Aleph number

 ( Cardinal Numbers ) 

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of . They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).

The smallest cardinality of an infinite set is that of the , denoted by $\backslash aleph\_0$ (read aleph-nought, aleph-zero, or aleph-null); the next larger cardinality of a well-order set is $\backslash aleph\_1,$ then $\backslash aleph\_2,$ then $\backslash aleph\_3,$ and so on. Continuing in this manner, it is possible to define an infinite cardinal number $\backslash aleph\_\{\backslash alpha\}$ for every ordinal number $\backslash alpha,$ as described below.

The concept and notation are due to Georg Cantor,

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who defined the notion of cardinality and realized that infinite sets can have different cardinalities.

The aleph numbers differ from the infinity ($\backslash infty$) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "divergent series to infinity" or "increases without bound"), or as an extreme point of the extended real number line.

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Aleph-zero $\backslash aleph\_0$ ( aleph-nought, 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 ordinal number, called $\backslash omega$ or $\backslash omega\_0$ (where $\backslash omega$ is the lowercase Greek letter omega), also has cardinality $\backslash aleph\_0$. A set has cardinality $\backslash aleph\_0$ 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 natural numbers, irrespective of including or excluding zero,
• the set of all ,
• any infinite subset of the integers, such as the set of all square numbers or the set of all prime numbers,
• the set of all ,
• the set of all constructible numbers (in the geometric sense),
• the set of all ,
• 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 of any given countably infinite set.

Among the countably infinite sets are certain infinite ordinals, including for example $\backslash omega$, $\backslash omega+1$, $\backslash omega\; \backslash cdot\; 2$, $\backslash omega^2$, $\backslash omega^\backslash omega$, and $\backslash varepsilon\_0$. For example, the sequence (with order type $\backslash omega\; \backslash cdot\; 2$) of all positive odd integers followed by all positive even integers $\backslash \{1,\; 3,\; 5,\; 7,\; 9,\; \backslash cdots;\; 2,\; 4,\; 6,\; 8,\; 10,\; \backslash cdots\backslash \}$ is an ordering of the set (with cardinality $\backslash aleph\_0$) of positive integers.

If the axiom of countable choice (a weaker version of the axiom of choice) holds, then $\backslash aleph\_0$ is smaller than any other infinite cardinal, and is therefore the (unique) least infinite ordinal.

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Aleph-one $\backslash aleph\_1$ is the cardinality of the set of all countable . This set is denoted by $\backslash omega\_1$ (or sometimes Ω). The set $\backslash omega\_1$ is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, $\backslash aleph\_1$ is the smallest cardinality that is larger than $\backslash aleph\_0,$ the smallest infinite cardinality.

The definition of $\backslash aleph\_1$ implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between $\backslash aleph\_0$ and $\backslash aleph\_1.$ If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus $\backslash aleph\_1$ is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set : Any countable subset of $\backslash omega\_1$ has an upper bound in $\backslash omega\_1$ (this follows from the fact that the union of a countable number of countable sets is itself countable). This fact is analogous to the situation in $\backslash aleph\_0$: Every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.

An example application of the ordinal $\backslash omega\_1$ 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 (, group theory, 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 $\backslash omega\_1.$

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Continuum hypothesis The cardinality of the set of (cardinality of the continuum) is 2^{$\backslash aleph\_0$}. It cannot be determined from ZFC (Zermelo–Fraenkel set theory augmented 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

$2^\{\backslash aleph\_0\}\; =\; \backslash aleph\_1$.

The CH states that there is no set whose cardinality is strictly between that of the natural numbers 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 consistency). 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.

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Aleph-omega Aleph-omega is $\backslash aleph\_\backslash omega\; =\; \backslash sup\backslash \{\backslash aleph\_n|\; n\; \backslash in\; \backslash omega\backslash \}\; =\; \backslash sup\backslash \{\backslash aleph\_n|\; n\; \backslash in\; \backslash \{0,\; 1,\; 2,\backslash cdots\backslash \}\backslash \}$ where the smallest infinite ordinal is denoted as $\backslash omega$. That is, the cardinal number $\backslash aleph\_\backslash omega$ is the least upper bound of $\backslash sup\backslash \{\backslash aleph\_n|\; n\; \backslash in\; \backslash \{0,\; 1,\; 2,\backslash cdots\backslash \}\backslash \}$.

Notably, $\backslash aleph\_\backslash omega$ 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 $2^\{\backslash aleph\_0\}$: For any natural number $n\; \backslash ge\; 1$, we can consistently assume that $2^\{\backslash aleph\_0\}\; =\; \backslash aleph\_n$, and moreover it is possible to assume that $2^\{\backslash aleph\_0\}$ is as least as large as any cardinal number we like. The main restriction ZFC puts on the value of $2^\{\backslash aleph\_0\}$ is that it cannot equal certain special cardinals with cofinality $\backslash aleph\_0$. An uncountably infinite cardinal $\backslash kappa$ having cofinality $\backslash aleph\_0$ means that there is a (countable-length) sequence $\backslash kappa\_0\; \backslash le\; \backslash kappa\_1\; \backslash le\; \backslash kappa\_2\; \backslash le\; \backslash cdots$ of cardinals whose limit (i.e. its least upper bound) is $\backslash kappa$ (see Easton's theorem). As per the definition above, $\backslash aleph\_\backslash omega$ is the limit of a countable-length sequence of smaller cardinals.

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Aleph-α for general α To define $\backslash aleph\_\backslash alpha$ for arbitrary ordinal number $\backslash alpha$, we must define the successor cardinal operation, which assigns to any cardinal number $\backslash rho$ the next larger cardinal $\backslash rho^\{+\}$ (if the axiom of choice holds, this is the (unique) next larger cardinal).

We can then define the aleph numbers as follows:

$\backslash aleph\_0\; =\; \backslash omega$

$\backslash aleph\_\{\backslash alpha+1\}\; =\; (\backslash aleph\_\{\backslash alpha\})^\{+\}$

for $\backslash lambda$ an infinite limit ordinal,

The $\backslash alpha$-th infinite initial ordinal is written $\backslash omega\_\backslash alpha$. Its cardinality is written $\backslash aleph\_\backslash alpha$.

Informally, the aleph function $\backslash aleph\; :\; \backslash text\{On\}\; \backslash rightarrow\; \backslash text\{Cd\}$ is a bijection from the ordinals to the infinite cardinals. Formally, in ZFC, $\backslash aleph$ is not a function, but a function-like class, as it is not a set (due to the Burali-Forti paradox).

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Fixed points of omega For any ordinal $\backslash alpha$ we have $\backslash alpha\; \backslash le\; \backslash omega\_\backslash alpha$.

In many cases $\backslash omega\_\backslash alpha$ is strictly greater than α. For example, it is true for any successor Ordinal number: $\backslash alpha\; +\; 1\; \backslash le\; \backslash omega\_\{\backslash alpha\; +\; 1\}$ 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

$\backslash omega,\; \backslash omega\_\{\backslash omega\},\; \backslash omega\_\{\backslash omega\_\{\backslash omega\}\},\; \backslash cdots$

which is sometimes denoted $\backslash omega\_\{\backslash omega\_\{\backslash ddots\}\}$.

Any weakly inaccessible cardinal is also a fixed point of the aleph function.

This can be shown in ZFC as follows. Suppose $\backslash kappa\; =\; \backslash aleph\_\{\backslash lambda\}$ is a weakly inaccessible cardinal. If $\backslash lambda$ were a successor ordinal, then $\backslash aleph\_\{\backslash lambda\}$ would be a successor cardinal and hence not weakly inaccessible. If $\backslash lambda$ were a limit ordinal less than $\backslash kappa$ then its cofinality (and thus the cofinality of $\backslash aleph\_\backslash lambda$) would be less than $\backslash kappa$ and so $\backslash kappa$ would not be regular and thus not weakly inaccessible. Thus $\backslash lambda\; \backslash ge\; \backslash kappa$ and consequently $\backslash lambda\; =\; \backslash kappa$ which makes it a fixed point.

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Role of axiom of choice 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 .

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 $\backslash text\{card\}(S)$ to be the set of sets with the same cardinality as $S$ of minimum possible rank. This has the property that $\backslash text\{card\}(S)\; =\; \backslash text\{card\}(T)$ if and only if $S$ and $T$ have the same cardinality. (The set $\backslash text\{card\}(S)$ does not have the same cardinality of $S$ in general, but all its elements do.)

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See also

• Beth number
• Gimel function
• Regular cardinal
• Infinity
• Transfinite number
• Ordinal number

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Notes

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External links

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