Transfinite number

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Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were, nevertheless, not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". However, the term "transfinite" also remains in use.

Definition

Any finite 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 third man from the left" or "the twenty-seventh day of January"). When extended to transfinite numbers, these two concepts become distinct. 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's ordered. The most notable ordinal and cardinal numbers are, respectively:

• ω (omega) is defined as the lowest transfinite ordinal number and is the order type of the natural numbers under their usual linear ordering.
• Aleph-naught, ${\displaystyle \aleph _{0},}$ is defined as the first transfinite cardinal number and is the cardinality of the infinite set of the natural numbers. If the axiom of choice holds, the next higher cardinal number is aleph-one, ${\displaystyle \aleph _{1}.}$ If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-naught. But in any case, there are no cardinals between aleph-naught and aleph-one.

The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the continuum (the set of real numbers): that is to say, aleph-one is the cardinality of the set of real numbers. (If Zermelo–Fraenkel set theory (ZFC) is consistent, then neither the continuum hypothesis nor its negation can be proven from ZFC.)

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 A such that the cardinality of A is ${\displaystyle {\mathfrak {m}}.}$
• ${\displaystyle {\mathfrak {m}}+1={\mathfrak {m}}.}$
• ${\displaystyle \aleph _{0}\leq {\mathfrak {m}}.}$
• There is a cardinal ${\displaystyle {\mathfrak {n}}}$ such that ${\displaystyle \aleph _{0}+{\mathfrak {n}}={\mathfrak {m}}.}$