Mouse (set theory)
Encyclopedia
In set theory
, a mouse is a small model
of (a fragment of) Zermelo-Fraenkel set theory with desirable properties. The exact definition depends on the context. In most cases, there is a technical definition of "premouse" and an added condition of iterability (referring to the existence of wellfounded iterated ultrapower
s): a mouse is then an iterable premouse. The notion of mouse generalizes the concept of a level of Gödel
's constructible hierarchy
while being able to incorporate large cardinals.
Mice are important ingredients of the construction of core model
s. The concept was isolated by Ronald Jensen
in the 1970s and has been used since then in core model constructions of many authors.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, a mouse is a small model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
of (a fragment of) Zermelo-Fraenkel set theory with desirable properties. The exact definition depends on the context. In most cases, there is a technical definition of "premouse" and an added condition of iterability (referring to the existence of wellfounded iterated ultrapower
Ultraproduct
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature...
s): a mouse is then an iterable premouse. The notion of mouse generalizes the concept of a level of Gödel
Godel
Godel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*Gödel...
's constructible hierarchy
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"...
while being able to incorporate large cardinals.
Mice are important ingredients of the construction of core model
Core model
In set theory, the core model is a definable inner model of the universe of all sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a class of inner models that under the right set theoretic assumptions have very special...
s. The concept was isolated by Ronald Jensen
Ronald Jensen
Ronald Björn Jensen is an American mathematician active in Europe, primarily known for his work in mathematical logic and set theory.-Career:...
in the 1970s and has been used since then in core model constructions of many authors.