Ehrenfeucht–Mostowski theorem
Encyclopedia
In model theory
, a field within mathematical logic
, the Ehrenfeucht–Mostowski theorem gives conditions for the existence of a model with indiscernibles
.
The Ehrenfeucht–Mostowski theorem states that
if T is a theory with an infinite model, then there is a model of T containing any given linearly ordered set X as a set of indiscernibles.
The proof uses Ramsey's theorem
.
to construct indiscernibles in the constructible universe
.
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, a field within mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, the Ehrenfeucht–Mostowski theorem gives conditions for the existence of a model with indiscernibles
Indiscernibles
In mathematical logic, indiscernibles are objects which cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas are considered...
.
Statement
A linearly ordered set X is called a set of indiscernibles of a model if the truth of a statement about elements of X depends only on their order.The Ehrenfeucht–Mostowski theorem states that
if T is a theory with an infinite model, then there is a model of T containing any given linearly ordered set X as a set of indiscernibles.
The proof uses Ramsey's theorem
Ramsey's theorem
In combinatorics, Ramsey's theorem states that in any colouring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs...
.
Applications
The Ehrenfeucht–Mostowski is used to construct models with many automorphisms. It is also used in the theory of zero sharpZero sharp
In the mathematical discipline of set theory, 0# is the set of true formulas about indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers , or as a subset of the hereditarily finite sets, or as a real number...
to construct indiscernibles in the constructible universe
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
.