FOM: Pratt's error on the hyperreals

[Martin Davis]

>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?
>

Non-Archimedean extensions of the reals R were well known long before
Robinson. His key contribution was such an extension R* for which the same
sentences that are true of R are true of R*.

Martin