| MadSci Network: Other |
Goedel's Incompleteness Theorems refer to a group of theorems which showed that there exist statements in some logical systems that can neither be logically arrived at nor disproven (their inverse logically arrived at) from the fundamental axioms of the system. The first of these theorems was the result of Goedel's attempting to show that classical set theory could be completely proven, just as he had shown that the predicate calculus could be completely proven a couple of years earlier. But in fact, there are a whole host of logical systems which are plagued by this logical incomplet- eness or 'undecideability'. There were those who proposed that Fermat's Last Theorem might be among the 'incomplete set' for number theory. But obviously, they could never demonstrate this. :)
For a more complete reference on the logical underpinnings of mathematics, Russell's _Principia Mathematica_, and the long story of logic and math, see the URL:
There's also an archive of a discussion mailing list on Russell, which has many more detailed references to Goedel and the great logic debate. This archive is somewhat hard to read, but the discussion mailing list itself might be of interest if you're concerned with this history further. The archive has a search engine at:
See for example: http://www.mcmaster.ca/russdocs/dig95jun.27
Last but not least, you might try the Dr. Math service mentioned on the Mad Scientists' Home Page. Thanks for the question!
Lew Gramer
Try the links in the MadSci Library for more information on Other.