Transfinite number

Notable work on transfinite numbers was done by Wacław Sierpiński, described in his 1928 book Leçons sur les nombres transfinis, much expanded into Cardinal and Ordinal Numbers in 1958, with a second slightly revised edition in 1965..

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 (e.g., "the fifth man from the left" or "the twenty-seventh day of January"). For finite numbers these concepts are in one-to-one correspondence: five ⇔ fifth, but when extended to transfinite numbers, the 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. The most notable ordinal and cardinal numbers are:

• ${\displaystyle \omega }$ (omega): the lowest transfinite ordinal number. It is also the order type of the natural numbers under their usual linear ordering.
• ${\displaystyle {\mathrm{\aleph }}_0}$ (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, ${\displaystyle {\mathrm{\aleph }}_1.}$ 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 ${\displaystyle {\mathrm{\aleph }}_0}$ and the cardinality of the continuum (the cardinality of the set of real numbers): or equivalently that ${\displaystyle {\mathrm{\aleph }}_1}$ 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:

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

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.