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Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology. Axiom of extensionality; Axiom of empty set; Axiom of pairing; Axiom of union; Axiom of infinity; Axiom schema of replacement; Axiom of power set ...
Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-empty intersection with S. We define a binary relation R on S by a R b :⇔ b ∈ S ∩ a {\textstyle aRb:\Leftrightarrow b\in S\cap a} , which is entire by assumption.
The mathematical statements discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be ...
Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2. [11] The collection of all sets that are obtained in this way, over all the stages, is known as V. The sets in V can be arranged into a hierarchy by assigning to each set the first stage at which that set was added to V.
The set {} is empty and thus not inhabited. Naturally, the example section thus focuses on non-empty sets that are not provably inhabited. It is easy to give such examples by using the axiom of separation, as with it logical statements can always be
This category is for axioms in the language of set theory; roughly speaking, ones that "talk about sets".Inclusion in this category does not necessarily imply that the axiom in question is an accepted axiom, or that it is believed to be true in the von Neumann universe of sets.
For any set A there is a function f such that for any non-empty subset B of A, f(B) lies in B. The negation of the axiom can thus be expressed as: There is a set A such that for all functions f (on the set of non-empty subsets of A), there is a B such that f(B) does not lie in B.
The axiom of global choice states that there is a global choice function τ, meaning a function such that for every non-empty set z, τ(z) is an element of z.. The axiom of global choice cannot be stated directly in the language of Zermelo–Fraenkel set theory (ZF) with the axiom of choice (AC), known as ZFC, as the choice function τ is a proper class and in ZFC one cannot quantify over classes.