Tuesday, October 9, 2007

Gödel's first incompleteness theorem

As humans, we all are incomplete. What ever we do, they all are incomplete. What ever we think, they all are incomplete. So, lets understand this and try to live the life in full before we die.

Do not wait for perfection... Do not wait for the perfect time... Do not wait for the perfect partner... simply because they all are incomplete... imperfect...

Lets accept this and move forward... Now is the best time for every thing.... We love to live in the past and dream about future... But why cant humans live in the present?

Just to give you an idea about this incompleteness, I'm quoting a theorem:


"For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete." - Kurt Gödel

Ok. Let me quote the explanation for this (Wikipedia):

"Here, "theory" refers to an infinite set of statements, some of which are taken as true without proof (these are called axioms), and others (the theorems) that are taken as true because they are implied by the axioms. "Provable in the theory" means "derivable from the axioms and primitive notions of the theory, using standard first-order logic". A theory is "consistent" if it never proves a contradiction. "Can be constructed" means that some mechanical procedure exists which can construct the statement, given the axioms, primitives, and first order logic. "Elementary arithmetic" consists merely of addition and multiplication over the natural numbers. The resulting true but unprovable statement
is often referred to as "the Gödel sentence" for the theory, although there are infinitely many other statements in the theory that share with the Gödel sentence the property of being true but not provable from the theory."

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