Review of Short Phrases and Links|
This Review contains major "Cardinality"- related terms, short phrases and links grouped together in the form of Encyclopedia article.
- Cardinality is a measure of the size of a set.
- Cardinality is also an area studied for its own sake as part of set theory, particularly in trying to describe the properties of large cardinal s.
- A cardinality is a text string with values chosen from the set .
- The cardinality is a simple number.
- Cardinality is based on possibility for "1-to-1 mapping".
- With Choice, all sets have cardinality, and the cardinal numbers can be well ordered.
- In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality.
- There are two approaches to cardinality --- one which compares sets directly using bijections and injections, and another which uses cardinal numbers.
- Any well-ordered set having that ordinal as its order type has the same cardinality.
- That is, there is a Dedekind infinite set A such that the cardinality of A is m.
- Without the axiom of choice, there might exist cardinalities incomparable to aleph-null (such as the cardinality of a Dedekind-finite infinite set).
- Sets A and B have the same cardinality if there is a one-to-one function f with domain A and range B. We denote this by | A | = | B |. Definition.
- This motivates the definition of an infinite set being any set which has a proper subset of the same cardinality; in this case is a proper subset of .
- The alephs are defined in terms of successor cardinality.
- Any set X with cardinality less than that of the natural numbers (| X | < | N |) is said to be a finite set.
- In set theory, The Hebrew aleph glyph is used as the symbol to denote the aleph numbers, which represent the cardinality of infinite sets.
- The transfinite numbers specify the cardinality of infinite sets.
- Without realizing it, the manager of the Hotel Infinity taught us an important lesson in the cardinality of infinite sets.
- Sameness of cardinality is sometimes referred to as equipotence, equipollence, or equinumerosity.
- This set clearly has cardinality (the natural bijection between the set of binary sequences and P( N) is given by the indicator function).
- Formally, assuming the axiom of choice, cardinality of a set X is the least ordinal α such that there is a bijection between X and α.
- Assuming the axiom of choice holds, the law of trichotomy holds for cardinality, so we have the following definitions.
- Cardinality as defined by my way are more general than cardinality defined by Von Neumann given that the axiom of choice is not in action.
- This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power set of the integers.
- Cantor's diagonal argument shows that the integers and the continuum do not have the same cardinality.
- The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets.
- The cardinality of the set of natural numbers, Aleph null, has many of the properties of a large cardinal.
- The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, .
- In mathematics, a countable set is a set with the same cardinality (i.e., number of elements) as some subset of the set of natural numbers.
- It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.
- Cantor first established cardinality as an instrument to compare finite sets; e.g.
- For finite sets the cardinality is given by a natural number, being simply the number of elements in the set.
- If a finite set S has cardinality n then the power set of S has cardinality 2 n.
- The concept goes back to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities.
- Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets.
- The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874 – 1884.
- It is the 2nd beth number, and is the result of cardinal exponentiation when 2 is raised to the power of c, the cardinality of the continuum.
- The second beth number, , is the cardinality of the set of all subsets of the real line.
- The last inequality implies that the cofinality of the cardinality of the continuum must be uncountable.
- The cofinality of A is the least cardinality of a cofinal subset.
- One might naturally wonder whether , the cardinality of the reals, is equal to or if it is strictly larger.
- The cardinality of R is often called the cardinality of the continuum and denoted by c or ( beth-one).
- Formally, an uncountable set is defined as one whose cardinality is strictly greater than ( aleph-null), the cardinality of the natural numbers.
- Similarly, the question of whether there exists a set whose cardinality is between s and P( s) for some s, leads to the generalized continuum hypothesis.
- The cardinality of the power set is given by the formula 2 N, where N is the number elements in the set.
- This new cardinal number, called the cardinality of the continuum, was termed c by Cantor.
- One can also easily study cardinality without referring to cardinal numbers.
- Aleph-one is the cardinality of the set of all countably infinite ordinal numbers.
- Any set of cardinality Aleph-null can be put into a direct one-to-one correspondence (see bijection) with the integers, and thus is a countably infinite set.
- The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers.
- It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described.
- The latter cardinal number is also often denoted by c; it is the cardinality of the set of real numbers, or the continuum, whence the name.
- Now in this model, all those sets of cardinality kappa which were proper classes in the former theory are now simply sets in this theory.
- Despite the paradoxes and some of the problems one faces in defining and dealing with cardinality, choice, proper classes, etc.
- It is thus said that two sets with the same cardinality are, respectively, equipotent, equipollent, or equinumerous.
- If two sets are given, then they either have the same cardinality, or one has a smaller cardinality than the other.
- For instance, two sets may each have an infinite number of elements, but one may have a greater cardinality.
- In fact, without AC, it's possible to have a family of sets of cardinality three without having a choice function for that family.
- The cardinality of a set A is defined as its equivalence class under equinumerosity.
- I'm not sure why the class of a transitive sets with cardinality omega should not be a set either.
- In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets.
- In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets.
- The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite.
- The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal.
- For example, the cardinality of -- 0 = -- is , which is also the cardinality of ---- or -- 0 (all are countable ordinals).
- A rooted Kripke frame of has cardinality i , a limit ordinal.
- Any cardinal number can be identified with the smallest ordinal number of cardinality .
- Any set X with cardinality greater than that of the natural numbers (| X | | N |, for example | R | = | N |) is said to be uncountable.
- Aleph-two is the cardinality of the set of all ordinal numbers of cardinality no greater than aleph-one.
- For instance, I distinguish a vastly greater number of different infinities than cardinality simply by ordering formulas on a unit infinity.
- Because these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.
- The cardinality of the natural numbers is (aleph-null, also aleph-naught or aleph-zero) the next larger cardinality is aleph-one , then and so on.
- Second, it is false that all sets have a larger cardinality than their proper subsets.
- The concepts of intersection, union, subset, cartesian product and cardinality of multisets are defined by the above formulas.
- Here you can find out something about sets, basic operations on sets, functions, relations, Cartesian product, denumerable sets and-cardinality.
- The ---th infinite initial ordinal is written -- --. Its cardinality is written .
- The smallest infinite cardinality is that of the natural numbers ().
- The smallest infinite cardinal is the cardinality of a countable set.
- This article incorporates material from cardinality of the continuum on PlanetMath, which is licensed under the GFDL.
- PlanetMath: cardinality of the continuum (Site not responding.
- Each set in this sequence has cardinality strictly greater than the one preceding it, because of Cantor's theorem.
- The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis.
- If D is uncountable, but of cardinality strictly smaller than and the poset has the countable chain condition, we can instead use Martin's axiom.
- The lowest transfinite ordinal number is --. The first transfinite cardinal number is aleph-null, , the cardinality of the infinite set of the integers.
- Cantor called this first transfinite cardinality א 0, It is now known as the continuum, and is denoted by the letter c.
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