2006-09-18
14:43
unprovables
Posted by Luke Schierer under evolution | Permalink | | Leave A Comment
Mr. George Gilder claims that Mr. (Dr.?) Kurt Gödel proved “in essence that every logical system, including mathematics, is dependent on premises that it cannot prove and that cannot be demonstrated within the system itself, or be reduced to it.”[1][2] Does anyone know if that is in fact the case? The proof is no doubt beyond me, but E.L. or J.D. would likely be able to understand it, were they interested. I am rather curious to know if this authors claims of the proof (that mathematics necessarily rests on unprovables) are true.
- Mr. George Gilder. “Evolution and Me.” National Review 2006-07-17. Viewed 2006-09-18 at http://www.discovery.org/scripts/viewDB/index.php?command=view&id=3631
- I have tagged this post as being “evolution.” Why I have done so will become clear if either you read the article referenced, or if I get around to posting _about_ the article, vs about a tiny bit of its content.
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Gödel’s theorems do not relate to the unprovability of a formal system’s axioms. The observation that you have to have certain statements that define the system you are reasoning about is not particularly profound and hardly merits a theorem stating it. The thing to bear in mind is that in any interesting formal system these axioms are just definitions of the concepts we will be reasoning about. Thus, the Peano axioms give you things like the reflexive, symmetric, and transitive properties that define what it means for two numbers to be “equal”.
Gödel’s First Incompleteness Theorem, on the other hand states that for any sufficiently powerful formal system (i.e., powerful enough to describe the standard laws of arithmetic) there are statements that are true within that system, but which are not provable as theorems. There is also a Second Incompleteness Theorem that states that any formal system that can prove its own consistency is perforce inconsistent. These theorems place fundamental limits on the power of formal systems, inasmuch as they assert that no formal system can capture every true statement about its domain of applicability.
If you really want to learn about the Incompleteness Theorems, you can hardly do better than Douglas Hofstadter’s book “Gödel, Escher, Bach”. It has excellent descriptions of the theorems and their proofs and is very accessible to non-mathematicians.
Hope that helps.