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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.
An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system (usually formulated within model theory) that describes a set of sentences that is closed under logical implication. [1] A formal proof is a complete rendition of a mathematical proof within a formal system.
The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. [b] Such systems come in two flavors, those whose ontology consists of: Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Fragments of ZFC include:
Some of "mainstream mathematics" (mathematics not directly connected with axiomatic set theory) is beyond Peano arithmetic and second-order arithmetic, but still, all such mathematics can be carried out in ZC (Zermelo set theory with choice), another theory weaker than ZFC. Much of the power of ZFC, including the axiom of regularity and the ...
An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions.
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
An axiom is called independent if it can not be proved or disproved from the other axioms of the axiomatic system. An axiomatic system is said to be independent if each of its axioms is independent. If a true statement is a logical consequence of an axiomatic system, then it will be a true statement in every model of that system. To prove that ...
Informal presentation of an axiomatic set theory, e.g. as in Naive Set Theory by Paul Halmos. Early or later versions of Georg Cantor's theory and other informal systems. Decidedly inconsistent theories (whether axiomatic or not), such as a theory of Gottlob Frege [6] that yielded Russell's paradox, and theories of Giuseppe Peano [7] and ...