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 numberℵα for every ordinal numberα, as described below.
The aleph numbers differ from the infinity (∞) 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 "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.
Aleph-zero
ℵ0 (aleph-zero, also aleph-nought 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 ω or ω0 (where ω is the lowercase Greek letter omega), has cardinality ℵ0. A set has cardinality ℵ0 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 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 ε0 are among the countably infinite sets.[6] For example, the sequence (with ordinalityω⋅2) of all positive odd integers followed by all positive even integers
{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.
ℵ1 is the cardinality of the set of all countable ordinal numbersω1 (or sometimes Ω). The set ω1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ℵ1 is distinct from ℵ0. The definition of ℵ1 implies (in ZF, Zermelo–Fraenkel set theorywithout the axiom of choice) that no cardinal number is between ℵ0 and ℵ1. If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus ℵ1 is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set ω1: Any countable subset of ω1 has an upper bound in ω1 (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 ℵ0: Every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.
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.
The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers.[8] 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.[7][9]
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 least upper bound of
{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 cofinalityℵ0. 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).
The α-th infinite initial ordinal is written ωα. Its cardinality is written ℵα.
Informally, the aleph functionℵ:On→Cd is a bijection from the ordinals to the infinite cardinals. Formally, in ZFC, ℵ is not a function, but a function-like class, as it is not a set (due to the Burali-Forti paradox).
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 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.
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.)
↑ In older mathematics books, the letter aleph is often printed upside down by accident – for example, in Sierpiński (1958)[3]:402 the letter aleph appears both the right way up and upside down – partly because a monotype matrix for aleph was mistakenly constructed the wrong way up.[4]
↑ Weisstein, Eric W. "Aleph". mathworld.wolfram.com. Retrieved 2020-08-12.
↑ Sierpiński, Wacław (1958). Cardinal and Ordinal Numbers. Polska Akademia Nauk Monografie Matematyczne. Vol.34. Warsaw, PL: Państwowe Wydawnictwo Naukowe. MR0095787.
↑ 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 (updateded.). Providence, RI: American Mathematical Society. p.16. ISBN0-8218-0053-1. MR0553111.
↑ Harris, Kenneth A. (April 6, 2009). "Lecture31"(PDF). Department of Mathematics. kaharris.org. Intro to Set Theory. University of Michigan. Math582. Archived from the original(PDF) on March 4, 2016. Retrieved September 1, 2012.
In mathematics, specifically set theory, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states that
there is no set whose cardinality is strictly between that of the integers and the real numbers,
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter (aleph) marked with subscript indicating their rank among the infinite cardinals.
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set contains 3 elements, and therefore has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set may also be called its size, when no confusion with other notions of size is possible.
In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than aleph-null, the cardinality of the natural numbers.
In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite was coined in 1895 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 numbers. Nevertheless, the term transfinite also remains in use.
The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely:
In mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear.
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that is a regular cardinal if and only if every unbounded subset has cardinality . Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular.
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers, conventionally written , where is the Hebrew letter beth. The beth numbers are related to the aleph numbers, but unless the generalized continuum hypothesis is true, there are numbers indexed by that are not indexed by .
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by or .
In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory T in a language we write I(T, κ) for the number of models of T (up to isomorphism) of cardinality κ. The spectrum problem is to describe the possible behaviors of I(T, κ) as a function of κ. It has been almost completely solved for the case of a countable theory T.
In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal.
In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal κ, a κ-Aronszajn tree is a tree of height κ in which all levels have size less than κ and all branches have height less than κ. They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by Kurepa (1935).
In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum and combinatorics on successors of singular cardinals.
In mathematics, the first uncountable ordinal, traditionally denoted by or sometimes by , is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum of all countable ordinals. When considered as a set, the elements of are the countable ordinals, of which there are uncountably many.
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals aimed to extend enumeration to infinite sets.
In the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between , and the cardinality of the continuum, that is, the cardinality of the set of all real numbers. The latter cardinal is denoted or . A variety of such cardinal characteristics arise naturally, and much work has been done in determining what relations between them are provable, and constructing models of set theory for various consistent configurations of them.
This is a glossary of set theory.
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