Infinity (math)

In mathematics, the adjective infinite is used to characterize some mathematical terms in more detail. As a rule, this results in a characterization that is complementary to the term finite .

overview

Examples of terms that include the term "infinite" are:

infinite set as a complementary term to the finite set ,
infinite ordinal number ,
infinite cardinal number ,
infinite-dimensional vector space as a complementary term to finite-dimensional vector space.

But there are also terms like an infinite product , the definition of which contains more extensive properties than the non-finiteness of the number of factors. It is similar with the concepts of an infinitely large non-standard number and an infinite limit value .

An example of a redundant use is the obsolete term infinite series, or series for short .

Infinite values are represented in mathematics by the infinity sign. This symbol was introduced in 1655 by the English mathematician John Wallis as a sign of an abstract infinite greatness. Examples of its use are: ${\ displaystyle \ infty}$

${\ displaystyle [a, \ infty) = \ {x \ in \ mathbb {R} \ mid x \ geq a \}}$ or for unlimited intervals , ${\ displaystyle (- \ infty, b]}$

{\ displaystyle \ lim _ {n \ to \ infty} a_ {n}}

for the limit of a convergent sequence ,

{\ displaystyle a_ {n}}

{\ displaystyle \ sum _ {n = 0} ^ {\ infty} \, a_ {n} = \ lim _ {m \ to \ infty} \ sum _ {n = 0} ^ {m} \, a_ {n }}

for the Limes of a series,

{\ displaystyle \ int _ {- \ infty} ^ {\ infty} \, f (t) \ \ mathrm {d} t = \ lim _ {t \ to - \ infty} \ int _ {t} ^ {0 } \, f (t) \ \ mathrm {d} t + \ lim _ {t \ to \ infty} \ int _ {0} ^ {t} \, f (t) \ \ mathrm {d} t}

in the case of an improper integral with an unlimited area of integration ,

{\ displaystyle \ | x \ | _ {\ infty}}

for the maximum norm or the supremum norm .

There are also gradations among the infinite quantities:

One says that a set contains almost all elements of an infinite basic set if it contains all elements with a finite number of exceptions. Closely related to this is the concept of an almost certain event in a measure space . Sets of these two types are always infinite sets (given infinite basic sets). But the reverse is not true.

Even among the infinite cardinal and ordinal numbers there are gradations that are reflected in size relationships. These correspond to the inclusion relations of the underlying quantities.

The term also occurs in topology - such as in Alexandroff compactification - and also in geometry - especially in the construction of projective geometries . Here, sets are expanded by elements, with the elements added in this way being referred to as infinitely distant points due to the intuitive notion .

In the previous examples of terms, the adjective infinite in its intuitive word meaning stood for infinitely large , infinitely many or infinitely far away . The infinitesimal calculus was named after the historically used infinitely small quantities, whereby it meant a great advance , especially due to Karl Weierstrass , to avoid such a construction.

Infinite amount

An infinite quantity is a quantity that is not finite. In the case of a set , this means that there is no natural number for which a bijection , i.e. a "one-to-one assignment", ${\ displaystyle M}$ ${\ displaystyle n}$

{\ displaystyle f \ colon M \ rightarrow \ {0, \ dotsc, n-1 \}}

exists.

The existence of infinite sets is the subject of the axiom of infinity of the axiomatically based Zermelo-Fraenkel set theory .

Analysis: certain divergence

A sequence of real numbers shows a certain divergence towards infinity if every arbitrarily given real number is exceeded by almost all sequence members. In this case, one also writes symbolically, in which one speaks of an improper convergence ${\ displaystyle x_ {1}, x_ {2}, x_ {3}, \ dotsc}$

${\ displaystyle \ lim _ {n \ to \ infty} x_ {n} = \ infty.}$

Analogously, one writes in the case of a certain divergence towards minus infinity

${\ displaystyle \ lim _ {n \ to \ infty} x_ {n} = - \ infty.}$

For such facts, "calculation rules" can also be formulated such as

${\ displaystyle a + \ infty = \ infty.}$

Such a calculation rule must, however, always be understood as a statement about improper limit values. The calculation rule just mentioned stands for the following facts:

Are and so that two sequences of real numbers against converges and diverges determined to infinity, then diverges also the result determined to infinity.

{\ displaystyle (a_ {n})}

{\ displaystyle (b_ {n})}

{\ displaystyle (a_ {n})}

{\ displaystyle a}

{\ displaystyle (b_ {n})}

{\ displaystyle (a_ {n} + b_ {n})}

Compactifications

Improper limit values can be understood as limit values from the perspective of the topology in the context of a so-called compactification, but now in a topological space in which the real numbers are expanded by two elements. The two added elements can be intuitively imagined as infinitely distant points on the number line . ${\ displaystyle \ mathbb {R} \ cup \ {+ \ infty, - \ infty \}}$ ${\ displaystyle \ mathbb {R}}$

It should be noted, however, that it is not possible to define arithmetic operations with the known calculation rules on the set expanded in this way. There are also other compactifications that lead to other statements with regard to the convergence of sequences. One example is the one-point compactification with only one additional, "infinite" element. There the sequence converges ${\ displaystyle \ mathbb {R} \ cup \ {\ infty \}}$

${\ displaystyle 1, -2.3, -4.5, -6, \ dotsc,}$

but which does not converge in compactification . ${\ displaystyle \ mathbb {R} \ cup \ {+ \ infty, - \ infty \}}$

A compactification of a topological space is also suitable for the investigation of continuous functions that are defined on this topological space. For this purpose, it must be possible to continuously continue the relevant functions on the compactification .

Stereographic projection : representation of the infinite plane as a finite sphere

In the field of function theory , one-point compactification is used to investigate holomorphic and meromorphic functions , which is also known as the Riemann number sphere . ${\ displaystyle \ mathbb {C} \ cup \ {\ infty \}}$

A generalization of the one-point compactification of the real numbers (and correspondingly of ) is the -dimensional projective space . It arises from the -dimensional Euclidean space through points that can be intuitively understood as infinitely far and which correspond to the straight lines through the zero point. ${\ displaystyle \ mathbb {R} \ cup \ {\ infty \}}$ ${\ displaystyle \ mathbb {C} \ cup \ {\ infty \}}$ ${\ displaystyle n}$ ${\ displaystyle \ mathbb {P} ^ {n} (\ mathbb {R})}$ ${\ displaystyle n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

For the construction of the projective space: is defined as the set of all straight lines through the origin of the . An embedding of Euclidean space to the projective space results from the fact that on the one hand, the a -affinity hyperplane in that does not include the zero point, are identified and on the other hand extending through the zero point straight in , which are not parallel to this affine hyperplane, with their intersections with this hyperplane. ${\ displaystyle \ mathbb {P} ^ {n} (\ mathbb {R})}$ ${\ displaystyle \ mathbb {R} ^ {n + 1}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle \ mathbb {P} ^ {n} (\ mathbb {R})}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle \ mathbb {R} ^ {n + 1}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

Projective plane

In the case of the Euclidean plane , the construction just described can also be interpreted in terms of affine geometry . Each straight line is extended by an “infinitely distant” point, namely by the point that is represented by the straight line through the zero point that runs parallel to the given straight line. Two parallels, which are extended from the Euclidean plane to the projective plane , intersect at this common, infinitely distant point . ${\ displaystyle \ mathbb {R} ^ {2}}$

Non-standard analysis

The basis of the nonstandard analysis is the ordered field of the hyper real numbers , which contains the real numbers as part of the field. The field of hyperreal numbers contains infinitesimally neighboring numbers as well as infinitely large numbers.

Infinitely large numbers also exist in the class of surreal numbers and in the subclass of combinatorial games . The hyperreal numbers form a subset of the surreal numbers.

Cardinal numbers

As with finite sets, two infinite sets can also be examined to see whether they have the same cardinality . By definition, this is the case if and only if there is a bijection between them. Sets of equal thickness are identified by a matching cardinal number. In a finite set, the cardinal number is the number of elements.

The two sets of natural and real numbers in particular have different thicknesses , which was first proven by Georg Cantor , the founder of set theory. The simplest proof uses Cantor's second diagonal argument , which can be generalized to the effect that a set always has a different power than its power set .

The cardinal number of the set of natural numbers has the designation using the Hebrew letter ( aleph ) . Sets of this cardinality are called countable . Infinite sets that are not countable are called uncountable . The cardinal number of the real numbers is because the set of real numbers is equal to the power set of natural numbers. ${\ displaystyle \ aleph}$ ${\ displaystyle \ aleph _ {0}}$ ${\ displaystyle 2 ^ {\ aleph _ {0}}}$

The two most important examples of countable sets are (besides the set of natural and whole numbers) the set of rational numbers (proof with Cantor's first diagonal argument ) and then the set of algebraic numbers . With the latter example and the knowledge that there are only countable algebraic numbers, one of the first great triumphs of Georg Cantor (found in 1874) and set theory is connected, since the set of real numbers is uncountable. Because this means that not all real numbers can be algebraic, and consequently leads to the proof that there must be transcendent numbers .

Cantor already suspected the so-called continuum hypothesis , according to which every subset of real numbers is either countable or equal to the set of real numbers.

Ordinal numbers

As with cardinal numbers, each ordinal number can be represented by a set. However, only well-ordered sets are considered , whereby two sets, between which an order isomorphism exists, define the same ordinal number. For example, the two represent sets with the elements

{\ displaystyle 1 <2 <3 <4 <\ cdots}

or.

{\ displaystyle 2 <4 <6 <8 <\ cdots}

the ordinal number is consistent . The ordinal number is represented by the amount that the elements ${\ displaystyle \ omega}$ ${\ displaystyle \ omega +1}$

{\ displaystyle 1 <2 <3 <4 <\ cdots <\ omega}

contains. The ordinal number can be represented by the amount that the elements (in view of their designation, which are ordered differently from the standard) ${\ displaystyle \ omega + \ omega}$

{\ displaystyle 1 <3 <5 <\ cdots <2 <4 <6 <\ cdots}

contains.

Controversies about the existence of infinite sets

The terms described above have been the subject of historical controversy over basic assumptions of mathematics in relation to whether, on what basis, and how such terms can be formally defined . Today, the Zermelo-Fraenkel set theory, including the axiom of choice (abbreviated to ZFC ), is considered a tried and tested and widely accepted framework for mathematics , even if its consistency cannot be proven due to Gödel's second incompleteness theorem . Apart from this “mainstream”, there are other “schools” of constructivists , finitists and ultrafinitists , who find their parallel justification in a renunciation of certain axioms or conclusions.

literature

Amir D. Aczel: The nature of infinity. Math, Kabbalah and the secret of the Aleph. Rowohlt-Taschenbuch-Verlag, Reinbek near Hamburg 2002, ISBN 3-499-61358-1 ( rororo - science 61358).
Heinz-Dieter Ebbinghaus : Introduction to set theory . 3rd, completely revised and expanded edition. BI-Wissenschaftsverlag, Mannheim (inter alia) 1994, ISBN 3-411-17113-8 .
Adolf Fraenkel : Introduction to set theory (= The basic doctrines of the mathematical sciences in individual representations . Volume 9 ). 3rd, revised and greatly expanded edition. Springer Verlag, Berlin (among others) 1928.
Eli Maor : To Infinity and Beyond. A Cultural History of the Infinite. Birkhäuser, Boston a. a. 1987, ISBN 0-8176-3325-1 .
Raymond Smullyan : Satan, Cantor and Infinity and 200 other amazing tinkering. Insel-Verlag, Frankfurt am Main a. a. 1997, ISBN 3-458-33599-4 ( Insel-Taschenbuch 1899).
Rudolf Taschner : The Infinite. Mathematicians struggle for a term. 2nd improved edition. Springer, Berlin a. a. 2006 (published: 2005), ISBN 3-540-25797-7 .
Nelly Tsouyopoulos : The concept of the infinite from Zeno to Galileo . In: Rete , 1 (1972), volume 3/4, pp. 245-272.

Web links

Wikiquote: Infinite (Math) Quotes

Compact dictionary of the infinite
Spektrum .de: From Infinity to Infinity October 13, 2017

Individual evidence

↑ David Hilbert : About the Infinite. In: Mathematical Annals. 95, 1926, ISSN 0025-5831 , pp. 161–190 ( online ( page no longer available , search in web archives ) Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. )@1@ 2

↑ Adolf Fraenkel : Introduction to set theory (= The basic doctrines of the mathematical sciences in individual representations . Volume 9 ). 3rd, revised and greatly expanded edition. Springer Verlag, Berlin (among others) 1928, p. 53-54 .

[1] David Hilbert : About the Infinite. In: Mathematical Annals. 95, 1926, ISSN 0025-5831 , pp. 161–190 ( online ( page no longer available , search in web archives ) Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. )@1@ 2

[2] Adolf Fraenkel : Introduction to set theory (= The basic doctrines of the mathematical sciences in individual representations . Volume 9 ). 3rd, revised and greatly expanded edition. Springer Verlag, Berlin (among others) 1928, p. 53-54 .