If the axiom of choice holds, every cardinal κ has a successor κ+ > κ, and there are no cardinals between κ and its successor. For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal.
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Anonymous2013-08-31 13:36
In real analysis, the symbol \infty, called "infinity", denotes an unbounded limit. x ightarrow \infty means that x grows without bound, and x o -\infty means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then
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Anonymous2013-08-31 14:22
Cognitive scientist George Lakoff considers the concept of infinity in mathematics and the sciences as a metaphor. This view is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1,2,3,...>.
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Anonymous2013-08-31 15:07
The field of effective descriptive set theory is between set theory and recursion theory. It includes the study of lightface pointclasses, and is closely related to hyperarithmetical theory. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable.
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Anonymous2013-08-31 15:53
orall X \left[ \emptyset
otin X \implies \exists f: X arr igcup X \quad orall A \in X \, ( f(A) \in A ) ight] \,.
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Anonymous2013-08-31 16:38
Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.
The additive groups of R and C are isomorphic.[13] and [14]
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Anonymous2013-08-31 18:08
Many important constructions in mathematics can be studied in this context. "Naturality" is a principle, like general covariance in physics, that cuts deeper than is initially apparent. An arrow between two functors is a natural transformation when it is subject to certain naturality or commutativity conditions.
These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, constructive mathematics. Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology.
We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. It can be shown that for finite cardinals these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic.
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Anonymous2013-08-31 20:45
Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties
Artist M. C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.
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Anonymous2013-08-31 22:15
Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.
Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle in constructive set theory.
The Nielsen–Schreier theorem, that every subgroup of a free group is free.
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Anonymous2013-09-01 1:16
Abstracting yet again, some diagrammatic and/or sequential constructions are often "naturally related" – a vague notion, at first sight. This leads to the clarifying concept of natural transformation, a way to "map" one functor to another.
Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundation of mathematics. A topos can also be considered as a specific type of category with two additional topos axioms.
Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a covariant functor on the opposite category C^\mathrm{op}. Some authors prefer to write all expressions covariantly. That is, instead of saying F: Cightarrow D is a contravariant functor, they simply write F: C^{\mathrm{op}} ightarrow D (or sometimes F:C ightarrow D^{\mathrm{op}}) and call it a functor.
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Anonymous2013-09-01 10:24
We can then extend this to an equality-style relation. Two sets X and Y are said to have the same cardinality if there exists a bijection between X and Y. By the Schroeder-Bernstein theorem, this is equivalent to there being both a one-to-one mapping from X to Y and a one-to-one mapping from Y to X. We then write |X| = |Y|. The cardinal number of X itself is often defined as the least ordinal a with |a| = |X|. This is called the von Neumann cardinal assignment; for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the well-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.
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Anonymous2013-09-01 11:10
The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus. He used the word apeiron which means infinite or limitless.
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Anonymous2013-11-30 7:59
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