# reciprocal of infinitesimals in nonstandard analysis

• Jun 15th 2011, 10:05 PM
KOSS
reciprocal of infinitesimals in nonstandard analysis
Had an issue with a post on physicsforums. Some dude was wondering about Zeno's paradoxes and brought up infinitesimals. Quoting a dud McLaughlin from New.Sci. 1994, page 84. He thought the reciprocal of some infinitesimal hyperreals are "unlimited" but finite. From the article:
Quote:

The third type of nonstandard number is simply the inverse of an infinites imal. Because an infinitesimal is very small, its inverse will be very large (in the standard realm, the inverse of one millionth is one million). This type of nonstandard number is called an unlimited number. The unlimited numbers, though large, are finite and hence smaller than the truly infinite numbers created in mathematics. These unlimited numbers live in a kind of twilight zone between the familiar standard numbers, which are finite, and the infinite ones.
The reference was to Nelson's work on Internal Set Theory, which I do not know much about, other than that is an attempt to axiomatize some of Robinson's nonstandard analysis.

In NSA I believe an infinitesimal is infinite when reciprocated. But what about in Nelson's scheme? Anyone know? Is it even a sensible question?

One respondent posted the idea that there are a class of finite unlimited numbers sandwiched between the finite natural numbers and the actual infinite. Is that the case with Internal Set theory, is it also and allowable class of number in conventional NSA?

One reason I ask is that I would naively think that if an infinitesimal does not yield an infinite number when reciprocated, then it seems likely a contradiction would arise somewhere down the road. Can anyone enlighten me?

I've never heard of these twilight numbers before. Are they anything like some of the surreals Conway invented, or quantities that might be formally identified with $\displaystyle \omega-1, \omega-2$, ,etc.?