Kurt Gödel

From Conservapedia

Kurt Gödel (1906-1978) was an Austrian mathematician who proved the greatest theorem of the 20th century: "Gödel's Incompleteness Theorems." This ended a century of attempts to place all of mathematics on an axiomatic basis.

Gödel published his remarkable proof in 1931. He showed that in any axiomatic mathematical system there are always propositions that cannot be proved or disproved using the axioms of the system. He additionally showed that it is impossible to prove the consistency of the axioms. This was the famous incompleteness theorem: any axiomatic system powerful enough to describe arithmetic on natural numbers cannot be both consistent and complete. Moreover, the consistency of the axioms cannot be proven within the system.

Godel's work abruptly ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.

This demonstrated the folly of the work of several liberal mathematical leaders. The self-described atheist Bertrand Russell had already published, in Principia Mathematica (1910-13), a massive attempt to do what Gödel later proved was impossible. Gödel's proof also disproved the entire "formalism" approach of David Hilbert.

Gödel's proof was a landmark for mathematics, demonstrated that it can never be a finished project as many mathematicians had believed. No one, not even the most powerful computer imaginable, can answer all mathematical questions.

http://www.conservapedia.com/Kurt_G%C3%B6del

Ricardo, are you posting this as a joke?

Russells' atheism had nothing to do with Principia Mathematica. His co-author (and fellow mathematician) Alfred North Whitehead did believe in God (though an unconventional one) and had a profound influence on 20th century theology (by way of process theology).

But Godel also proved the Completeness theorem of first order model theory which showed that in any mathematical language, a theory is provable if and only if it has a model. Thats why I say that mathematics is objective. Whether you accept something like the axiom of choice is irrelivant. The axiom of choice is just one of your assumptions. The world of mathematics without the axiom of choice still works through the rules of mathematics. The reason that mathematics is not a science is because science is based on theories and experiments. Mathematics is based purely on rules of logic like modus ponens: assuming we have the two statements, P and P -> Q, then we will have the statement Q as well.

quote:

Originally posted by FrankLee:
Ricardo, are you posting this as a joke?

Yeah. I stumbled across this a couple of days ago. I think that the conservapedia is supposed to be some sort of alternative wikipedia project for home schooled children of right wing Christians.

quote:

Originally posted by Blake Manner:
But Godel also proved the Completeness theorem of first order model theory which showed that in any mathematical language, a theory is provable (I think you mean consistent here) if and only if it has a model. Thats why I say that mathematics is objective. Whether you accept something like the axiom of choice is irrelivant. The axiom of choice is just one of your assumptions. The world of mathematics without the axiom of choice still works through the rules of mathematics. The reason that mathematics is not a science is because science is based on theories and experiments. Mathematics is based purely on rules of logic like modus ponens: assuming we have the two statements, P and P -> Q, then we will have the statement Q as well.

(continuing the discussion of ZFC vs ZF+AD from the other thread...)

Yes, but when Godel proved his completeness theorem for first order logic, he did that within ZFC (or some closely related set theory with choice).

What I mean is that he imagined himself as living in a set theoretical universe that satisfied the axiom of choice as his meta-theory. He them applied the metamathematical axiom of choice to show that for every syntactically consistent set of first order sentences, he could construct a model that satisfied them all. (More narrowly, today we would say that he only needed to apply the Prime Ideal theorem for Boolean Algebras, which has been proven to be strictly weaker than the Axiom of Choice, assuming that ZF itself is consistent.)

However, if I imagine myself as living an a metamathematical set theoretical universe which satisfies the axiom of determinateness, then I can construst a syntactically consistent set of first order sentences that has no model, thereby giving a counterexample to Godel's Completeness "Theorem".

The subjectivity here is at the meta level, with sentences and models being ordinary mathematical objects constructed out of sets.

I simply don't buy the completeness theorem itself, because I can prove that it is false. On the other hand, you believe it because you can prove that it is true.

Remeber, while you may believe in choice, or at least be a prime idealist, I am a determinist, at least for the duration of this little parable.

This message has been edited. Last edited by: ricardomath,

Clowns to the left of me...
Jokers to the right...
Here I am, stuck in the middle with you.