```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 specifiedfirst-order language, and the homomorphisms have to respect thedenotation of the ground terms (maybe there's a special case if thereare no constants). The set of ground terms can be taken as the domainof 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
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