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Re: Thinning of theories?
Posted:
Apr 15, 2013 11:28 PM


On Monday, April 15, 2013 5:16:34 PM UTC4, zuhair wrote: > Define (thinned to): Theory T is thinned to theory D iff D is a proper > > subtheory 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 ("Noncovering 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.
scattered



