Posts Tagged ‘Kurt Gödel

25
Aug
06

Gödel Proof

All consistent axiomatic formulations of number theory include undecidable propositions …
Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved …

How can you figure out if you are sane? … Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self-fulfilling prophecies, though the process is by no means inevitable. Everyone knows that the insane interpret the world via their own peculiarly consistent logic; how can you tell if your own logic is “peculiar’ or not, given that you have only your own logic to judge itself? I don’t see any answer. I am reminded of Gödel’s second theorem, which implies that the only versions of formal number theory which assert their own consistency are inconsistent.

The other metaphorical analogue to Gödel’s Theorem which I find provocative suggests that ultimately, we cannot understand our own mind/brains … Just as we cannot see our faces with our own eyes, is it not inconceivable to expect that we cannot mirror our complete mental structures in the symbols which carry them out? All the limitative theorems of mathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally.

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25
Aug
06

Gödel’s Incompleteness Theorem

In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn’t be proven either true or false using the rules and axioms … of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you’ll only create a larger system with its own unprovable statements.

godel.jpg

The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.

Gödel’s Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths … It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express ideas. And it has been taken to imply that you’ll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.




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