Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Search Thread: Advanced Search

 Messages: [ Previous | Next ]
 scattered Posts: 92 Registered: 6/21/12
Re: Thinning of theories?
Posted: Apr 15, 2013 11:28 PM
 Plain Text Reply

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.

-scattered

Date Subject Author
4/15/13 Zaljohar@gmail.com
4/15/13 Butch Malahide
4/15/13 fom
4/15/13 Butch Malahide
4/15/13 fom
4/15/13 Zaljohar@gmail.com
4/15/13 Zaljohar@gmail.com
4/15/13 Butch Malahide
4/15/13 Zaljohar@gmail.com
4/15/13 fom
4/15/13 William Elliot
4/15/13 scattered

© The Math Forum at NCTM 1994-2018. All Rights Reserved.