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 ...
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 ... Equivalently, a set is hereditarily finite if and only if its transitive closure is finite ...
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 ... 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 ... 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 ...