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Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; ... Higher Set Theory (PDF). Lecture Notes in Mathematics. Vol. 669.
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory.It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom, which states that for each set there is a "Tarski universe" it belongs to (see below).
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.
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers.Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ω α).
Lectures in set theory, with particular emphasis on the method of forcing, Springer-Verlag Lecture Notes in Mathematics 217 (1971) (ISBN 978-3540055648) The axiom of choice, North-Holland 1973 (Dover paperback edition ISBN 978-0-486-46624-8) (with K. Hrbáček) Introduction to set theory, Marcel Dekker, 3rd edition 1999 (ISBN 978-0824779153)
In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about 1992.. A Berkeley cardinal is a cardinal κ in a model of Zermelo–Fraenkel set theory with the property that for every transitive set M that includes κ and α < κ, there is a nontrivial elementary embedding of M into M with α < critical ...
Every set is a projective object in Set (assuming the axiom of choice). The finitely presentable objects in Set are the finite sets. Since every set is a direct limit of its finite subsets, the category Set is a locally finitely presentable category. If C is an arbitrary category, the contravariant functors from C to Set are often an important ...
Models of ZF-Set Theory. Lecture Notes in Mathematics. Springer-Verlag. ISBN 3-540-05591-6. Gödel, Kurt (1938). "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. 24 (12). National Academy of Sciences: 556–557.