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CHAT: math

From:John Cowan <cowan@...>
Date:Saturday, November 18, 2000, 23:56
On Sat, 18 Nov 2000, H. S. Teoh wrote:

> For example, between every two arbitrarily close "polynomial magnitudes", > which correspond with real numbers, there are sub-polynomial magnitudes > ("infinitesimals") that arise from logarithms. And between every two > arbitrarily close logarithmic magnitudes, there are sub-logarithmic > magnitudes ("sub-infinitesimals"?) that arise from iterated logarithms, > and so on, ad infinitum.
It sounds like you are just describing the (dense, continuous) real numbers themselves, which are by no means only polynomial (= algebraic).
> The same holds for the "large" magnitudes (which correspond with > "infinite" real numbers, if such things exist). Entire hierarchies of > these things arise from exponentials, tetration functions, etc.. > > > Wouldn't it be cool if there was a finite proof for G? Nobody actually > > knows if it's true -- but if it were, nonstandard numbers would be > > *hard-wired* into number theory, willy-nilly. > > Hmm, I'd like to find out more about these "non-standard" numbers -- any > web resources on this? :-)
Terse version (doesn't mention Goedel's Theorem) at . NSA home page at . But IMHO the best approach to GT and nonstandard stuff of every kind is Hofstadter's book _Goedel, Escher, Bach: An Eternal Golden Braid_.
> > "Almost all proofs have bugs, but almost all theorems are true." > > --A math/CS friend of mine > > !!! This is such a good quote, I'm gonna "steal" it for my signatures file > :-)
For the record, his name is Paul Pedersen. -- John Cowan One art/there is/no less/no more/All things/to do/with sparks/galore --Douglas Hofstadter