Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/26/05 

Date: 25 Jan 2005 16:13:14 -0800

Check this out, from searching for "infinitesimal measure", there are
many results, one is this paper from the Bulletin of Symbolic Logic, H.
Enderton et al. eds., Erik Palmgren's "Developments in Constructive
Nonstandard Analysis."

http://66.102.7.104/search?q=cache:aMxxjC7fi6sJ:www.math.ucla.edu/~asl/bsl/0403/0403-001.ps+%22infinitesimal+measure%22+theory&hl=en&ie=UTF-8

Palmgren says Schmieden and Laugwitz might have been already
considering the nonstandard reals as a partially ordered ring, with a
"rather restricted transfer principle", before Robinson and his
hyperreals a decade or so. Then he goes on to say he's a fan of Bishop
and Cheng. Ready results lead to Peter Zahn and Palmgren as
contemporary nonstandard constructivists, among the many nonstandard
constructivists.

http://mathworld.wolfram.com/TransferPrinciple.html

http://www.mathematik.uni-muenchen.de/~antipode/abstracts.html

There is lots of stuff about nonstandard analysis and infinitesimals,
greater than zero and thus between zero and one. Compare to "Continuum
Hypothesis". There's a lot in real analysis.

I guess the idea here is that measure theory doesn't differentiate
between empty/zero and countable/non-zero, calling them both zero.
Infinite sets are equivalent.

Regards,

Ross Finlayson

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