Aleph number
In
The cardinality of the natural numbers is ℵ0 (read aleph-nought or aleph-zero or aleph-null), the next larger cardinality of a well-ordered set is aleph-one ℵ1, then ℵ2 and so on. Continuing in this manner, it is possible to define a cardinal number ℵα for every ordinal number α, as described below.
The concept and notation are due to Georg Cantor,[5] who defined the notion of cardinality and realized that
The aleph numbers differ from the
Aleph-zero
ℵ0 (aleph-zero, also aleph-nought or aleph-null) is the cardinality of the set of all natural numbers, and is an
- the set of natural numbers, irrespective of including or excluding zero,
- the set of all integers,
- any infinite subset of the integers, such as the set of all prime numbers,
- 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 computable functions,
- the set of all binary strings of finite length, and
- the set of all finite subsets of any given countably infinite set.
These infinite ordinals: ω, ω + 1, ω⋅2, ω2, ωω, and
- {1, 3, 5, 7, 9, ...; 2, 4, 6, 8, 10, ...}
is an ordering of the set (with cardinality ℵ0) of positive integers.
If the axiom of countable choice (a weaker version of the axiom of choice) holds, then ℵ0 is smaller than any other infinite cardinal, and is therefore the (unique) least infinite ordinal.
Aleph-one
ℵ1 is the cardinality of the set of all countable
The ordinal ω1 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 σ-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, etc. 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 ω1.
Continuum hypothesis
The
- 2ℵ0 = ℵ1.[7]
The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers.
Aleph-omega
Aleph-omega is
- ℵω = sup{ ℵn | n ∈ ω } = sup{ ℵn | n ∈ {0, 1, 2, ...} }
where the smallest infinite ordinal is denoted ω. That is, the cardinal number ℵω is the
- { ℵn | n ∈ {0, 1, 2, ...} }.
Notably, ℵω 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 2ℵ0: For any natural number n ≥ 1, we can consistently assume that 2ℵ0 = ℵn, and moreover it is possible to assume that 2ℵ0 is as least as large as any cardinal number we like. The main restriction ZFC puts on the value of 2ℵ0 is that it cannot equal certain special cardinals with cofinality ℵ0. An uncountably infinite cardinal κ having cofinality ℵ0 means that there is a (countable-length) sequence κ0 ≤ κ1 ≤ κ2 ≤ ... of cardinals κi < κ whose limit (i.e. its least upper bound) is κ (see Easton's theorem). As per the definition above, ℵω is the limit of a countable-length sequence of smaller cardinals.
Aleph-α for general α
To define ℵα for arbitrary ordinal number α, we must define the successor cardinal operation, which assigns to any cardinal number ρ the next larger well-ordered cardinal ρ+ (if the axiom of choice holds, this is the (unique) next larger cardinal).
We can then define the aleph numbers as follows:
- ℵ0 = ω
- ℵα+1 = (ℵα)+
- ℵλ = ⋃{ ℵα | α < λ } for λ an infinite limit ordinal,
The α-th infinite
Informally, the aleph function ℵ: On → Cd is a bijection from the ordinals to the infinite cardinals. Formally, in
Fixed points of omega
For any ordinal α we have
- α ≤ ωα.
In many cases ωα is strictly greater than α. For example, it is true for any successor ordinal: α + 1 < ωα+1 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
- ω, ωω, ωωω, ...,
which is sometimes denoted ωω....
Any weakly inaccessible cardinal is also a fixed point of the aleph function.[10] This can be shown in ZFC as follows. Suppose κ = ℵλ is a weakly inaccessible cardinal. If λ were a successor ordinal, then ℵλ would be a successor cardinal and hence not weakly inaccessible. If λ were a limit ordinal less than κ then its cofinality (and thus the cofinality of ℵλ) would be less than κ and so κ would not be regular and thus not weakly inaccessible. Thus λ ≥ κ and consequently λ = κ which makes it a fixed point.
Role of axiom of choice
The cardinality of any infinite
Each finite set is well-orderable, but does not have an aleph as its cardinality.
Over ZF, the assumption that the cardinality of each infinite set is an aleph number is equivalent 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.)
See also
Notes
Citations
- ^ "Aleph". Encyclopedia of Mathematics.
- ^ Weisstein, Eric W. "Aleph". mathworld.wolfram.com. Retrieved 2020-08-12.
- MR 0095787.
- ^
Swanson, Ellen; O'Sean, Arlene Ann; Schleyer, Antoinette Tingley (1999) [1979]. Mathematics into type: Copy editing and proofreading of mathematics for editorial assistants and authors (updated ed.). Providence, RI: MR 0553111.
- ^
Miller, Jeff. "Earliest uses of symbols of set theory and logic". jeff560.tripod.com. Retrieved 2016-05-05; who quotes
Dauben, Joseph Warren (1990). Georg Cantor: His mathematics and philosophy of the infinite. Princeton University Press. 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 ...
- Springer-Verlag.
- ^ a b Szudzik, Mattew (31 July 2018). "Continuum Hypothesis". Wolfram Mathworld. Wolfram Web Resources. Retrieved 15 August 2018.
- ^ Weisstein, Eric W. "Continuum Hypothesis". mathworld.wolfram.com. Retrieved 2020-08-12.
- ^ Chow, Timothy Y. (2007). "A beginner's guide to forcing". ].
- ^ Harris, Kenneth A. (April 6, 2009). "Lecture 31" (PDF). Department of Mathematics. kaharris.org. Intro to Set Theory. University of Michigan. Math 582. Archived from the original (PDF) on March 4, 2016. Retrieved September 1, 2012.
External links
- "Aleph-zero", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Weisstein, Eric W. "Aleph-0". MathWorld.