FOM: More questions about infinitesimals

Vaughan R. Pratt pratt at cs.Stanford.EDU
Thu Nov 13 01:18:53 EST 1997

Originally I'd planned to sit out the NSA discussion as not something I
could usefully contribute to.  But Jon Barwise's question

>Indeed, Robinson gives us a whole family of non-isomorphic models of it.
>That is the rub.  What are we to make of this multiplicity of models? Do
>they correspond to different concpetions?  If so, are some conceptions of
>infinitesimal better than others?  How so?

had a certain appeal that lured me in, and somehow (aphasia? lack of
sleep?) I wound up inappropriately contradicting Martin in an area he
knows more about than I, for which my apologies.

Nevertheless it was worthwhile (for me anyway) because, after all my
stumbling around in the dark, I got out of it the sense that language
is a crucial parameter for Jon's question, a point that had not
previously sunk in on my few brief encounters with NSA.

A language that contains all functions seems so wildly nonconstructive
that any constructive notion of infinitesimal would appear out of the

At the other extreme, if an ordered field will do then the field of
Laurent series offers a constructive notion of infinitesimal, easily
seen.  And I was very glad to learn from Dave Marker about the field of
Puisex series with its square roots, less easily seen.

This suggests the following cousins of Jon Barwise's question.

(i) What functions on R are compatible with a suitably constructive
notion of infinitesimal?

(ii) The same with "compatible" replaced by "incompatible".  In
particular, what useful functions are incompatible with constructive
notions of infinitesimal?

(iii) What other issues besides selection of functions are relevant to
the constructive definition of infinitesimals?


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