Continuum hypothesis

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following: There is no set whose size is strictly between that of the integers and that of the real numbers. Or mathematically speaking, noting that the cardinality for the integers is ("aleph-null") and the cardinality of the real numbers is , the continuum hypothesis says: ...

Including:

• Continuum hypothesis - The size of a set
• Continuum hypothesis - Impossibility of proof and disproof
• Continuum hypothesis - Arguments pro and con
• Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia - Continuum hypothesis

To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if there exists a bijection . Intuitively, this means that it is possible to "pair off" elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Hence, the set {banana, appl ...

See also:

Continuum hypothesis, Continuum hypothesis - The size of a set, Continuum hypothesis - Impossibility of proof and disproof, Continuum hypothesis - Arguments pro and con, Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia II - Continuum hypothesis - The size of a set

To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if there exists a bijection . Intuitively, this means that it is possible to "pair off" elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}. With infinite sets such as the set of intege ...

See also:

Continuum hypothesis, Continuum hypothesis - The size of a set, Continuum hypothesis - Impossibility of proof and disproof, Continuum hypothesis - Arguments pro and con, Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia II - Continuum hypothesis - The size of a set

In mathematics, the cardinality of a set is a measure of the "number of elements of the set". There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. Cardinality - Comparing sets. We say that two sets A and B have the same cardinality if there exists a bijection, i.e. a injective and surjective function, from A to B. For example, the set E = {2, 4, 6, ...} of positi ...

Including:

• Cardinality - Comparing sets
• Cardinality - Countable and uncountable sets
• Cardinality - Cardinal numbers
• Cardinality - Examples and other properties

Read more here: » Cardinality: Encyclopedia - Cardinality

In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. Not all epistemologists agree that any axioms, understood in that sense, exist. In mathematics, an axiom is not necessarily a self-evident truth but rather, a formal logical expression used in a deduction to yield further results. Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms. Axiom - Etymology. The word axiomIncluding:

• Axiom - Etymology
• Axiom - Mathematics
• Axiom - Logical axioms
• Axiom - Non-logical axioms
• Axiom - Role in mathematical logic
• Axiom - Further discussion

Read more here: » Axiom: Encyclopedia - Axiom

In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph (). The cardinality of the natural numbers is aleph-null () (also aleph-naught, aleph-nought); 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 α, as will be described below. The concept goes back to Georg Cantor, who defined the notion of cardinality and realized t ...

Including:

• Aleph number - Aleph-null
• Aleph number - Aleph-one
• Aleph number - The continuum hypothesis
• Aleph number - Aleph-ω
• Aleph number - Aleph-α for general α

Read more here: » Aleph number: Encyclopedia - Aleph number

is the cardinality of the set of all countably infinite ordinal numbers, called ω1 or Ω. This definition implies (already in ZF, Zermelo-Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice (AC) 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 AC we can show one of the most useful properties of the set Ω (the standard example of a set o ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α, Aleph number - Fixed points of aleph, Aleph number - Popular culture

Read more here: » Aleph number: Encyclopedia II - Aleph number - Aleph-one

In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms. Axiom - Logical axioms. These are certain formulas in a language that are universally valid, that is, formulas that are satisfied by every structure under every variable assignment function . More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values. Usually one takes as logical axioms some minimal set of tautologies that is sufficient for proving all ...

See also:

Axiom, Axiom - Etymology, Axiom - Mathematics, Axiom - Logical axioms, Axiom - Non-logical axioms, Axiom - Role in mathematical logic, Axiom - Further discussion

Read more here: » Axiom: Encyclopedia II - Axiom - Mathematics

is the cardinality of the set of all countably infinite ordinal numbers, called ω1 or Ω. This definition implies (already in ZF, Zermelo-Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice (AC) 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 AC we can show one of the most useful properties of the set Ω (the standard example of a set o ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α

Read more here: » Aleph number: Encyclopedia II - Aleph number - Aleph-one

We say that two sets A and B have the same cardinality if there exists a bijection, i.e. a injective and surjective function, from A to B. For example, the set E = {2, 4, 6, ...} of positive even numbers has the same cardinality as the set N = {1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E. We say that a set A has cardinality greater than or equal to the cardinality of B (and B has cardinality l ...

See also:

Cardinality, Cardinality - Comparing sets, Cardinality - Countable and uncountable sets, Cardinality - Cardinal numbers, Cardinality - Examples and other properties

Read more here: » Cardinality: Encyclopedia II - Cardinality - Comparing sets

Continuum can refer to: The continuous black body radiation spectrum. continuum, a "fretless keyboard" continuum, or mathematical sets (such as the real line) that can be contrasted with the properties of discrete spaces Continuum, a game client for the SubSpace computer game Continuum, a publishing house Continuum, an album by John Mayer Continuum, a musical composition by György Ligeti Continuum, a branch of physics that deals with continuous ...

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CH can mean: Ch (digraph), considered a single letter in several Latin-alphabet languages CH (television system) four Global Network stations in Canada Ch interpreter, an interpreted superset of the C programming language Conservation Halton, in Ontario, Canada Continuum hypothesis, in set theory Horsepower (ch, from the French chevaux, "horses") Order of the Companions of Honour, post-nominal People's Republic of China (FIPS and NATO country code CH) Switzerland (licence plate and ISO cou

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For any ordinal α we have In many cases ℵα 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 aleph function. The first such is the limit of the sequence Any inaccessible cardinal is a fixed point of the aleph function as well. ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α, Aleph number - Fixed points of aleph, Aleph number - Popular culture

Read more here: » Aleph number: Encyclopedia II - Aleph number - Fixed points of aleph

Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rig ...

Including:

• Axiomatic set theory - The origins of rigorous set theory
• Axiomatic set theory - Axioms for set theory
• Axiomatic set theory - Independence in ZFC
• Axiomatic set theory - Set theory ZFC foundations for mathematics
• Axiomatic set theory - Well-foundedness and hypersets
• Axiomatic set theory - Objections to set theory

Read more here: » Axiomatic set theory: Encyclopedia - Axiomatic set theory

The cardinality of the set of real numbers is . It is not clear where this number fits in the aleph number hierarchy. It follows from ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that the celebrated continuum hypothesis, CH, is equivalent to the identity CH is independent of ZFC: it can be neither proven nor disproven within the context of that axiom system. That it is consistent with ZFC was demonstrated by Kurt Gödel in 1940; that it is i ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α, Aleph number - Fixed points of aleph, Aleph number - Popular culture

Read more here: » Aleph number: Encyclopedia II - Aleph number - The continuum hypothesis

To define aleph-α for arbitrary ordinal number α, we need the successor cardinal operation, which assigns to any cardinal number ρ the next bigger cardinal ρ + . We can then define the aleph numbers as follows and for λ, an infinite limit ordinal, ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α, Aleph number - Fixed points of aleph, Aleph number - Popular culture

Read more here: » Aleph number: Encyclopedia II - Aleph number - Aleph-α for general α

Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number is the smallest upper bound of Aleph-ω 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 th ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α

Read more here: » Aleph number: Encyclopedia II - Aleph number - Aleph-ω

The cardinality of the set of real numbers is . It is not clear where this number fits in the aleph number hierarchy. It follows from ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that the celebrated continuum hypothesis, CH, is equivalent to the identity CH is independent of ZFC: it can be neither proven nor disproven within the context of that axiom system. That it is consistent with ZFC was demonstrated by Kurt Gödel in 1940; that it is i ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α

Read more here: » Aleph number: Encyclopedia II - Aleph number - The continuum hypothesis

Note that, up until this point, we have only defined the term "cardinality" in a strictly functional role: we have not actually defined the "cardinality" of a set as a specified object itself. We now outline such an approach. The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all those sets which have the same cardinality as A. There are then two main approaches to the def ...

See also:

Cardinality, Cardinality - Comparing sets, Cardinality - Countable and uncountable sets, Cardinality - Cardinal numbers, Cardinality - Examples and other properties

Read more here: » Cardinality: Encyclopedia II - Cardinality - Cardinal numbers

To define aleph-α for arbitrary ordinal number α, we need the successor cardinal operation, which assigns to any cardinal number ρ the next bigger cardinal ρ + . We can then define the aleph numbers as follows and for λ, an infinite limit ordinal, ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α

Read more here: » Aleph number: Encyclopedia II - Aleph number - Aleph-α for general α