Transfinite number

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In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. [1] [2] The term transfinite was coined in 1895 by Georg Cantor, [3] [4] [5] [6] who wished to avoid some of the implications of the word infinite in connection with these objects, which were, nevertheless, not finite.[ citation needed ] Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as infinite numbers. Nevertheless, the term transfinite also remains in use.

Contents

Notable work on transfinite numbers was done by Wacław Sierpiński: Leçons sur les nombres transfinis (1928 book) much expanded into Cardinal and Ordinal Numbers (1958, [7] 2nd ed. 1965 [8] ).

Definition

Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set [9] (e.g., "the third man from the left" or "the twenty-seventh day of January"). When extended to transfinite numbers, these two concepts are no longer in one-to-one correspondence. A transfinite cardinal number is used to describe the size of an infinitely large set, [2] while a transfinite ordinal is used to describe the location within an infinitely large set that is ordered. [9] [ failed verification ] The most notable ordinal and cardinal numbers are, respectively:

The continuum hypothesis is the proposition that there are no intermediate cardinal numbers between and the cardinality of the continuum (the cardinality of the set of real numbers): [2] or equivalently that is the cardinality of the set of real numbers. In Zermelo–Fraenkel set theory, neither the continuum hypothesis nor its negation can be proved.

Some authors, including P. Suppes and J. Rubin, use the term transfinite cardinal to refer to the cardinality of a Dedekind-infinite set in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold. Given this definition, the following are all equivalent:

Although transfinite ordinals and cardinals both generalize only the natural numbers, other systems of numbers, including the hyperreal numbers and surreal numbers, provide generalizations of the real numbers. [10]

Examples

In Cantor's theory of ordinal numbers, every integer number must have a successor. [11] The next integer after all the regular ones, that is the first infinite integer, is named . In this context, is larger than , and , and are larger still. Arithmetic expressions containing specify an ordinal number, and can be thought of as the set of all integers up to that number. A given number generally has multiple expressions that represent it, however, there is a unique Cantor normal form that represents it, [11] essentially a finite sequence of digits that give coefficients of descending powers of .

Not all infinite integers can be represented by a Cantor normal form however, and the first one that cannot is given by the limit and is termed . [11] is the smallest solution to , and the following solutions give larger ordinals still, and can be followed until one reaches the limit , which is the first solution to . This means that in order to be able to specify all transfinite integers, one must think up an infinite sequence of names: because if one were to specify a single largest integer, one would then always be able to mention its larger successor. But as noted by Cantor,[ citation needed ] even this only allows one to reach the lowest class of transfinite numbers: those whose size of sets correspond to the cardinal number .

See also

Related Research Articles

In mathematics, specifically set theory, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states:

"There is no set whose cardinality is strictly between that of the integers and the real numbers."

<span class="mw-page-title-main">Cardinal number</span> Size of a possibly infinite set

In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter (aleph) marked with subscript indicating their rank among the infinite cardinals.

<span class="mw-page-title-main">Cardinality</span> Definition of the number of elements in a set

In mathematics, cardinality describes a relationship between sets which compares their relative size. For example, the sets and are the same size as they each contain 3 elements. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two notions often used when referring to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set may also be called its size, when no confusion with other notions of size is possible.

In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.

In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than aleph-null, the cardinality of the natural numbers.

<span class="mw-page-title-main">Aleph number</span> Infinite cardinal number

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).

<span class="mw-page-title-main">Limit ordinal</span> Infinite ordinal number class

In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ. Every ordinal number is either zero, or a successor ordinal, or a limit ordinal.

The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely:

In mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear.

In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that is a regular cardinal if and only if every unbounded subset has cardinality . Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular.

In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers, conventionally written , where is the Hebrew letter beth. The beth numbers are related to the aleph numbers, but unless the generalized continuum hypothesis is true, there are numbers indexed by that are not indexed by .

In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by or

In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations.

In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory T in a language we write I(T, κ) for the number of models of T (up to isomorphism) of cardinality κ. The spectrum problem is to describe the possible behaviors of I(T, κ) as a function of κ. It has been almost completely solved for the case of a countable theory T.

In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that satisfy the equation

In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations. However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not ; various more-concrete ways of defining ordinals that definitely have notations are available.

In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by . Georg Cantor proved that the cardinality is larger than the smallest infinity, namely, . He also proved that is equal to , the cardinality of the power set of the natural numbers.

<span class="mw-page-title-main">Ordinal number</span> Generalization of "n-th" to infinite cases

In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals aimed to extend enumeration to infinite sets.

In the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between , and the cardinality of the continuum, that is, the cardinality of the set of all real numbers. The latter cardinal is denoted or . A variety of such cardinal characteristics arise naturally, and much work has been done in determining what relations between them are provable, and constructing models of set theory for various consistent configurations of them.

In the mathematical fields of set theory and proof theory, the Takeuti–Feferman–Buchholz ordinal (TFBO) is a large countable ordinal, which acts as the limit of the range of Buchholz's psi function and Feferman's theta function. It was named by David Madore, after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as using Buchholz's psi function, an ordinal collapsing function invented by Wilfried Buchholz, and in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman. It is the proof-theoretic ordinal of several formal theories:

References

  1. "Definition of transfinite number | Dictionary.com". www.dictionary.com. Retrieved 2019-12-04.
  2. 1 2 3 "Transfinite Numbers and Set Theory". www.math.utah.edu. Retrieved 2019-12-04.
  3. "Georg Cantor | Biography, Contributions, Books, & Facts". Encyclopedia Britannica. Retrieved 2019-12-04.
  4. Georg Cantor (Nov 1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)". Mathematische Annalen. 46 (4): 481–512. Open Access logo PLoS transparent.svg
  5. Georg Cantor (Jul 1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49 (2): 207–246. Open Access logo PLoS transparent.svg
  6. Georg Cantor (1915). Philip E.B. Jourdain (ed.). Contributions to the Founding of the Theory of Transfinite Numbers (PDF). New York: Dover Publications, Inc. English translation of Cantor (1895, 1897).
  7. Oxtoby, J. C. (1959), "Review of Cardinal and Ordinal Numbers (1st ed.)", Bulletin of the American Mathematical Society , 65 (1): 21–23, doi: 10.1090/S0002-9904-1959-10264-0 , MR   1565962
  8. Goodstein, R. L. (December 1966), "Review of Cardinal and Ordinal Numbers (2nd ed.)", The Mathematical Gazette , 50 (374): 437, doi:10.2307/3613997, JSTOR   3613997
  9. 1 2 Weisstein, Eric W. (3 May 2023). "Ordinal Number". mathworld.wolfram.com.
  10. Beyer, W. A.; Louck, J. D. (1997), "Transfinite function iteration and surreal numbers", Advances in Applied Mathematics, 18 (3): 333–350, doi: 10.1006/aama.1996.0513 , MR   1436485
  11. 1 2 3 John Horton Conway, (1976) On Numbers and Games . Academic Press, ISBN 0-12-186350-6. (See Chapter 3.)

Bibliography