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Set Theory: An Introduction to Independence Proofs is a textbook and reference work in set theory by Kenneth Kunen. It starts from basic notions, including the ZFC axioms, and quickly develops combinatorial notions such as trees , Suslin's problem , , and Martin's axiom .
Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes. There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets.
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
The Metamath Proof Explorer (recorded in set.mm) is the main database. It is based on classical first-order logic and ZFC set theory (with the addition of Tarski-Grothendieck set theory when needed, for example in category theory). The database has been maintained for over thirty years (the first proofs in set.mm are dated September 1992). The ...
In the early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency. In 1900, Hilbert posed a famous list of 23 problems for the next century.
Further, since set theory was seen as the basis for an axiomatic development of all other branches of mathematics, Russell's paradox threatened the foundations of mathematics as a whole. This motivated a great deal of research around the turn of the 20th century to develop a consistent (contradiction-free) set theory.
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. [3] Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language.
While von Neumann–Bernays–Gödel set theory is a conservative extension of Zermelo–Fraenkel set theory (ZFC, the canonical set theory) in the sense that a statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC, Morse–Kelley set theory is a proper extension of ZFC. Unlike von Neumann–Bernays–Gödel ...