Formally, the order among cardinal numbers is defined as follows: |X| ≤ |Y| means that there exists an injective function from X to Y. The Cantor–Bernstein–Schroeder theorem states that if |X| ≤ |Y| and |Y| ≤ |X| then |X| = |Y|. The axiom of choice is equivalent to the statement that given two sets X and Y, either |X| ≤ |Y| or |Y| ≤ |X|
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Anonymous2013-08-31 8:10
The infinity symbol \infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ infinity (HTML: ∞ ∞) and in LaTeX as \infty.
In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating point data type, the infinity values might still be accessible and usable as the result of certain operations.
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Anonymous2013-08-31 9:40
Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and the study of extensions of Ramsey's theorem such as the Erdős–Rado theorem.
A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f(s) is an element of s. With this concept, the axiom can be stated:
In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the Law of excluded middle (unlike in Martin-Löf type theory, where it does not). Thus the axiom of choice is not generally available in constructive set theory.
When considering these large objects, we might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets. It happens that it doesn't; by considering the above example we can see that if some object "one greater than infinity" exists, then it must have the same cardinality as the infinite set we started out with. It is possible to use a different formal notion for number, called ordinals, based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers.
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Anonymous2013-08-31 20:40
However, recent readings of the Archimedes Palimpsest have hinted that Archimedes at least had an intuition about actual infinite quantities.
In logic an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
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Anonymous2013-08-31 22:10
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic.
To give an informal example, for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks (assumed to have no distinguishing features), such a selection can be obtained only by invoking the axiom of choice.
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Anonymous2013-08-31 23:40
As discussed above, in ZFC, the axiom of choice is able to provide "nonconstructive proofs" in which the existence of an object is proved although no explicit example is constructed. ZFC, however, is still formalized in classical logic. The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.
In this context, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism.
There are many constructions in mathematics which would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functor F from C to D as a mapping that
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Anonymous2013-09-01 3:27
abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category Grp of all groups with group homomorphisms as morphisms. If (G,*) is a group, we define its opposite group (Gop,*op) as follows: Gop is the same set as G, and the operation *op is defined by a *op b = b * a. All multiplications in Gop are thus "turned around". Forming the opposite group becomes a (covariant!) functor from Grp to Grp if we define fop = f for any group homomorphism f: G → H. Note that fop is indeed a group homomorphism from Gop to Hop: