Transfinite number

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 {{Em|five}} marbles), whereas ordinal numbers specify the order of a member within an ordered set{{Cite web|url=http://mathworld.wolfram.com/OrdinalNumber.html|title=Ordinal Number|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|date=3 May 2023}} (e.g., "the {{Em|third}} man from the left" or "the {{Em|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, while a transfinite ordinal is used to describe the location within an infinitely large set that is ordered.{{not in citation given|date=May 2021}} The most notable ordinal and cardinal numbers are, respectively:

• $\backslash omega$ (Omega): the lowest transfinite ordinal number. It is also the order type of the natural numbers under their usual linear ordering.
• $\backslash aleph\_0$ (Aleph-null): the first transfinite cardinal number. It is also the cardinality of the natural numbers. If the axiom of choice holds, the next higher cardinal number is aleph-one, $\backslash aleph\_1.$ If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-null. Either way, there are no cardinals between aleph-null and aleph-one.

The continuum hypothesis is the proposition that there are no intermediate cardinal numbers between $\backslash aleph\_0$ and the cardinality of the continuum (the cardinality of the set of real numbers): or equivalently that $\backslash aleph\_1$ 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:

• $\backslash mathfrak\{m\}$ is a transfinite cardinal. That is, there is a Dedekind infinite set $A$ such that the cardinality of $A$ is $\backslash mathfrak\; \{m\}.$
• $\backslash mathfrak\{m\}\; +\; 1\; =\; \backslash mathfrak\{m\}.$
• $\backslash aleph\_0\; \backslash leq\; \backslash mathfrak\{m\}.$
• There is a cardinal $\backslash mathfrak\{n\}$ such that $\backslash aleph\_0\; +\; \backslash mathfrak\{n\}\; =\; \backslash mathfrak\{m\}.$

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.{{citation

| last1 = Beyer | first1 = W. A.

| last2 = Louck | first2 = J. D.

| doi = 10.1006/aama.1996.0513

| issue = 3

| journal = Advances in Applied Mathematics

| mr = 1436485

| pages = 333–350

| title = Transfinite function iteration and surreal numbers

| volume = 18

| year = 1997| doi-access = free

}}

In Cantor's theory of ordinal numbers, every integer number must have a successor.John Horton Conway, (1976) On Numbers and Games. Academic Press, ISBN 0-12-186350-6. (See Chapter 3.) The next integer after all the regular ones, that is the first infinite integer, is named $\backslash omega$. In this context, $\backslash omega+1$ is larger than $\backslash omega$, and $\backslash omega\backslash cdot2$, $\backslash omega^\{2\}$ and $\backslash omega^\{\backslash omega\}$ are larger still. Arithmetic expressions containing $\backslash omega$ 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, essentially a finite sequence of digits that give coefficients of descending powers of $\backslash omega$.

Not all infinite integers can be represented by a Cantor normal form however, and the first one that cannot is given by the limit $\backslash omega^\{\backslash omega^\{\backslash omega^\{...\}\}\}$ and is termed $\backslash varepsilon\_\{0\}$. $\backslash varepsilon\_\{0\}$ is the smallest solution to $\backslash omega^\{\backslash varepsilon\}=\backslash varepsilon$, and the following solutions $\backslash varepsilon\_\{1\},\; ...,\backslash varepsilon\_\{\backslash omega\},\; ...,\backslash varepsilon\_\{\backslash varepsilon\_\{0\}\},\; ...$ give larger ordinals still, and can be followed until one reaches the limit $\backslash varepsilon\_\{\backslash varepsilon\_\{\backslash varepsilon\_\{...\}\}\}$, which is the first solution to $\backslash varepsilon\_\{\backslash alpha\}=\backslash alpha$. 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|date=May 2021}} even this only allows one to reach the lowest class of transfinite numbers: those whose size of sets correspond to the cardinal number $\backslash aleph\_\{0\}$.

{{Wiktionary|transfinite}}

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• Actual infinity
• Aleph number
• Beth number
• Epsilon number
• Infinitesimal
• Transfinite induction

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{{Reflist}}

• Levy, Azriel, 2002 (1978) Basic Set Theory. Dover Publications. {{isbn|0-486-42079-5}}
• O'Connor, J. J. and E. F. Robertson (1998) "[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html Georg Ferdinand Ludwig Philipp Cantor]," MacTutor History of Mathematics archive.
• Rubin, Jean E., 1967. "Set Theory for the Mathematician". San Francisco: Holden-Day. Grounded in Morse–Kelley set theory.
• Rudy Rucker, 2005 (1982) Infinity and the Mind. Princeton Univ. Press. Primarily an exploration of the philosophical implications of Cantor's paradise. {{isbn|978-0-691-00172-2}}.
• Patrick Suppes, 1972 (1960) "[https://books.google.com/books?id=sxr4LrgJGeAC Axiomatic Set Theory]". Dover. {{isbn|0-486-61630-4}}. Grounded in ZFC.

{{Large numbers}}

{{Infinity}}

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Category:Basic concepts in infinite set theory

Category:Cardinal numbers

Category:Ordinal numbers