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.
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 8:58
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,...>.
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 10:28
orall X \left[ \emptyset
otin X \implies \exists f: X arr igcup X \quad orall A \in X \, ( f(A) \in A ) ight] \,.
His continuum hypothesis is the proposition that ???? is the same as ℵ1. This hypothesis has been found to be independent of the standard axioms of mathematical set theory; it can neither be proved nor disproved from the standard assumptions.
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Anonymous2013-08-31 13:10
After my scanner stopped working (I guess 2000 pages was too much for it?), I kind of just lost motivation.
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Anonymous2013-08-31 13:22
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).
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Anonymous2013-08-31 13:55
most free country in the world. we gots states.
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Anonymous2013-08-31 14:07
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 14:40
I have to work with Jewish Calendar in Python. I want to know if there is a Jewish Calendar in Python or if there is import in python
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Anonymous2013-08-31 14:53
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 16:23
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.
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Anonymous2013-08-31 17:08
Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint (the Freyd adjoint functor theorem).
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Anonymous2013-08-31 17:53
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.
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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Anonymous2013-08-31 19:52
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.
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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Anonymous2013-08-31 21:22
If, on the other hand, the universe were not curved like a sphere but had a flat topology, it could be both unbounded and infinite. The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation.
An enrichment of ZFC called Internal Set Theory was proposed by Edward Nelson in 1977.
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Anonymous2013-08-31 22:52
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that "the product of a collection of non-empty sets is non-empty". More explicitly, it states that for every indexed family (S_i)_{i \in I} of nonempty sets there exists an indexed family (x_i)_{i \in I} of elements such that x_i \in S_i for every i \in I.
It is possible to prove many theorems using neither the axiom of choice nor its negation; such statements will be true in any model of Zermelo–Fraenkel set theory (ZF), regardless of the truth or falsity of the axiom of choice in that particular model.
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Anonymous2013-09-01 0:23
If the set A is infinite, then there exists an injection from the natural numbers N to A (see Dedekind infinite).
Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects; by studying these morphisms, we are able to learn more about the structure of the objects. In the case of groups, the morphisms are the group homomorphisms.
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Anonymous2013-09-01 1:53
Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically arises from a construction defined by a universal property; this can be seen as a more abstract and powerful view on universal properties.
F(\mathrm{id}_{X}) = \mathrm{id}_{F(X)}\,\! for every object X \in C
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Anonymous2013-09-01 3:24
An infranatural transformation η from F to G is simply a family of morphisms ηX: F(X) → G(X). Thus a natural transformation is an infranatural transformation for which ηY ∘ F(f) = G(f) ∘ ηX for every morphism f : X → Y. The naturalizer of η, nat(η), is the largest subcategory of C containing all the objects of C on which η restricts to a natural transformation.
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Anonymous2013-09-01 10:36
Addition is commutative κ + μ = μ + κ.
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Anonymous2013-09-01 11:22
\int_{-\infty}^{\infty} \, f(t)\ dt \ = a means that the total area under f(t) is finite, and equals a