FOM: Simplicity, Calculus, Need for new axioms

David Ross ross at
Fri Nov 14 12:23:18 EST 1997

I should emphasize that this result *does* have a simple nonstandard proof
(sans epsilonics), but that the nonstandard machinery used is marginally
deeper than for easier results (e.g., a continuous function of a continuous
function is continuous) and so is an interesting first test of the utility of
'more concrete' infinitesimals.

- David (ross at

(J Shipman wrote:
> 2) Thanks to David Ross for his example.  OK, nonstandard analysts: is the
> theorem "continuous function attains a max on closed interval" true in any
> concrete models of infinitesimals such as the surreals or the surreals up to
> ordinal stage alpha for some infinite alpha?  If so, great!  If not, how hard
> is it to prove that it still holds for the subfield of standard reals without
> simply duplicating the epsilonics of standard analysis?

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