CHAT: math

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
http://perso.wanadoo.fr/naidon/pascal/science/nonstandE.html .
NSA home page at http://members.tripod.com/PhilipApps/nonstandard.html .

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                                   cowan@ccil.org
One art/there is/no less/no more/All things/to do/with sparks/galore
        --Douglas Hofstadter