart says.

To see how silly this is, by the same argument you can conclude that there are no mathematical theorems that _really_ say “2 + 2 = 4”.

Try for a

rigorousproof of something simple, say that all integers have an essentially unique factorisation into prime numbers. It will take you at least two pages, if you don’t make any slips (that includes proving any lemmas to the desired rigour)My take on Godel is that you can only prove a statement at a given level of language by going to a higher level of language (a meta level, as they say in the trade).

In practical terms, and in scientific discourse outside artificial or formalised languages, it means that you have to keep shifting to a higher meta level, which the skeptics claim is an infinte regress, but it can be viewed as an infinite progress,so long as you are prepared to accept a conjectural or fallibist epistemology.