Hi Bas,
thank you - indeed this is a very interesting observation. This was
the proof I had in mind when I first claimed that the axiom of choice
is provable in OTT. However, I was deluded in believing that OTT/
predicative topoi exactly characterize the theory of the setoid
model. As you clearly point out this is not the case. To me this is
an incompleteness of the formal system (OTT/predicative topoi) wrt
the intended interpretation. This should be fixable, leading to OTT+X
or predicative topoi + X which should be complete wrt the setoid
model. More categorically we could say that X characterizes the
predicative topoi obtained by an exact completion of an LCCC - I
think.
First of all, I am not sure whether a "fix" is needed. My recent
experience in
constructive mathematics is that it is actually quite pleasant to work
without countable choice. Initial experience shows that it may lead
to better
algorithms implicit in the proofs. (Most of this is in my work with
Thierry
Coquand.) As you know, choice breaks abstract datatypes. See for
instance the
encoding of ADT as existential types by Mitchell and Plotkin. This
is not
possible in dependent type theory precisely because we have choice
(this was
pointed out to me by Jesper Carlstrom some time ago). I understand one
motivation for your work on OTT as an attempt to bring back (some?)
ADTs in
type theory. It would be a pity to throw them out again without a good
reason.
I'd like to understand this better - in the moment I am unconvinced.
Instead of "choice", I'd like
f : Pi a:A.(B a)/(~ a)
-------------------------
lift f: (Pi a:A.B a)/~'
for "discrete" types A, which means that I work in the internal
language of the setoid model. Quotient types, even with lift, seem to
be a good way to capture ADTs.
Also if you want to reject lift, it would be good to have a model
construction (maybe like the setoid model) which refutes it. Other
things which you want may become true in this model.
Having said that here are some quick remarks on your proposal below.
Having choice for N->N as somewhat uncommon. It is not present in
Bishop's
constructive mathematics, i.e. more or less the setoid model of ML
type
theory. However, it is present in Brouwerian intuitionism and in some
realizability models. Brouwer does not have choice for (N->N)->N.
Martin Hofmann has some discussion on choice and extensionality in
his thesis,
but uses a syntactic criterion somewhat like the one that you outline.
I alreay realized that I should reread Martin's thesis.
One semantic candidate for X could be `all types have a projective
cover'.
This is what is used in realizability theory and what Erik Palmgren
translated to type theory. A type P is projective if for all f:A->B
and every
function g:P->B there is a function h:P->A with f o h = g. Projective
types "have choice". The idea is that the projective types are
Bishop's
pre-sets or the types in the setoid construction.
I'll think about this.
Cheers,
Thorsten
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