Re: [HoTT] Bishop's work on type theory

[Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>]

On 05/05/2018, 05:27, "homotopyt...@googlegroups.com on behalf of
Michael Shulman" <homotopyt...@googlegroups.com on behalf of
shu...@sandiego.edu> wrote:

>3. He includes the axiom of choice (p12) formulated in terms of his
>(proof-irrelevant) propositions, as well as what seems to be a Hilbert
>choice operator (though it's not clear to me whether this applies in
>open contexts or not).  Since he has powerclasses with propositional
>extensionality, I think this means that Diaconescu's argument proves
>LEM, which he obviously wouldn't want.  It's harder for me to guess
>how this should be fixed, since without some kind of AC, setoids don't
>satisfy the principle of unique choice.

Why not? If we identify propositions with setoids that are internally
propositions (all elements are equal) and identify propositions upto
logical equality we get unique choice.

What do I miss here?
Thorsten


>




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