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.