# Naive Set Theory (from the book `"Naïve Set Theory"` by Paul R. Halmos, 1960) Axioms of naive set theory * axiom of extension * axiom of unions * axiom of specification * axiom of pairing * axiom of powers * axiom of infinity * axiom of substitution * axiom of choice The term "naïve" qualifies the presentation of set theory as being given informally, without rigorous compliance to a set of axioms. Axioms of set theory (by Paul Halmos) 1. *Axiom of extension*: two sets are equal iff they have the same elements 2. *Axiom of unions*: for every collection of sets there exists a set that contains all the elements that belong to at least one of them 3. *Axiom of specification*: to every set `S` and to every condition `P(a)`, there corresponds a set `A` whose elements are exactly those elements `a ∈ S` for which `P(a)` holds. 4. *Axiom of pairing*: for any two sets there is a set whom they both belong to 5. *Axiom of powers*: for each set there is a collection of sets that contains, as elements, all the subsets of the given set 6. *Axiom of infinity*: there is a set containing 0 and the successor of each of its elements 7. *Axiom of substitution*: if `S(a,b)` is a sentence such that `∀a ∈ A` the set `{ b | S(a,b) }` can be formed, then there exists a function `f` with domain `A` such that `f(a) = { b | S(a,b) }` for each `a ∈ A` (a reasonable operation over the element of a set yields a set) 8. *Axiom of choice*: the Cartesian product of a non-empty family of non-empty sets is non-empty --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/200-set-theory/set-theories/naive-set-theory-by-halmos.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.