The continuum hypothesis (CH) states that there are no cardinals strictly between ℵ0 and 2ℵ0. The latter cardinal number is also often denoted by ????; it is the cardinality of the continuum (the set of real numbers). In this case 2ℵ0 = ℵ1. The generalized continuum hypothesis (GCH) states that for every infinite set X, there are no cardinals strictly between | X | and 2| X |. The continuum hypothesis is independent of the usual axioms of set theory, the Zermelo-Fraenkel axioms together with the axiom of choice (ZFC).
Physicists however require that the end result be physically meaningful. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called renormalization.
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Anonymous2013-08-31 22:02
General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets;
ZF set theory does not formalize the notion of classes. They can instead be described in the metalanguage, as equivalence classes of logical formulas. For example, if \mathcal A is a structure interpreting ZF, then the metalanguage expression \{x\mid x=x \} is interpreted in \mathcal A by the collection of all the elements from the domain of \mathcal A; that is, all the sets in \mathcal A. So we can identify the "class of all sets" with the predicate x=x or any equivalent predicate.
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Anonymous2013-08-31 23:32
Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proven to exist using the axiom of choice, it is consistent that no such set is definable.
Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint (the Freyd adjoint functor theorem).
The most accessible example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However, the objects of a category need not be sets, and the arrows need not be functions; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category, and all the results of category theory will apply to it.
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Anonymous2013-09-01 1:47
Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find universal properties that uniquely determine the objects of interest.
In a functor category, the morphisms are natural transformations.
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Anonymous2013-09-01 3:19
Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limit and colimit generalize several of the above.
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Anonymous2013-09-01 10:04
He graduated from Berlin's Luisenstädtisches Gymnasium in 1889. He then studied mathematics, physics and philosophy at the universities of Berlin, Halle and Freiburg. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations (Untersuchungen zur Variationsrechnung). Zermelo remained at the University of Berlin, where he was appointed assistant to Planck, under whose guidance he began to study hydrodynamics. In 1897, Zermelo went to Göttingen, at that time the leading centre for mathematical research in the world, where he completed his habilitation thesis in 1899.
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Anonymous2013-09-01 10:50
κ0 = 1 (in particular 00 = 1), see empty function.