Talk:O-minimal theory

Page moved here from "o-minimality"

I think it's much easier to avoid name clashes by choosing adjective + "theory" or adjective + "structure", whatever is correct. If both notions exist, I would prefer adjective + "theory". If the other notion is really more general, we can split later. --Hans Adler (talk) 19:09, 23 November 2007 (UTC)[reply]

I rescued this from the old talk page:

Hello, I started this article some time ago.

I thought I would outline how I think the page could be productively expanded:

1. Explain the notion of "cells" and why they are nice.

2. Explain cellular decomposition of definable sets.

3. Talk about the nice properties of definable families in o-minimal structures.

4. Explain monotonicity of definable functions.

It may be a concern that some "supporting notions" for these ideas have scanty representation on wikipedia as yet, eg the notion of "definability". The present article on definability is simply wrong in some respects, and is generally unhelpful.

As much research seems to be happening at present in the theory of weakly o-minimal structures, it might be useful to interleave 1-4 with comments about the analogous results known to hold in the weakly o-minimal case.

Also, it would be nice if there were a page on weak o-minimality. :)

Hunter

O-minimality vs. existence of quantifier elimination

Right above the Contents there is a sentence "In other words, any set definable in M by an arbitrary formula is also definable via a quantifier free formula using only the ordering. " I understand this as "o-minimality implies quantifier elimination," which I think is not correct. O-minimality does not depend on the chosen language while quantifier elimination does. As an example consider $(R,\cdot ,<)$ , which is - such as $(R,0,1,+,-,\cdot ,<)$ - o-minimal. But $(R,\cdot ,<)$ does not admit quantifier elimination: for instance, $\forall y(x\cdot y=x)$ defines the set {0}, which cannot be defined without quantifiers.

Thomas-sturm (talk) 18:13, 16 June 2009 (UTC)[reply]

You are right, it's misleading. "In M" is a hint that we are talking about unary sets, i.e. subsets of M¹, not M², M³,..., and this hint is definitely too subtle. This article seems to be a bit neglected. I will have a look later today. --Hans Adler (talk) 18:38, 16 June 2009 (UTC)[reply]

Notice that my counterexample is about a subset of M¹, which is defined in terms of the only free variable x.--Thomas-sturm (talk) 06:01, 17 June 2009 (UTC)[reply]

Definable in the article means definable with parameters, another thing that's not emphasised in the lead. Your example is thus definable by x = 0, and any definable set in the domain is definable with the end points of the intervals and the individual points as parameters. Chenxlee (talk) 07:11, 17 June 2009 (UTC)[reply]

No, you cannot because I have removed 0 from the language. This is in fact essentially my point.-- Thomas-sturm (talk) 09:59, 17 June 2009 (UTC)[reply]

Parameters don't have to come from the interpretation of the language's constants, though, just some subset of Mⁿ. Chenxlee (talk) 10:20, 17 June 2009 (UTC)[reply]

Sorry, I didn't even look at your example. Chenxlee is right, "with parameters" means we add a constant to the language for every element of the model. --Hans Adler (talk) 12:47, 17 June 2009 (UTC)[reply]

With such a non-countable language you probably loose results like the positive semi-decidability of the (complete) first-order theory (or at least straightforward proofs for this). Anyway, I can imagine that this makes sense in the context of o-minimality. I would consider it important to point at the fact that the existence of constants for all elements of the universe is assumed. I come from a quantifier elimination background, and this is very much a game about picking the right language.-- Thomas-sturm (talk) 15:57, 17 June 2009 (UTC)[reply]

We don't actually change the language. It would make no sense to do that because we look at all models of the theory. (At least the proper model theorists do; there are "applied" model theorists who keep the model fixed.) We keep the original language fixed, but nevertheless work with "formulas" such as

\phi ({\bar {x}},{\bar {b}})

, where

{\bar {b}}\in M

is any tuple from M. An equivalent formulation of o-minimality without doing this: For every formula

\phi (x,{\bar {y}})

(one variable x, many

{\bar {y}}

) there is a formula

\psi (x,{\bar {z}})

such that

M\models \forall {\bar {y}}\exists {\bar {z}}\forall x(\phi (x,{\bar {y}})\leftrightarrow \psi (x,{\bar {z}}))

. --Hans Adler (talk) 18:03, 17 June 2009 (UTC)[reply]

I think the text is good now. Thanks for the constructive discussion. I think I have learned quite a bit.-- Thomas-sturm (talk) 06:12, 19 June 2009 (UTC)[reply]