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Topic: Thinning of theories?
Replies: 11   Last Post: Apr 15, 2013 11:28 PM

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Posts: 92
Registered: 6/21/12
Re: Thinning of theories?
Posted: Apr 15, 2013 11:28 PM
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On Monday, April 15, 2013 5:16:34 PM UTC-4, zuhair wrote:
> Define (thinned to): Theory T is thinned to theory D iff D is a proper
> sub-theory of T and D interpret T.
> Can a theory be thinned infinitely?
> I mean can we have an infinite sequence of theories T1,T2,T3,.... such
> that each Ti is thinned to Ti+1?
> Zuhair

Mycielski proved that theories form a lattice under an appropriate notion of interpretation (see "A Lattice of Interpretability Types of Theories" in the Journal of Symbolic Logic 1977). Your question (if I understand it correctly -- which is a big if) involves the existence or nonexistence of atoms in this lattice. I don't know the answer to that question, but Walter Taylor asked a similar question about the lattice of interpretability types of varieties (= equational theories). McKenzie and Swierczkowski proved that atoms don't exist in this lattice ("Non-covering in the interpretability lattice of equational theories", Algebra Universalis, 1993), so if you stick to algebraic theories the answer is yes -- there are algebraic theories that can be "thinned" to other algebraic theories indefinitely. This perhaps isn't exactly what you were asking for but might give you a few ideas.


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