type
Let be a first order language.Let be an http://planetmath.org/node/3384-structure.Let , and let .Then we define the type of over to be the set of -formulas
with parameters from so that .A collection
of -formulas is a complete
-type over iff it is of the above form for some and .
We call any consistent collection of formulas in variables with parameters from a partial -type over . (See criterion for consistency of sets of formulas.)
Note that a complete -type over is consistent so is in particular a partial type over . Also is maximal in the sense that for every formula over we have either or .In fact, for every collection of formulas in variables the following are equivalent:
- •
is the type of some sequence of elements over in some model
- •
is a maximal consistent set of formulas.
For we define to be the set of complete -types over .
Some authors define a collection of formulas to be a -type iff is a partial -type. Others define to be a type iff is a complete -type.
A type (resp. partial type/complete type) is any -type (resp. partial type/complete type) for some .
Title | type |
Canonical name | Type |
Date of creation | 2013-03-22 13:22:45 |
Last modified on | 2013-03-22 13:22:45 |
Owner | ratboy (4018) |
Last modified by | ratboy (4018) |
Numerical id | 6 |
Author | ratboy (4018) |
Entry type | Definition |
Classification | msc 03C07 |
Related topic | Formula |
Related topic | DefinableType |
Related topic | TermsAndFormulas |
Defines | type |
Defines | complete type |
Defines | partial type |