Date: Feb 13, 2008 12:59 PM
Author: Alan Smaill
Subject: Re: Initial Model Theorem

David C. Ullrich <dullrich@sprynet.com> writes:

> (sci.logic added)
>
> On Tue, 12 Feb 2008 08:42:13 -0800 (PST), Tjark Weber
> <tjark.weber@gmx.de> wrote:
>
>>Hi,
>>
>>the Stanford Encyclopedia of Philosophy, in its entry on first-order
>>model theory (http://plato.stanford.edu/entries/modeltheory-fo/),
>>states the following "initial model theorem":
>>
>>"Let T be a theory consisting of strict universal Horn sentences. Then
>>T has a model A with the property that for every model B of T there is
>>a unique homomorphism from A to B. (Such a model A is called an
>>initial model of T. It is unique up to isomorphism.)"
>
> So the axioms of T are all universal quantifications of formulas of 
> the form
>
> P
>
> or
>
> (P_1 & ... & P_n) -> Q
>
> where P, P_j and Q are atomic.
>
> I must be missing something - the theorem as stated seems clearly
> false. Say the only axiom in T is Ax P(x). Then any map from
> any model of T to any other model of T is a homomorphis,
> and since T has models containing more than one element in
> the universe there cannot exist a model A such that for every
> model B there is a unique homomorphism from A to B
> (if B has more than one element then for every A there are
> at least two homomorphisms from A to B.)

Presumably the idea is that associated with T is a specified
first-order language, and the homomorphisms have to respect the
denotation of the ground terms (maybe there's a special case if there
are no constants). The set of ground terms can be taken as the domain
of the initial model.

> On the other hand, if we change the statement to
>
> "Let T be a theory consisting of strict universal Horn sentences. Then
> T has a model A with the property that for every model B of T there is
> a unique homomorphism from A to B. (Such a model A is called an
> initial model of T. It is unique up to isomorphism.)"

I can't see any change ...

> then the theorem seems trivial: Say A is the set of atomic formulas,
> and let X consist of all the mappings f : A -> {true, false} which are
> compatible with T (in what I suspect is the obvious sense - I can
> be more explicit if this is not clear). Then it seems clear that X 
> becomes a model of T with the required property, if for every P
> we let the interpretation of P be the set of all f such that f(P) =
> true.
>
> ???
>
>>Can anybody provide a reference (book/article) for this theorem and
>>its proof?
>>
>>Thanks in advance,
>>Tjark
>
> David C. Ullrich

-- 
Alan Smaill