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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". The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more".
In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets.
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Alexander Sotirios Kechris is a set theorist and logician at the California Institute of Technology.
In set theory, is the least nonzero ordinal such that there is no surjection from the reals onto .
Stephen Craig Jackson is an American set theorist at the University of North Texas. Much of his most notable work has involved the descriptive set-theoretic consequences of the axiom of determinacy. In particular he is known for having calculated the values of all the projective ordinals under the assumption that the axiom of determinacy holds.
The Association for Symbolic Logic (ASL) is an international organization of specialists in mathematical logic and philosophical logic. The ASL was founded in 1936, and its first president was Curt John Ducasse. The current president of the ASL is Phokion Kolaitis.
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Penelope Maddy is an American philosopher. Maddy is Emerita UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the philosophy of mathematics, where she has worked on mathematical realism and mathematical naturalism.
In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wadge.
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Akihiro Kanamori is a Japanese-born American mathematician who won the Abel Prize. He specializes in set theory and is the author of the monograph on large cardinals, The Higher Infinite. He has written several essays on the history of mathematics, especially set theory.
In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set of points in some Polish space. Scales were originally isolated as a concept in the theory of uniformization, but have found wide applicability in descriptive set theory, with applications such as establishing bounds on the possible lengths of wellorderings of a given complexity, and showing that there are largest countable sets of certain complexities.
Stevo Todorčević, is a Yugoslavian mathematician specializing in mathematical logic and set theory. He holds a Canada Research Chair in mathematics at the University of Toronto, and a director of research position at the Centre national de la recherche scientifique in Paris.
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References
- Maddy, Penelope (1988a). "Believing the Axioms I" (PDF). The Journal of Symbolic Logic . 53 (2): 481–511. doi:10.1017/s0022481200028425.
- Maddy, Penelope (1988b). "Believing the Axioms II" (PDF). The Journal of Symbolic Logic. 53 (3): 736–764. doi:10.2307/2274569. JSTOR 2274569.
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