FOM: Barwise should be moved

[Harvey Friedman]

This is a very brief response to Barwise, 11/30/97, 11:49.

The analogy with i and -i and the complex numbers is entirely
inappropriate. The complex numbers also comes with natural additional
structure - real and imaginary parts - and with this natural additional
structure i becomes definable. Moreover, this additional structure is not
an artificial construct by me in order to argue with Barwise. You can't do
serious complex analysis without this additional structure.

The fact that the hperreals are an elementary extension of the reals is not
nearly sufficient to conclude that you can't define an infinitesmial. It
only implies that you can't define an infinitesimal in the hyperreals with
the usual structure. The undefinability results and conjectures of mine on
the FOM concern the deeper matter of unrestricted set theoretic
definability. (See my postings of 11/16-11/18/97).

This Barwise posting suggests yet another new twist to the investigations
that I discussed on the FOM. Namely, *can we show that there is no natural
additional structure that allows one to define an infinitesimal?* Of
course, this may be a tricky question to formalize.

As far as students are concerned, I continue to maintain that they always
want examples and should want examples, and 1/N is not an example. This is
especially true of undergraduate computer science students and engineers.

Now Barwise is such a likeable guy, and presumably such a splendid teacher,
with such charisma, that he no doubt can get around this problem of no
example. However, what about the rest of us??

Rather than model the paradox stuff by some technical theory of
infinitesimals, one is much better off considering formal systems which
formalize intuitive reasoning about "largeness" and the like, rather than
to pretend that there is some objective reality of infinitesimals. Or
pretend that this is in some sense comparable to the fundamental notions of
f.o.m. such as set theory or computability. Jon - please stop pretending.

One proves conservative extension results, and some other things I am
thinking about. This is an interesting and fruitful common ground that we
can both stand on. I have a number of positive ideas about this, including
ones that I haven't yet posted. And I was anxious to hear more from Rick
Sommer!

As far as using this to replace standard analysis in teaching, I am
obviously with the silent majority.

To the FOM: sorry for being away so long from the FOM. I've been cooking up
some goodies, and will be back shortly. Miss me?