Godel, Kurt (1906-1978). Born in Brünn, Austria-Hungary currently Brno, Czech Republic, Godel’s “Undecidability Theorem” demonstrated that at some point a mathematical system based on axioms and operational rules becomes self-referential and cannot prove nor disprove the original axioms upon which subsequent statements are derived. Furthermore, Godel’s “Incompleteness Theorem” revealed that a mathematical system, if taken far enough, begins to generate contradictory statements. These theorems suggest that mathematics is always an extension of original axioms. Rules and procedures within a given system merely extend the original axioms. This presents problems for mathematics as a whole and particularly for those who wish to integrate its various branches. Godel suggests that if each branch stems from different core assumptions, all branches cannot be artificially pasted onto a single body of assumptions. A mathematician at IBM, Gregory Chaitin, says that Godel’s work, along with uncertain, unpredictable elements in quantum physics, demonstrates the limits of mathematics in particular and scientific knowledge in general.
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Well, I don’t know how to phrase it nicely but this article does not at all reflect Gödel’s work. Let’s do it point by point.
– “Godel’s “Undecidability Theorem””:
Gödel produced 2 theorems about “incompleteness” and that’s usually how people refer to it.
– demonstrated that at some point a mathematical system based on axioms and operational rules becomes self-referential …
Rather, he used self-referent expressions to demonstrate his theorems. Formal systems (the “mathematical system based on axioms and operational rules”) don’t “become” self-referential
–“cannot prove nor disprove the original axioms upon which subsequent statements are derived”
By definition axioms are never to be proved. Those are the original starting points from which subsequent derivations (the proofs of the theorems) are made.
– “revealed that a mathematical system, if taken far enough, begins to generate contradictory statements.”
No. The first Incompleteness theorem just states that within any formal system that includes the number theory and that is consistent (meaning one cannot demonstrate one thing and its negation) one can find true propositions/statements (“true” meaning that we can demonstrate they are true) that cannot be demonstrated/derived in *this* formal system; hence the name “incompleteness”: not all “truths” for a given system can be demonstrated *in* this system.
“These theorems suggest that mathematics is always an extension of original axioms.”
Formal demonstrations are always derivations from axiomatic systems. It’s not what is suggested by Gödel at all, it’s just the way mathematicians are building formal systems.
“This presents problems for mathematics as a whole and particularly for those who wish to integrate its various branches”
I don’t understand this argument
Hope this is useful.
jm
Comment by Jean-Marc — December 11, 2007 @ 10:48 pm |
Thanks… I’m going to have to look over my sources for this again and also ask mathematics people to continue with this thread.
I appreciate the time you’ve taken to comment on this.
Comment by earthpages — December 11, 2007 @ 11:23 pm |