FOM: Natural examples/evolution of subjects

Harvey Friedman friedman at
Mon Aug 2 08:41:00 EDT 1999

Reponse to Shipman 1:17PM 8/2/99.

>Thanks, Harvey, for two very thought-provoking posts.  An observation
>and a comment:
>1) It seems a little unfair to restrict your domain for "naturalness" to
>sets of integers so that you can say that there are no naturally
>occuring examples of r.e. sets that are not recursive.  There are
>certainly some naturally occurring sets of FINITE OBJECTS which are r.e.
>but not recursive (integer polynomials with a zero, piecewise linear
>spaces homeomorphic to the 5-sphere, presentations of the trivial
>group).  It's bad enough that there is essentially only ONE canonical

I went on to mention integer polynomials with a zero. I am aware of these
other contexts too. The point is that if you are looking for natural
nonrecursive sets, you do seem to have to go to somewhat complicated spaces
- far more complicated than the space of integers - and this is quite
curious. Also, I think that the problem of finding a natural nonrecursive
set of integers is interesting.

>2) I agree that the four traditional branches of mathematical logic,
>considered as individual subjects, are not at stage six yet; but can you
>provide an example of a mathematical field which is?  (Even better would
>be two examples, one where the stage six status is deserved and one
>where it is undeserved.)
Without commenting on deserved/undeserved, there is general topology and
(general) category theory.

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