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Something Different Than Being Derived

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Posted by Richard Ruquist on January 25, 2002 15:05:11 UTC

Forgive my ignorance. I naively thought that when a theorem is derived from axioms, it was automatically proven. Apparently that is a mistaken notion. I have no idea what the concept of decidibility means. Maybe something different from a proof.

You can try to educate me if you like and I will appreciate the effort. But basically it is my responsibility to educate myself on such matters.

However, I do not understand where the following quote from your post came from:

"I don't see how one can say that he
was against theorems being derived from their axioms."

Did I say that? If so, I certainly did not mean to convey the impression that I thought Godel was against derivation.
I have now read Godel's paper that you referenced and confess that I have almost no understanding of it. I'll have to rely on you and Penrose for interpretation.


Regarding that first paragraph: whether Penrose's take- that there are solutions or theorems that are true yet not derivable from axioms- depends on what Godel means by the word 'problem'. If problem implies solution, or some kind of correct mathematical statement, then Penrose would appear to be correct. That it confounded Hilbert's program is also correct and beside the point.

My original point was that these underivable theorems are emergent properties of any math system. Godel seems to label them as (x)F(x) in his paper, whatever that means. So mathematics gives us something for nothing, to use a cliche.

The Sante Fe people like to consider such emergent properties as a possible explanation of consciousness. My hypothesis is that quantum waves of any kind are consciousness and that Godel's emergent solutions are a possible means for disparate quantum systems to communicate.



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