Ryll-Nardzewski theorem

This article deals with the Ryll-Nardzewski theorem from model theory, for the Ryll-Nardzewski theorem from functional analysis see Ryll-Nardzewski's Fixed Point Theorem .

The omega-categorical theory is a set of the model theory , a branch of mathematical logic. He characterizes - categorical theories . It is named after the Polish mathematician Czesław Ryll-Nardzewski . ${\displaystyle \omega }$

Ryll-Nardzewski theorem

Be a complete theory about a countable language . The space of complete types is designated with. ${\displaystyle {\mathcal {T}}}$ ${\displaystyle S_{n}(T)}$ ${\displaystyle n}$

Then it is equivalent:

• ${\displaystyle {\mathcal {T}}}$is -categorical. ${\displaystyle \omega }$
• ${\displaystyle S_{n}(T)}$is finite for everyone . ${\displaystyle n}$
• Except for equivalence, there are only finitely many formulas for each ${\displaystyle {\mathcal {T}}}$ ${\displaystyle n}$ ${\displaystyle \phi (x_{1},\ldots ,x_{n}).}$

Further equivalences

Under the same conditions as in Ryll-Nardzewski's theorem, it holds that it is equivalent:

• ${\displaystyle {\mathcal {T}}}$is -categorical. ${\displaystyle \omega }$
• Every countable model of is saturated . ${\displaystyle {\mathcal {T}}}$

Examples

Dense linear order without end points

Let be a model of the theory of dense linear order without endpoints and ${\displaystyle M}$

${\displaystyle A=\{a_{1},\ldots a_{n}\}\subset M}$

and without loss of generality

${\displaystyle i<j\Rightarrow a_{i}<a_{j}}$

A full type over is either given by a formula of the form: ${\displaystyle A}$

${\displaystyle \phi _{j}(x):=x=a_{j}\ (1\leq j\leq n)}$

or the shape

${\displaystyle \phi _{i,j}(x):=a_{i}<x<a_{j}\ (1\leq i<j\leq n)}$

generated. This can be proven by quantifier elimination .

The set of types is finite, so the theory is -categorical. ${\displaystyle \omega }$

Theory with an infinite number of constant symbols

The theory of the language with the axioms has countably many complete 1-types: the types produced by the formula are the isolated types, the type produced by the set is the only non-isolated type. The theory is therefore non- categorical. (But it is -categorical.) ${\displaystyle {\mathcal {T}}}$ ${\displaystyle \{c_{i}\mid i<\omega \}}$ ${\displaystyle \{c_{i}\neq c_{j}\mid i<j\}}$ ${\displaystyle \phi _{i}:=x=c_{i}}$ ${\displaystyle \{x\neq c_{i}\mid i<\omega \}}$ ${\displaystyle \omega }$ ${\displaystyle \omega _{1}}$

Web links

• Martin Ziegler: Script Model Theory 1 . (PDF; 649 kB)

literature

• Wilfrid Hodges : Model theory. Cambridge University Press, 1993, ISBN 0-521-30442-3 .
• Chang, Chen C., Keisler, H. Jerome: Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. ISBN 0-444-88054-2
• Philipp Rothmaler: Introduction to Model Theory. Spektrum Akademischer Verlag, 1995, ISBN 978-3-86025-461-5 .