Just to answer the question-mark in (2.1), I mechanised a proof that
new_type_definition is model-theoretically conservative for standard models
and arbitrary base theories. To be clear, the proof assumes the base theory
contains "fun", "bool", and "=", and that it has a model that interprets
these types/constants in the standard way. I presume this counts as
"arbitrary", but maybe it doesn't :) (The proof is at
https://github.com/CakeML/cakeml/blob/master/candle/standard/semantics/
holExtensionScript.sml#L366.)
On 25 October 2016 at 09:06, Andrei Popescu <a.pope...@mdx.ac.uk> wrote:
> The counterexample I had in mind is due to Makarius Wenzel (
> https://www4.in.tum.de/~wenzelm/papers/axclass-TPHOLs97.pdf, page 8): The
> theory T containing the single HOL formula "no type has cardinal 3" has a
> Henkin model M; yet, M has no expansion to the theory T extended with the
> definition of the type {1,2,3}. But actually this extension is not
> proof-theoretically conservative either (as it even breaks consistency) ...
>
>
> In fact, now I see that I have not clearly spelled out all the assumptions
> of the statements in my summary. So let me try again, also factoring in
> the base (i.e., to-be-extended) theory:
>
>
> (1) The constant definition mechanisms (including the more general ones)
> are known to be:
> (1.1) model-theoretic conservative w.r.t. standard (Pitts) models and
> arbitrary base theories
> (1.2) model-theoretic conservative w.r.t. Henkin models and arbitrary
> base theories
> (1.3) proof-theoretic conservative and arbitrary base theories
>
> (2) The type definition mechanism is known to be:
> (2.1) model-theoretic conservative w.r.t. standard models and
> arbitrary(?) base theories
>
> and known *not* to be:
> (2.2) model-theoretic conservative w.r.t. Henkin models and arbitrary
> base theories
> (2.3) proof-theoretic conservative w.r.t. Henkin models and arbitrary
> base theories
>
> On the other hand, it is of course legitimate to lower the expectation for
> typedefs, so we could ask what happens with (2.2) and (2.3) if we restrict
> to base theories that are themselves definitional. Here, the above
> counterexample does not work. And yes, Rob, without being able to follow
> your Heyting arithmetic analogy, I do see the similarity between a possible
> semantic proof of definitional-base-(2.2) and a possible syntactic proof
> of definitional-base-(2.3) (both revolving around the notion of
> relativization to sets).
>
> But I am surprised that a lot of attention has been given to
> the conservativity of constant definitions/specifications, but not to that
> of the old and venerable typedef.
>
> Best,
> Andrei
>
>
>
>
>
>
> ------------------------------
> *From:* Rob Arthan <r...@lemma-one.com>
> *Sent:* 24 October 2016 21:37
> *To:* Ondřej Kunčar
> *Cc:* Andrei Popescu; Prof. Andrew M. Pitts; Prof. Thomas F. Melham;
> cl-isabelle-us...@lists.cam.ac.uk; Roger Bishop Jones; Prof. Peter B.
> Andrews; HOL-info list
> *Subject:* Re: conservativity of HOL constant and type definitions
>
> Ondrej,
>
> > On 24 Oct 2016, at 20:32, Ondřej Kunčar <kun...@in.tum.de> wrote:
> >
> > On 10/24/2016 09:16 PM, Rob Arthan wrote:
> >> I am pretty sure nothing has been published and, if you are right about
> (2.2),
> >> then I don't think type definitions can be proof-theoretically
> conservative.
> I made that sound too strong: I was just making a conjecture: for "think"
> read "feel".
>
> > They could. You can try to argue by "unfolding" the type definitions.
>
> "Unfolding" of types is exactly what I had in mind when I mentioned
> the methods used in connect with Heyting arithmetic.
>
> > Again, the model-theoretic conservativity is stronger than the
> proof-theoretic in general. And here you don't have an existential
> quantifier for type constructors so you [can't] use the approach as you did
> for constants.
>
> Yes, but if the unfolding approach works, you would have reduced the
> essential properties of the type definition to a statement about the
> existence
> of a certain subset of the representation type bearing a relationship with
> some
> siubsets of the parameter types and you would then be able to deduce
> model-theoretic conservativeness. That's why I felt, that if Andrei is
> right that
> the type definition principle is not model-theoretically conservative
> w.r.t.
> Henkin models (his point (2.2)), then it won't be proof-theoretically
> conservative
> either, because the unfolding argument must break down somewhere.
> It would be very useful to see an example of a type definition that is not
> conservative w.r.t. Henkin models.
>
> Regards,
>
> Rob.
>
>
>
>
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