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Re: Initial Model Theorem
Posted:
Feb 14, 2008 6:18 AM
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On Wed, 13 Feb 2008 17:59:53 +0000, Alan Smaill
<smaill@SPAMinf.ed.ac.uk> wrote:
>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.
Ok, if that's the idea...
>> 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 ...
Aargh. I meant to change the "homomorphism from A to B" to
"homomorphism from B to A".
>> 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
David C. Ullrich
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