Math
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The cardinality of the set { x , y , z }, is three, while there are eight elements in its power set, ordered in respect to inclusion (3 < 2 3 =8) In elementary set theory , Cantor's theorem states that, for any set A , the set of all subsets of A (the power set of A ) has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can be seen to be true by a much simpler proof than that given below, since in addition to subsets of A with just one member, there are others as well, and since n < 2 n for all natural numbers n . But the theorem is true of infinite sets as well. In particular, the power set of a countably infinite set is uncountably infinite . The theorem is named for German mathematician Georg Cantor , who first stated and proved it.
Cantor's theorem
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic . The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics . The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem . The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an " effective procedure " (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers ( arithmetic ). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.
In differential geometry , Stokes' theorem (also called the generalized Stokes' theorem ) is a statement about the integration of differential forms on manifolds , which both simplifies and generalizes several theorems from vector calculus . Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e. This modern form of Stokes' theorem is a vast generalization of a classical result first discovered by Lord Kelvin , who communicated it to George Stokes in a letter dated July 2, 1850. [ 1 ] [ 2 ] [ 3 ] Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. [ 4 ] [ 3 ] This classical Kelvin–Stokes theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ:
Stokes' theorem
In the philosophy of mathematics , constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption , one still has not found the object and therefore not proved its existence, according to constructivism. This viewpoint involves a verificational interpretation of the existence quantifier, which is at odds with its classical interpretation. There are many forms of constructivism. [ 1 ] These include the program of intuitionism founded by Brouwer , the finitism of Hilbert and Bernays , the constructive recursive mathematics of Shanin and Markov , and Bishop 's program of constructive analysis . Constructivism also includes the study of constructive set theories such as IZF and the study of topos theory .
Constructivism (mathematics)
The language of mathematics is the system used by mathematicians to communicate mathematical ideas among themselves. This language consists of a substrate of some natural language (for example English ) using technical terms and grammatical conventions that are peculiar to mathematical discourse (see Mathematical jargon ), supplemented by a highly specialized symbolic notation for mathematical formulas . Like natural languages in general, discourse using the language of mathematics can employ a scala of registers .
Language of mathematics
Logicism
Logicism is one of the schools of thought in the philosophy of mathematics , putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. [ 1 ] Bertrand Russell and Alfred North Whitehead championed this theory, created by mathematicians Richard Dedekind and Gottlob Frege . Dedekind's path to logicism had a turning point when he was able to reduce the theory of real numbers to the rational number system by means of set theory. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of sets; furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real numbers published in the year 1872.
Fictionalism is the view in philosophy according to which statements that appear to be descriptions of the world should not be construed as such, but should instead be understood as cases of "make believer", of pretending to treat something as literally true (a "useful fiction"). Two important strands of fictionalism are modal fictionalism developed by Gideon Rosen , which states that possible worlds , regardless of whether they exist or not, may be a part of a useful discourse, and mathematical fictionalism advocated by Hartry Field , which states that talk of numbers and other mathematical objects is nothing more than a convenience for doing science. Also in meta-ethics , there is an equivalent position called moral fictionalism. Many modern versions of fictionalism are influenced by the work of Kendall Walton in aesthetics. Fictionalism consists in at least the following three theses:
Fictionalism
Structuralism (philosophy of mathematics)
Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their place in such structures, consequently having no intrinsic properties . For instance, it would maintain that all that needs to be known about the number 1 is that it is the first whole number after 0. Likewise all the other whole numbers are defined by their places in a structure, the number line .
A category with objects X , Y , Z and morphisms f , g , g ∘ f , and three identity morphisms (not shown) 1 X , 1 Y and 1 Z . Category theory 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 . Several terms used in category theory, including the term "morphism", differ from their uses within mathematics itself.
Category theory
Axiom of dependent choice
In mathematics , the axiom of dependent choices , denoted DC , is a weak form of the axiom of choice (AC) which is still sufficient to develop most of real analysis . Unlike full AC, DC is insufficient to prove (given ZF ) that there is a nonmeasurable set of reals , or that there is a set of reals without the property of Baire or without the perfect set property . The axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X , there is a sequence ( x n ) in X such that x n R x n +1 for each n in N . (Here an entire binary relation on X is one such that for each a in X there is a b in X such that aRb .) Note that even without such an axiom we could form the first n terms of such a sequence, for any natural number n ; the axiom of dependent choices merely says that we can form a whole sequence this way.