# Re: [isabelle] conservativity of HOL constant and type definitions

```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 <rda at 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-users at 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 <kuncar at 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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