In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, a discipline within mathematics, an admissible set is a
transitive set
In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold:
* whenever x \in A, and y \in x, then y \in A.
* whenever x \in A, and x is not an urelement, then x is a subset of A.
Simil ...
such that
is a
model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
of
Kripke–Platek set theory
The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek.
The theory can be thought of as roughly the predicative part of ZFC and is considerably weaker than it.
Axioms
In its fo ...
(Barwise 1975).
The smallest example of an admissible set is the set of
hereditarily finite set
In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to t ...
s. Another example is the set of
hereditarily countable set In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. This inductive definition is well-founded and can be expressed in the language of first-order set theory. A set is hereditarily count ...
s.
See also
*
Admissible ordinal In set theory, an ordinal number ''α'' is an admissible ordinal if L''α'' is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, ''α'' is admissible when ''α'' is a limit ordinal and L''α'' ⊧ Σ0- ...
References
*
Barwise, Jon (1975). ''Admissible Sets and Structures: An Approach to Definability Theory'', Perspectives in Mathematical Logic, Volume 7, Springer-Verlag
Electronic versionon
Project Euclid Project Euclid is a collaborative partnership between Cornell University Library and Duke University Press which seeks to advance scholarly communication in theoretical and applied mathematics and statistics through partnerships with independent an ...
.
Set theory
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