Talk:Type (model theory)

message: The example about \sqrt{2} confuses me. I have corrected the obvious problem: If x=\sqrt{2} (say, in reals) then for y smaller than -\sqrt 2 it is not true that y^2>2 \implies y>x (there is nothing about considering just positive reals in the article).
The definition of type that I know requests that if \phi(x) and \psi(x) belong to the type in a certain model, then also \exists x (\phi(x) \wedge \psi(x)) holds in the model (that is,  \phi \wedge \psi  is consistent with the model). Thus, if the example statements about \sqrt 2 are meant as a type in model of rationals then we would need examples of two conditions that are also met by some rational number. Or maybe what is meant by the example is a type in the theory of arithmetics, because being consistent with a model of a theory and being consistent with a theory are differnt things?
I am not an expert on this, so a second opinion would be helpful.82.208.2.227 15:23, 25 September 2006 (UTC)
thread_title:  Example of a type 

message: I agree that the article as it stands lacks sufficient context, and I've tried to add a better lede, but what I'm missing is _why_ types are significant, or some kind of analogy as to what they're _like_ -- for example it was explained to me
that the diagram of a structure is 'analogous' to the multiplication table for a group -- I cannot find any similar useful and correct analogy for types that would  provide motivation for why we study them.  The rest of the article does (I think) a good job explaining what types are and what you can do with them.  Help! Zero sharp (talk) 18:33, 15 August 2008 (UTC)
thread_title:  lede needs help 

message: There should also be a link/mention of saturation since that concept is
defined in terms of types Zero sharp (talk) 18:36, 15 August 2008 (UTC)
thread_title:  saturated structures 

message: Given a complete n-type p one can ask if there is a model of the theory that omits p, in other words there is no n-tuple in the model which realizes p. If p is an isolated point in the Stone space, i.e., if {p} is an open set. It is easy to see that every model realizes p (at least if the theory is complete). The omitting types theorem says that conversely if p is not isolated then there is a countable model omitting p (provided that the language is countable).
This is really unhelpful to me and I know at least basic model theory.
There needs to be an explanation of the relationship between topology and model theory - the link to the topological notion of an open point doesn't help me to understand how an element of a model's domain can be isolated. What exactly are the conditions under which a type is omitted?  —Preceding unsigned comment added by Dr satsuma (talk • contribs) 12:10, 12 March 2010 (UTC)
thread_title:  Omitting Types 

message: :In model theory and related areas of mathematics, a type is a set of first-order formulas in a language L with free variables x1, x2,…, xn which are true of a sequence of elements of an L-structure \mathcal{M}.
This line is hard to understand because it is complicated and imprecise.
The way it is stated, it can be interpreted as "a type is just a set of satisfiable formulas".
Is the "sequence of elements" fixed for a given type? Is a type a set of _all_ formulas that are true of a sequence?
Given paragraph Model theory#Types, I guess that the following is meant:
:Given a language L, an L-structure \mathcal{M} and a sequence b = (b_1, b_2, \dots, b_n) of elements of the universe of \mathcal{M}, the type of b is the set of all first-order formulas with free variables x_1, x_2, \dots, x_n which are satisfied by b_1, b_2, \dots, b_n.
Is this correct?
Anyway, wouldn't it make more sense to talk about the basic idea of types in the first sentence in order to build up some intuition before giving a formal definition? That would mitigate such imprecisions intrinsic to natural language.
-- 188.192.83.224 (talk) 06:13, 20 June 2012 (UTC)
thread_title:  First line is hard to understand and imprecise