Using König's theorem, one can prove κ < κcf(κ) and κ < cf(2κ) for any infinite cardinal κ, where cf(κ) is the cofinality of κ.
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Anonymous2013-08-31 21:14
The formula would be useful neither to compute the force between two objects of finite mass nor to compute their motions. If an infinite mass object were to exist, any object of finite mass would be attracted with infinite force (and hence acceleration) by the infinite mass object, which is not what we can observe in reality.
The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose ontology consists of:
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Anonymous2013-08-31 22:44
One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free complete lattice.
The reason that we are able to choose least elements from subsets of the natural numbers is the fact that the natural numbers are well-ordered: every nonempty subset of the natural numbers has a unique least element under the natural ordering.
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Anonymous2013-09-01 0:14
Examples of category-theoretic statements which require choice include:
Category theory[1] is used to formalize mathematics and its concepts as a collection of objects and arrows (also called morphisms). Category theory can be used to formalize concepts of other high-level abstractions such as set theory, field theory, and group theory.
The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that ηX is an isomorphism for every object X in C.
One can compose functors, i.e. if F is a functor from A to B and G is a functor from B to C then one can form the composite functor G∘F from A to C. Composition of functors is associative where defined. Identity of composition of functors is identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the category of small categories.
It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described. This can be visualized using Cantor's diagonal argument; classic questions of cardinality (for instance the continuum hypothesis) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals. In more recent times mathematicians have been describing the properties of larger and larger cardinals.
The Indian mathematical text Surya Prajnapti (c. 3rd–4th century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
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Anonymous2013-09-01 11:23
Infinity is also used to describe infinite series: