[FOM] pathology

[Frank Waaldijk]

Martin wrote:

It is certainly worthwhile to study mathematical entities whose existence
> can be proved but for which it can be proved that explicit examples (using
> acceptable mathematical language) can not be given.


There is a somewhat analogous situation in intuitionistic / constructive
mathematics with regard to the existence of non-recursive functions from N
to N (or non-recursive real numbers, whichever you prefer).

The axioms of INT (for instance formalized by Kleene in his book
Foundations of Intuitionistic Mathematics, in an axiom scheme usually
called FIM) prove that not all real numbers can be recursive. (Since this
would contradict the Fan Theorem FT, which ensures compactness of the unit
interval).

But FIM also has the property that any FIM-definable real number is
recursive (I believe this is called the Church-Kleene rule) so it can be
proved that an explicit example of a non-recursive real number cannot be
given in FIM.

(Notably Kleene, Kreisel, Troelstra and Joan Moschovakis worked on these
results, please consult their work for the details, which are above my
head).

In RUSS, the branch of recursive (constructive) mathematics, there are only
recursive sequences, but compactness fails. This failure of compactness is
widely considered undesirable, and efforts have been made (e.g. in formal
topology) to restore compactness without having to accept FT. This in my
eyes indicates a common constructive `belief' in non-recursive sequences.
So even in constructive mathematics it would seem that
non-explicitly-definable entities play an important role.

(My own doubts on this subject can be found in my papers.)

Best wishes,
Frank Waaldijk

http://www.fwaaldijk.nl/mathematics.html
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