In mathematics, and in particular set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They are named after the symbol used to denote them, the Hebrew letter aleph (ℵ) (though in older mathematics books the letter aleph is often printed upside down by accident,[nb 1] partly because a monotype matrix for aleph was mistakenly constructed the wrong way up).
The cardinality of the natural numbers is (read aleph-naught or aleph-zero; the term aleph-null is also sometimes used), the next larger cardinality is aleph-one , then and so on. Continuing in this manner, it is possible to define a cardinal number for every ordinal number , as described below.
The aleph numbers differ from the infinity () commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line.
(aleph-naught, also aleph-zero or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called or (where is the lowercase Greek letter omega), has cardinality . A set has cardinality if and only if it is countably infinite, that is, there is a bijection (one-to-one correspondence) between it and the natural numbers. Examples of such sets are
- the set of all square numbers, the set of all cubic numbers, the set of all fourth powers, ...
- the set of all perfect powers, the set of all prime powers,
- the set of all even numbers, the set of all odd numbers,
- the set of all prime numbers, the set of all composite numbers,
- the set of all integers,
- the set of all rational numbers,
- the set of all constructible numbers (in the geometric sense),
- the set of all algebraic numbers,
- the set of all computable numbers,
- the set of all definable numbers,
- the set of all binary strings of finite length, and
- the set of all finite subsets of any given countably infinite set.
is an ordering of the set (with cardinality ) of positive integers.
is the cardinality of the set of all countable ordinal numbers, called or sometimes . This is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, is distinct from . The definition of implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number. Using the axiom of choice we can show one of the most useful properties of the set : any countable subset of has an upper bound in . (This follows from the fact that the union of a countable number of countable sets is itself countable, one of the most common applications of the axiom of choice.) This fact is analogous to the situation in : every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.
is actually a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the -algebra generated by an arbitrary collection of subsets (see e.g. Borel hierarchy). This is harder than most explicit descriptions of "generation" in algebra (vector spaces, groups, etc.) because in those cases we only have to close with respect to finite operations—sums, products, and the like. The process involves defining, for each countable ordinal, via transfinite induction, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of .
The cardinality of the set of real numbers (cardinality of the continuum) is . It cannot be determined from ZFC (Zermelo–Fraenkel set theory with the axiom of choice) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis, CH, is equivalent to the identity
The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers. CH is independent of ZFC: it can be neither proven nor disproven within the context of that axiom system (provided that ZFC is consistent). That CH is consistent with ZFC was demonstrated by Kurt Gödel in 1940 when he showed that its negation is not a theorem of ZFC. That it is independent of ZFC was demonstrated by Paul Cohen in 1963 when he showed, conversely, that the CH itself is not a theorem of ZFC by the (then novel) method of forcing. 
where the smallest infinite ordinal is denoted ω. That is, the cardinal number is the least upper bound of
is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers; for any positive integer n we can consistently assume that , and moreover it is possible to assume is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality , meaning there is an unbounded function from to it (see Easton's theorem).
Aleph- for general Edit
To define for arbitrary ordinal number , we must define the successor cardinal operation, which assigns to any cardinal number the next larger well-ordered cardinal (if the axiom of choice holds, this is the next larger cardinal).
We can then define the aleph numbers as follows:
and for λ, an infinite limit ordinal,
Fixed points of omegaEdit
For any ordinal α we have
In many cases is strictly greater than α. For example, for any successor ordinal α this holds. There are, however, some limit ordinals which are fixed points of the omega function, because of the fixed-point lemma for normal functions. The first such is the limit of the sequence
Any weakly inaccessible cardinal is also a fixed point of the aleph function. This can be shown in ZFC as follows. Suppose is a weakly inaccessible cardinal. If were a successor ordinal, then would be a successor cardinal and hence not weakly inaccessible. If were a limit ordinal less than , then its cofinality (and thus the cofinality of ) would be less than and so would not be regular and thus not weakly inaccessible. Thus and consequently which makes it a fixed point.
Role of axiom of choiceEdit
The cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal. Any set whose cardinality is an aleph is equinumerous with an ordinal and is thus well-orderable.
Each finite set is well-orderable, but does not have an aleph as its cardinality.
The assumption that the cardinality of each infinite set is an aleph number is equivalent over ZF to the existence of a well-ordering of every set, which in turn is equivalent to the axiom of choice. ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality (i.e. is equinumerous with its initial ordinal), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.
When cardinality is studied in ZF without the axiom of choice, it is no longer possible to prove that each infinite set has some aleph number as its cardinality; the sets whose cardinality is an aleph number are exactly the infinite sets that can be well-ordered. The method of Scott's trick is sometimes used as an alternative way to construct representatives for cardinal numbers in the setting of ZF. For example, one can define card(S) to be the set of sets with the same cardinality as S of minimum possible rank. This has the property that card(S) = card(T) if and only if S and T have the same cardinality. (The set card(S) does not have the same cardinality of S in general, but all its elements do.)
- For example, in (Sierpiński 1958, p.402) the letter aleph appears both the right way up and upside down
- Swanson, Ellen; O'Sean, Arlene Ann; Schleyer, Antoinette Tingley (1999) , Mathematics into type: Copy editing and proofreading of mathematics for editorial assistants and authors (updated ed.), Providence, R.I.: American Mathematical Society, p. 16, ISBN 0-8218-0053-1, MR 0553111
- Jeff Miller. "Earliest Uses of Symbols of Set Theory and Logic". jeff560.tripod.com. Retrieved 2016-05-05. Miller quotes Joseph Warren Dauben (1990). Georg Cantor:His Mathematics and Philosophy of the Infinite. ISBN 9780691024479. : "His new numbers deserved something unique. ... Not wishing to invent a new symbol himself, he chose the aleph, the first letter of the Hebrew alphabet...the aleph could be taken to represent new beginnings..."
- Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag
- Dales H.G., Dashiell F.K., Lau A.TM., Strauss D. (2016) Introduction. In: Banach Spaces of Continuous Functions as Dual Spaces. CMS Books in Mathematics (Ouvrages de mathématiques de la SMC). Springer, Cham
- Szudzik, Mattew (31 July 2018). "Continuum Hypothesis". Wolfram Mathworld. Wolfram Web Resources. Retrieved 15 August 2018.
- aleph numbers at PlanetMath.org.
- Harris, Kenneth (April 6, 2009). "Math 582 Intro to Set Theory, Lecture 31" (PDF). Department of Mathematics, University of Michigan. Archived from the original (PDF) on March 4, 2016. Retrieved September 1, 2012.