# [Agda] Classical Mathematics for a Constructive World

Altenkirch Thorsten psztxa at exmail.nottingham.ac.uk
Wed Nov 28 10:57:13 CET 2012

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On 28/11/2012 08:52, "Peter Hancock" <hancock at spamcop.net> wrote:

>Now, my question, such as it was.  Something I got from rereading an
>ancient
>book by Troelstra, is essentially this:
>
>   SCT + EXT2 + AC are jointly inconsistent over HAomega
>
>EXT2 is extensionality at type (N->N)->N.
>I also saw something like this in Beeson's book.  Troelstra indicates that
>from these 3 ingredients one could solve the halting problem.  He mentions
>that a proof can also be constructed from the Kreisel-Lacombe-Shoenfield
>theorem (from 1959).
>I was surprised that SCT is so hostile, even to extensionality.
>Univalence apparantly
>implies EXT2 inter alia, and is compatible with AC.

Isn't this just the observation that from these 3 assumptions we can
decide extensional equality of functions f : N -> N because using SCT + AC
we get a function code : (N -> N) -> N  such that code f = code g <-> f =
g?

>Similar things can be formulated for continuity at type 2, where the
>thing that is said to "exist"
>is roughly speaking a well-founded countably-branching tree that
>represents the type2 functional
>a la Ghani/Hancock/Pattinson or Escardo.

I agree that continuity is similar to CT because it reflects the fact that
we talking about computable functions only. One may think that they are
inconsistent because the FAN principle and CT are jointly inconsistent
(due to the construction of the Kleene tree). However, does this also work
when we have WCT or WhCT? Indeed CT corresponds to the ability to observe
directly the well-founded tree corresponding to a computation of type 2
which is also inconsistent with Ext.

Thorsten

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