### 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

... 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

... 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 ...

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