# Hereditarily

### Some articles on hereditarily:

Hereditarily Countable Set
... In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets ... A set is hereditarily countable if and only if it is countable, and every element of its transitive closure is countable ... If the axiom of countable choice holds, then a set is hereditarily countable if and only if its transitive closure is countable ...
Finite Set - Foundational Issues
... its relative consistency the universe of hereditarily finite sets constitutes a model of Zermelo–Fraenkel set theory with the Axiom of Infinity replaced by its negation ... One can interpret the theory of hereditarily finite sets within Peano arithmetic (and certainly also vice-versa), so the incompleteness of the theory of Peano arithmetic implies that of the ... A seeming paradox, non-standard models of the theory of hereditarily finite sets contain infinite sets --- but these infinite sets look finite from within the model ...
Hereditarily Finite Set - Formal Definition
... A recursive definition of a hereditarily finite set goes as follows Base case The empty set is a hereditarily finite set ... Recursion rule If a1...ak are hereditarily finite, then so is {a1...ak} ... The set of all hereditarily finite sets is denoted Vω ...
Hereditary Property - In Set Theory
... A couple of notions are defined analogously A hereditarily finite set is defined as a finite set consisting of zero or more hereditarily finite sets ... Equivalently, a set is hereditarily finite if and only if its transitive closure is finite ... A hereditarily countable set is a countable set of hereditarily countable sets ...
Hereditarily Finite Set - Discussion
... The hereditarily finite sets are a subclass of the Von Neumann universe ... Notice that there are countably many hereditarily finite sets, since Vn is finite for any finite n (its cardinality is n−12, see tetration), and the union of countably many finite sets is countable ... Equivalently, a set is hereditarily finite if and only if its transitive closure is finite ...