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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 ...
Many different equivalent complete axiom systems have been formulated. They differ in the choice of basic connectives used, which in all cases have to be functionally complete (i.e. able to express by composition all n-ary truth tables), and in the exact complete choice of axioms over the chosen basis of connectives.
The first of these, called the theory of true arithmetic, cannot be written as the set of logical consequences of any enumerable set of axioms. The theory of (R, +, ×, 0, 1, =) was shown by Tarski to be decidable; it is the theory of real closed fields (see Decidability of first-order theories of the real numbers for more).
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.
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 ...
New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. The system of Kripke–Platek set theory is closely related to generalized recursion theory. Two famous statements in set theory are the axiom of choice and the continuum hypothesis. The axiom of ...
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.
List or describe a set of sentences in the language L σ, called the axioms of the theory. Give a set of σ-structures, and define a theory to be the set of sentences in L σ holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite fields.