FOM: non-uniqueness of the hyperreals
Vaughan R. Pratt
pratt at cs.Stanford.EDU
Wed Nov 12 10:39:08 EST 1997
>From: Jon Barwise <barwise at phil.indiana.edu>
>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?
>I would suggest that the answer to the first two questions is "Yes". The
>core concpet is just the one I mentioned above. But there is a richer
>concept, corresponding to countable saturation. The richer concept is more
>useful than the core concept alone, as various applications have shown.
>See, e.g. Keisler's AMS Memoir. Perhaps there is no final unique concept
>here, but a family of increasingly rich notions.
Mulling over Jon's reasonable concern, and wondering why there should
be no canonical notion of hyperreal, it occurred to me that polynomials
with real coefficients, more precisely formal power series (since
1/(1+x) must be infinite: 1 - x + x^2 - x^3 + ...) which are allowed to
begin at a negative power of x (since 1/x must be x^{-1}), ought to be
able to serve as the canonical model Jon is looking for.
These form a field, which one orders by ordering the sequences of
coefficients lexicographically. The standard reals are the constants,
those whose only nonzero coefficient if any is that of x^0. The finite
reals are those with no negative powers of x and at least one positive
power of x; the lexicographic ordering interleaves these between the
standard reals. The infinite reals are those with a negative power of
x, which the lexicographic ordering places at oo or -oo according to
the sign of the leading coefficient.
Since this is so absurdly simple it is clearly wrong, since NSA is well
known not to be that easy. Obviously I'm just overlooking some bug in
this proposal, where's the bug?
If it weren't for that bug I'd say this was reasonably canonical.
Vaughan
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