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 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 countable if and only if it is countable, and every element of its
transitive closure
In mathematics, the transitive closure of a binary relation on a set is the smallest relation on that contains and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite ...
is countable. If the
axiom of countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function ''A'' with domain N (where ...
holds, then a set is hereditarily countable if and only if its transitive closure is countable.
The
class of all hereditarily countable sets can be proven to be a set from the axioms of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
(ZF) without any form of the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, and this set is designated
. The hereditarily countable sets form a model 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 ...
with the
axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing th ...
(KPI), if the axiom of countable choice is assumed in the
metatheory
A metatheory or meta-theory is a theory whose subject matter is theory itself, aiming to describe existing theory in a systematic way. In mathematics and mathematical logic, a metatheory is a mathematical theory about another mathematical theory. ...
.
If
, then
.
More generally, a set is hereditarily of cardinality less than κ if it is of
cardinality less than κ, and all its elements are hereditarily of cardinality less than κ; the class of all such sets can also be proven to be a set from the axioms of ZF, and is designated
. If the axiom of choice holds and the cardinal κ is regular, then a set is hereditarily of cardinality less than κ if and only if its transitive closure is of cardinality less than κ.
See also
*
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 ...
*
Constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...
External links
"On Hereditarily Countable Sets"by
Thomas Jech
Thomas J. Jech ( cs, Tomáš Jech, ; born January 29, 1944 in Prague) is a mathematician specializing in set theory who was at Penn State for more than 25 years.
Life
He was educated at Charles University (his advisor was Petr Vopěnka) and from ...
Set theory
Large cardinals
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