About: Reflection principle

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In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that resemble the class of all sets. There are several different forms of the reflection principle depending on exactly what is meant by "resemble". Weak forms of the reflection principle are theorems of ZF set theory due to , while stronger forms can be new and very powerful axioms for set theory. The name "reflection principle" comes from the fact that properties of the universe of all sets are "reflected" down to a smaller set.

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rdfs:label	Reflexionsprinzip (Mengenlehre) (de) Principio de reflexión (es) Reflection principle (en)
rdfs:comment	Das Reflexionsprinzip ist ein mathematischer Satz aus dem Gebiet der Mengenlehre. Die Kernaussage lautet, dass es keinen in der Sprache der Mengenlehre formulierbaren Satz über das Mengenuniversum, das heißt über die Klasse aller Mengen, gibt, der nicht bereits in einer geeigneten Menge „gespiegelt“ (siehe unten) würde, woraus sich der Name Reflexionsprinzip erklärt. Der Satz geht auf Richard Montague (1957) und Azriel Levy (1960) zurück. (de) In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that resemble the class of all sets. There are several different forms of the reflection principle depending on exactly what is meant by "resemble". Weak forms of the reflection principle are theorems of ZF set theory due to , while stronger forms can be new and very powerful axioms for set theory. The name "reflection principle" comes from the fact that properties of the universe of all sets are "reflected" down to a smaller set. (en) En teoría de conjuntos, una rama de las matemáticas, el principio de reflexión dice que es posible encontrar conjuntos que representen la clase de todos los conjuntos. Existen varias formas diferentes del principio de reflexión según se determine con exactitud el significado de «representar». La forma débil del principio de reflexión lo constituyen los axiomas de Zermelo-Fraenkel, mientras que formas más fuertes pueden resultar en axiomas nuevos y muy poderosos para la teoría de conjuntos. (es)
dcterms:subject	Mathematical principles Set theory
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Link from a Wikipage to another Wikipage	Mizar system Model (logic) Paul Bernays Cumulative hierarchy Von Neumann–Bernays–Gödel set theory Wholeness axiom Mathematics Rank-into-rank Class (set theory) Georg Cantor Erdős cardinal Löwenheim–Skolem theorem Subsets Closure (mathematics) Club set Zermelo–Fraenkel set theory Mahlo cardinal Peter Koellner Axiomatic set theory Mathematical principles Set theory Wilhelm Ackermann Large cardinal Absolute Infinite Akihiro Kanamori Cardinal number Quantifier (logic) Ackermann set theory Kenneth Kunen Transfinite number Transitive set Axiom of choice Axiom of extensionality Axiom of infinity Axiom of pairing Axiom of union Huge cardinal Inaccessible cardinal Kurt Gödel Ordinal number Set theory Set Theory: An Introduction to Independence Proofs Von Neumann universe Morse–Kelley set theory Pacific Journal of Mathematics Subset Axiom of foundation Axiom of replacement Axiom of specification Powerset Erdös cardinal Free variables Set model Inconsistent Regular ordinal Absolute Infinity Absoluteness (mathematical logic) Infinitary logics
Link from a Wikipage to an external page	http://projecteuclid.org/euclid.pjm/1103038638 http://mizar.org/version/current/html/zf_refle.html
sameAs	Reflection principle Reflection principle Reflection principle Reflection principle Reflection principle Reflection principle
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text	All the principles for setting up the axioms of set theory should be reducible to Ackermann's principle: The Absolute is unknowable. The strength of this principle increases as we get stronger and stronger systems of set theory. The other principles are only heuristic principles. Hence, the central principle is the reflection principle, which presumably will be understood better as our experience increases. Meanwhile, it helps to separate out more specific principles which either give some additional information or are not yet seen clearly to be derivable from the reflection principle as we understand it now. (en)

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