Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X.

Definitions of the hyper-real line within non-standard analysis ( the subject area dealing with such numbers ) overwhelmingly include the usual, uncountable set of real numbers as a subset.

An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset.

* If an uncountable set X is a subset of set Y, then Y is uncountable.

The Cantor set is an uncountable subset of R. The Cantor set is a fractal and has Hausdorff dimension greater than zero but less than one ( R has dimension one ).

This is an example of the following fact: any subset of R of Hausdorff dimension strictly greater than zero must be uncountable.

Since T is uncountable, this subset of R must be uncountable.

Or let X be an uncountable set, and let a subset of X be negligible if it is countable.

On the other hand, it has no derivative at any point in an uncountable subset of the Cantor set containing the interval endpoints described above.

Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset with the property that all of its translates by G are disjoint from X and from each other.

The so-called anti-diagonal is an uncountable discrete subset of this space, and this is a non-separable subset of the separable space.

If is a regular, uncountable cardinal, is a stationary subset of, and is regressive ( that is, < math > f (

In mathematics, particularly in set theory, if is a regular uncountable cardinal then, the filter of all sets containing a club subset of, is a-complete filter closed under diagonal intersection called the club filter.

A continuous random variable maps outcomes to values of an uncountable set ( e. g., the real numbers ).

Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable.

In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable.

But Cantor's diagonal argument proves that the real numbers ( and therefore also the complex numbers ) are uncountable ; so the set of all transcendental numbers must also be uncountable.

The best known example of an uncountable set is the set R of all real numbers ; Cantor's diagonal argument shows that this set is uncountable.

The number of elements ( either real numbers or points ) in all the above-mentioned sets is uncountable, as it is strictly greater than the number of natural numbers.

is a nowhere dense set ( it is without interior points ) and an uncountable set ( of the same cardinality as the real numbers ).

A constructive version of " the famous theorem of Cantor, that the real numbers are uncountable " is: " Let

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.

The set of possible input values may be infinitely large, and may possibly be continuous and therefore uncountable ( such as the set of all real numbers, or all real numbers within some limited range ).

*" Theorem 1 is the famous theorem of Cantor, that the real numbers are uncountable.

This theory consists of a finite theory characterizing the real numbers as a complete Archimedean ordered field plus an axiom saying that the domain is of the first uncountable cardinality.

It follows from the theorem that the theory of ( N, +, ×, 0, 1 ) ( the theory of true first-order arithmetic ) has uncountable models, and that the theory of ( R, +, ×, 0, 1 ) ( the theory of real closed fields ) has a countable model.

Another consequence that was considered particularly troubling is the existence of a countable model of set theory, which nevertheless must satisfy the sentence saying the real numbers are uncountable.

For example, the set of rational numbers is countable ( and has measure zero, and is a meagre set ), but the set of irrational numbers is uncountable ( and has full measure, and is a comeagre set ): " almost all " real numbers are irrational, in these senses.

* Conservation: a proof that any result about " real objects " obtained using reasoning about " ideal objects " ( such as uncountable sets ) can be proved without using ideal objects.

An example of a space which is not first-countable is the cofinite topology on an uncountable set ( such as the real line ).

The choice of the Polish space in the third clause above is not very important ; it could be replaced in the definition by a fixed uncountable Polish space, say Baire space or Cantor space or the real line.

" Kronecker even objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum.

The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.

The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers and the set of all subsets of the set of natural numbers.

A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by Ω or ω < sub > 1 </ sub >.

Because these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.

Not every countably compact space is compact ; an example is given by the first uncountable ordinal with the order topology.

The four-color theorem applies not only to finite planar graphs, but also to infinite graphs that can be drawn without crossings in the plane, and even more generally to infinite graphs ( possibly with an uncountable number of vertices ) for which every finite subgraph is planar.

The converse does not hold: there exist metric spaces that are not second countable, for example, an uncountable set endowed with the discrete metric.

In linguistics, a mass noun ( also uncountable noun or non-count noun ) is a noun with the syntactic property that any quantity of it is treated as an undifferentiated unit, rather than as something with discrete subsets.

As with English, most uncountable nouns are grammatically treated as singular, though some are plural, such as les mathématiques (" mathematics "), and some nouns that are uncountable in English are countable in French, such as une information (" a piece of information "), or une nouvelle (" a piece of news, a news item ").

The set of all integer sequences is uncountable ( with cardinality equal to that of the continuum ); thus, almost all integer sequences are uncomputable and cannot be defined.

Thus T is uncountable: it cannot be placed in one-to-one correspondence with the set of natural numbers.

Note that neither of these statements implies the other, since there are complete metric spaces which are not locally compact ( the irrational numbers with the metric defined below ; also, any Banach space of infinite dimension ), and there are locally compact Hausdorff space which are not metrizable ( for instance, any uncountable product of non-trivial compact Hausdorff spaces is such ; also, several function spaces used in Functional Analysis ; the uncountable Fort space ).

BCT1 also shows that every complete metric space with no isolated points is uncountable.

Throughout continental Europe with the exception of Germany, peperone is a common word for various types of capsicum, including bell peppers and a small, spicy and often pickled pepper known as peperoncino or also sometimes peperone piccante peperoncini or banana peppers in the U. S. Unlike in Europe, the English word pepperoni is used as a singular uncountable noun.

However S < sup > 2 </ sup > × R cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of solvable non-unimodular Bianchi groups, most of which give model geometries with no compact representatives.

Then 0 < sup >#</ sup > is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with c < sub > i </ sub > interpreted as the uncountable cardinal ℵ < sub > i </ sub >.

Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter.

The Roman poet Ovid, however, states in his Metamorphoses that Morpheus is a son of Hypnos, rather than his brother ( it does not mention the identity of the mother ), and multiplies the Oneiroi into an uncountable host of spirits, with Morpheus, Icelus and Phantasos being merely the most prominent among them.

We cannot eliminate the Hausdorff condition ; a countable set with the indiscrete topology is compact, has more than one point, and satisfies the property that no one point sets are open, but is not uncountable.

In practice étale cohomology is used mainly for constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with a little care it can be constructed in this case without using any uncountable sets, and this can easily be done in ZFC ( and even in much weaker theories ).

* The space of ordinals at most equal to the first uncountable ordinal with the order topology is a compact topological space.

Options included up to five different styles of wheels, various suspension set-ups, at least three different seat styles ( comfort, sport, racing ), uncountable upholstery options including the possibility to have almost any interior element of the car covered with leather, wood or carbon fiber, and various hi-fi systems including digital sound processing.

In 1970, Solovay constructed Solovay's model, which shows that it is consistent with standard set theory, excluding uncountable choice, that all subsets of the reals are measurable.