Limits of Gödel's Incompletness Theorems

[ste]

I am not a pure mathematician, and I would appreciate an explanation of what does and doesn't apply to Gödel's famous theorems. Do any of them apply to Euclidean Geometry, and if not, why? What is the broader implication to mathematics as a result of the theorems?

[Epsilon=One]

I am not a pure mathematician, and I would appreciate an explanation of what does and doesn't apply to Gödel's famous theorems. Do any of them apply to Euclidean Geometry, and if not, why? What is the broader implication to mathematics as a result of the theorems?I consider the Incompleteness Theory (www.CQthus.com/PT/GIT) (GIT) as important a 20th century theory as the theory of relativity.

If it is mathematics or any theory/philosophy that is based upon mathematical logic, GIT applies.

Concerning Kurt Gödel's (www.CQthus.com/PT/Go) Incompleteness Theorem: The Incompleteness Theory (www.CQthus.com/PT/GIT) (GIT) is a very difficult theorem to read and understand for most persons. Thus, I’ll paraphrase and over simplify a bit. Few people really understand it, Ernest Nagel and James R. Newman have probably best written about it; even Gödel has little commented about it. I consider Nagel and Newman’s “Gödel’s Proof,” 1958 as the definitive interpretation, as Gödel never refuted it.

As an example of how difficult GIT is to understand: one of my best friends and confidant, who is a heavily cited writer in physics’ journals wrote a sizeable paper concerning GIT that was published in a prestigious journal. He sent me his paper immediately upon its publication, when I read his paper; I realized he had badly misinterpreted GIT; I phoned him; pointed out his mistake; he was in disbelief concerning my interpretation; I immediately FAXed him (there was no e-mail at the time) the paragraph that he overlooked; I received a curt “thank you”; and, we never spoke of it again. It was the following, in a bit more detail, that I based my assertion:“Gödel's proof...does not mean...there are truths which are... incapable of becoming known...

It does not mean...there are "ineluctable limits to human reason."

It does mean that...intellect... and...new principles...await... discovery.

...mathematical propositions which cannot be established by...deduction... may...be established by meta-mathematical reasoning.

It would be irresponsible to claim...indemonstrable truths...by meta-mathematical arguments are based on...bare appeals to intuition.”Ernest Nagel and James R. Newman
Gödel’s Proof, 1958This is not too surprising in that it brought down probably the most powerful philosophy of the twentieth century . . . logical positivism (The Vienna Circle, Carnap, Frank, Neurath, etc., even Karl Popper). I believe that philosophy has “wandered” ever since.

GIT also established the concept that all mathematics was basically unprovable. Again to simplify: because all numbers are based on the value of "One" that cannot be established from within a given system. The value must be obtained from outside the system/set from which all the values of the set are determined. No academic person that I know of has ever successfully challenged GIT for about 75 years.

Thus, mathematics is unprovable; and, mathematical description is what I call the fifth foundational leg of theoretical physics; thus, theoretical physics is a “bit metaphysical” on its fifth leg (entirely metaphysical on the other four); though, as you can see from the above quotation, there was a bit of “wiggle” room for any “proof” that may be based upon metaphysics.