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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 ...
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second-order logic.
The following particular axiom set is from Kunen (1980). The axioms in order below are expressed in a mixture of first order logic and high-level abbreviations. Axioms 1–8 form ZF, while the axiom 9 turns ZF into ZFC. Following Kunen (1980), we use the equivalent well-ordering theorem in place of the axiom of choice for axiom 9.
Weak forms of the reflection principle are theorems of ZF set theory due to Montague (1961), while stronger forms can be new and very powerful axioms for set theory. The name "reflection principle" comes from the fact that properties of the universe of all sets are "reflected" down to a smaller set.
A proof requiring the axiom of choice may establish the existence of an object without explicitly defining the object in the language of set theory. For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no individual well-ordering of the ...
Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction. The axiom of induction tends to be used in place of the axiom of regularity in intuitionistic theories (ones that do not accept the law of the excluded middle), where the two axioms are not equivalent.
With the additional contributions of Abraham Fraenkel, Zermelo set theory developed into the now-standard Zermelo–Fraenkel set theory (commonly known as ZFC when including the axiom of choice). The main difference between Russell's and Zermelo's solution to the paradox is that Zermelo modified the axioms of set theory while maintaining a ...
In many popular versions of axiomatic set theory, the axiom schema of specification, [1] also known as the axiom schema of separation (Aussonderungsaxiom), [2] subset axiom [3], axiom of class construction, [4] or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.