4 ms·

There is no foundation for Mathematics [1] :-) [1]: http://en.wikipedia.org/wiki/Godel%27s_incompleteness_theorem http://en.wikipedia.org/wiki/Godel%27s_incomp

There is no foundation for Mathematics [1] :-) [1]: http://en.wikipedia.org/wiki/Godel%27s_incompleteness_theorem http://en.wikipedia.org/wiki/Godel%27s_incompleteness_theore...

praptak 15y ago
Goedel's theorem is not about nonexistance of foundation in mathematics, it's about existence of true but nonproveable statements in every nontrivial formal system. One could imagine a hypothetical stronger result - maybe every nontrivial set of axioms can actually derive p^(not p)?
- singular 15y ago
  IANAM, but I do know it was an unassailable problem in Russell + White's efforts to found mathematics on a firm basis in Principia Mathematica [1]. [1]:http://en.wikipedia.org/wiki/Principia_mathematica#Consistency_and_criticisms http://en.wikipedia.org/wiki/Principia_mathematica#Consisten...
  - oelewapperke 15y ago
    You want to actually understand mathematics, here's what to do. Reading any theory, any theory at all. Look at the author. Do a wikipedia search. If the article starts with what a great philosopher they are, burn the article, wash your hands in holy water and beg for forgiveness. Their articles and books do not actually have a basis for their consistency except "LGTM", they're less than worthless, and can easily convince you that idiocies are real if you're not careful. And if you're careful you'll find a massive unproven leap in logic in the first sentence of the first page and you'll be unable to continue reading. Avoid philosophers like the plague. Including Russell, including Principia mathematica. Rip them to shreds if you find them in a bookstore. You'll be doing the world a big favor.
- yequalsx 15y ago
  Provided the axiom set is recursively enumerable. The second order Peano axioms for the natural numbers are complete.
  - oelewapperke 15y ago
    Euhm ... sorry to doubt you here, but it does seem like you're flat-out wrong. Second-order peano arithmetic contains all the axioms you'd need for Godel's proof ... And frankly, though not 100% proved (yet?), it seems to me very clear that any numbering and arithmetic system that contains any form of infinity is doomed to be incomplete. So the only useful arithmetical fields that escape Godel are Zn.

There is no foundation for Mathematics [1] :-) [1]: http://en.wikipedia.org/wiki/Godel%27s_incompleteness_theorem http://en.wikipedia.org/wiki/Godel%27s_incompleteness_theore...

praptak 15y ago

Goedel's theorem is not about nonexistance of foundation in mathematics, it's about existence of true but nonproveable statements in every nontrivial formal system. One could imagine a hypothetical stronger result - maybe every nontrivial set of axioms can actually derive p^(not p)?

singular 15y ago
IANAM, but I do know it was an unassailable problem in Russell + White's efforts to found mathematics on a firm basis in Principia Mathematica [1]. [1]:http://en.wikipedia.org/wiki/Principia_mathematica#Consistency_and_criticisms http://en.wikipedia.org/wiki/Principia_mathematica#Consisten...
- oelewapperke 15y ago
  You want to actually understand mathematics, here's what to do. Reading any theory, any theory at all. Look at the author. Do a wikipedia search. If the article starts with what a great philosopher they are, burn the article, wash your hands in holy water and beg for forgiveness. Their articles and books do not actually have a basis for their consistency except "LGTM", they're less than worthless, and can easily convince you that idiocies are real if you're not careful. And if you're careful you'll find a massive unproven leap in logic in the first sentence of the first page and you'll be unable to continue reading. Avoid philosophers like the plague. Including Russell, including Principia mathematica. Rip them to shreds if you find them in a bookstore. You'll be doing the world a big favor.
yequalsx 15y ago
Provided the axiom set is recursively enumerable. The second order Peano axioms for the natural numbers are complete.
- oelewapperke 15y ago
  Euhm ... sorry to doubt you here, but it does seem like you're flat-out wrong. Second-order peano arithmetic contains all the axioms you'd need for Godel's proof ... And frankly, though not 100% proved (yet?), it seems to me very clear that any numbering and arithmetic system that contains any form of infinity is doomed to be incomplete. So the only useful arithmetical fields that escape Godel are Zn.