75 Sentences With "uncountably" | Random Sentence Generator

"There are uncountably many solar systems out there and the search for life beyond ours needs direction," Sugita said.

It is as though nature has an inclination to generate its random surfaces using a very particular kind of die (one with an uncountably infinite number of sides).

The set of all real numbers is an uncountably infinite set. The set of all irrational numbers is also an uncountably infinite set.

We build an uncountably categorical but not countably categorical theory whose only computably presentable model is the saturated one.

The existence of transcendental numbers was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that π is transcendental. Finally, Cantor showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite, so there is an uncountably infinite number of transcendental numbers.

Deleting any of the subsets of Z from the series for λ creates uncountably many distinct Liouville numbers, whose nth roots are U-numbers of degree n.

He is also sometimes known as Wanli'er, which has similar meaning, as the Chinese word wàn—like the English "myriad"—simultaneously means the number 10,000 and "innumerable" or "uncountably vast".

In the case of Rn with n ≠ 4, the number of these types is one, whereas for n = 4, there are uncountably many such types. One refers to these by exotic R4.

A continuous game is a mathematical concept, used in game theory, that generalizes the idea of an ordinary game like tic-tac-toe (noughts and crosses) or checkers (draughts). In other words, it extends the notion of a discrete game, where the players choose from a finite set of pure strategies. The continuous game concepts allows games to include more general sets of pure strategies, which may be uncountably infinite. In general, a game with uncountably infinite strategy sets will not necessarily have a Nash equilibrium solution.

In a Cartesian product of uncountably many compact Hausdorff spaces with more than one point, a point is never a Baire set, in spite of the fact that it is closed, and therefore a Borel set.

If there is a surjection from to that is not injective, then no surjection from to is injective. In fact no function of any kind from to is injective. This is not true for infinite sets: Consider the function on the natural numbers that sends 1 and 2 to 1, 3 and 4 to 2, 5 and 6 to 3, and so on. There is a similar principle for infinite sets: If uncountably many pigeons are stuffed into countably many pigeonholes, there will exist at least one pigeonhole having uncountably many pigeons stuffed into it.

Only a subset of such real number strings (albeit a countably infinite subset) contains the entirety of Hamlet (assuming that the text is subjected to a numerical encoding, such as ASCII). Meanwhile, there is an uncountably infinite set of strings which do not end in such repetition; these correspond to the irrational numbers. These can be sorted into two uncountably infinite subsets: those which contain Hamlet and those which do not. However, the "largest" subset of all the real numbers are those which not only contain Hamlet, but which contain every other possible string of any length, and with equal distribution of such strings.

A Whiteheadian process is most importantly characterized by extension in space-time, marked by a continuum of uncountably many points in a Minkowski or a Riemannian space-time. The word 'event', indicating a Whiteheadian actual entity, is not being used in the sense of a point event.

'' Their central idea is that, just as Turing modelled the human computer in 1936 by a Turing machine, they model a technician, performing an experimental procedure that governs an experiment, by a Turing machine. They show that the mathematics of computation imposes fundamental limits on what can be measured in classical physics: ::There is a simple Newtonian experiment to measure mass, based upon colliding particles, for which there are uncountably many masses m such that for every experimental procedure governing the equipment it is only possible to determine finitely many digits of m, even allowing arbitrary long run times for the procedure. In particular, there are uncountably many masses that cannot be measured.

Alternatively there is a sharper form of the conjecture that states that any countable complete T with uncountably many countable models will have a perfect set of uncountable models (as pointed out by John Steel, in "On Vaught's conjecture". Cabal Seminar 76—77 (Proc. Caltech-UCLA Logic Sem., 1976—77), pp.

In 1874, he showed that the set of all real numbers is uncountably infinite, but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. For more, see Cantor's first uncountability proof.

In mathematics, an uncountable set (or uncountably infinite set) 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 that of the set of all natural numbers.

Izbicki extended these results in 1959 and showed that there were uncountably many infinite graphs realizing any finite symmetry group. Finally, Johannes de Groot and Sabidussi in 1959/1960 independently proved that any group (dropping the assumption that the group be finite) could be realized as the group of symmetries of an infinite graph.

The analogy is to an egg being scrambled forever, or to typical pairs of atoms behaving in this way. A set S is called a scrambled set if every pair of distinct points in S is scrambled. Scrambling is a kind of mixing. (2) There is an uncountably infinite set S that is scrambled.

In particular, the Ornstein isomorphism theorem does not apply to K-systems, and so the entropy is not sufficient to classify such systems – there exist uncountably many non-isomorphic K-systems with the same entropy. In essence, the collection of K-systems is large, messy and uncategorized; whereas the B-automorphisms are 'completely' described by Ornstein theory.

Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the computable numbers. Because formal languages can have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers. However, by Cantor's diagonal argument, there are uncountably many real numbers, so almost every real number is undefinable.

He constructed a smooth 7-dimensional manifold (called now Milnor's sphere) that is homeomorphic to the standard 7-sphere but not diffeomorphic to it. There are, in fact, 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is the total space of a fiber bundle over the 4-sphere with the 3-sphere as the fiber). More unusual phenomena occur for 4-manifolds. In the early 1980s, a combination of results due to Simon Donaldson and Michael Freedman led to the discovery of exotic R4s: there are uncountably many pairwise non-diffeomorphic open subsets of R4 each of which is homeomorphic to R4, and also there are uncountably many pairwise non- diffeomorphic differentiable manifolds homeomorphic to R4 that do not embed smoothly in R4.

Temporal logics such as computation tree logic (CTL) can be used to specify some LT properties. All linear temporal logic (LTL) formulae are LT properties. By a counting argument, we see that any logic in which each formula is a finite string cannot represent all LT properties, as there must be countably many formulae but there are uncountably many LT properties.

In the mathematical study of stochastic processes, a Harris chain is a Markov chain where the chain returns to a particular part of the state space an unbounded number of times. Harris chains are regenerative processes and are named after Theodore Harris. The theory of Harris chains and Harris recurrence is useful for treating Markov chains on general (possibly uncountably infinite) state spaces.

Moreover, every permutation of the index set leads to a natural isomorphism; they are uncountably many! Another example. A structure of a (simple) graph on a set V = of vertices may be described by means of its adjacency matrix, a (0,1)-matrix of size n×n (with zeros on the diagonal). More generally, for arbitrary V an adjacency function on V × V may be used.

There are uncountably many different tilings of the hyperbolic plane by these tiles, even when they are modified by adding protrusions and indentations to force them to meet edge-to-edge. None of these different tilings are periodic (having a cocompact symmetry group), although some (such as the one in which there exists a line that is completely covered by tile edges) have a one-dimensional infinite symmetry group.

The number of chiefdoms known to the Chinese had been reduced from over a hundred before"Over a hundred" could mean uncountably many. the war to around thirty at the time of Himiko.Thirty is the number of chiefdoms ruled by Himiko. The rebellion also led to the formation of an early polity under Himiko's rule and as such is considered as a turning point between Yayoi and Kofun period.

A single outcome can be a part of many different events. Typically, when the sample space is finite, any subset of the sample space is an event (i.e. all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is uncountably infinite (most notably when the outcome must be some real number).

The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, they cannot both be countable.

Assuming that they form a set in the model, the real numbers definable in the language of set theory over a particular model of ZFC form a field. Each set model M of ZFC set theory that contains uncountably many real numbers must contain real numbers that are not definable within M (without parameters). This follows from the fact that there are only countably many formulas, and so only countably many elements of M can be definable over M. Thus, if M has uncountably many real numbers, we can prove from "outside" M that not every real number of M is definable over M. This argument becomes more problematic if it is applied to class models of ZFC, such as the von Neumann universe . The argument that applies to set models cannot be directly generalized to class models in ZFC because the property "the real number x is definable over the class model N" cannot be expressed as a formula of ZFC.

Set-Theoretic Foundations. A knob on a radio does not take on an uncountably infinite number of possible values -- it takes a finite number of possible values fully limited by the mechanical, physical, nature of the knob itself. There exists no one-to-one mapping between the continuous mathematics used for engineering applications and the physical product(s) produced by the engineering. Indeed, this is one of the core open problems within Philosophy of Mathematics.

There are only countably many algebraic numbers, but there are uncountably many real numbers, so in the sense of cardinality most real numbers are not algebraic. This nonconstructive proof that not all real numbers are algebraic was first published by Georg Cantor in his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers". Non-algebraic numbers are called transcendental numbers. Specific examples of transcendental numbers include π and Euler's number e.

When there is an uncountably infinite collection of formulas, the Axiom of Choice (or at least some weak form of it) is needed. Using the full AC, one can well-order the formulas, and prove the uncountable case with the same argument as the countable one, except with transfinite induction. Other approaches can be used to prove that the completeness theorem in this case is equivalent to the Boolean prime ideal theorem, a weak form of AC.

In classical statistical mechanics, the number of microstates is actually uncountably infinite, since the properties of classical systems are continuous. For example, a microstate of a classical ideal gas is specified by the positions and momenta of all the atoms, which range continuously over the real numbers. If we want to define Ω, we have to come up with a method of grouping the microstates together to obtain a countable set. This procedure is known as coarse graining.

Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining. So the Cantor set is not empty, and in fact contains an uncountably infinite number of points (as follows from the above description in terms of paths in an infinite binary tree). It may appear that only the endpoints of the construction segments are left, but that is not the case either. The number 1⁄4, for example, has the unique ternary form 0.020202... = .

For example the complex numbers C form a two-dimensional vector space over the real numbers R. Likewise, the real numbers R form a vector space over the rational numbers Q which has (uncountably) infinite dimension, if a Hamel basis exists. If V is a vector space over F it may also be regarded as vector space over K. The dimensions are related by the formula :dimKV = (dimFV)(dimKF) For example Cn, regarded as a vector space over the reals, has dimension 2n.

An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry. A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence. More generally, a theorem of Arthur Harold Stone states that the product of uncountably many non-compact metric spaces is never normal.

Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented. However a theorem of Graham Higman states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group. From this we can deduce that there are (up to isomorphism) only countably many finitely generated recursively presented groups. Bernhard Neumann has shown that there are uncountably many non- isomorphic two generator groups.

"... any finite patch that we choose in a tiling will lie inside a single inflated tile if we continue moving far enough up in the inflation hierarchy. This means that anywhere that tile occurs at that level in the hierarchy, our original patch must also occur in the original tiling. Therefore, the patch will occur infinitely often in the original tiling and, in fact, in every other tiling as well." This shows in particular that the number of distinct Penrose tilings (of any type) is uncountably infinite.

He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets).A countable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this terminology is not universally followed, and sometimes "denumerable" is used as a synonym for "countable".

Zermelo set theory (Z) is Zermelo–Fraenkel set theory without the axiom of replacement. It differs from ZF in that Z does not prove that the power set operation can be iterated uncountably many times beginning with an arbitrary set. In particular, Vω \+ ω, a particular countable level of the cumulative hierarchy, is a model of Zermelo set theory. The axiom of replacement, on the other hand, is only satisfied by Vκ for significantly larger values of κ, such as when κ is a strongly inaccessible cardinal.

A computer with access to an infinite tape of data may be more powerful than a Turing machine: for instance, the tape might contain the solution to the halting problem or some other Turing-undecidable problem. Such an infinite tape of data is called a Turing oracle. Even a Turing oracle with random data is not computable (with probability 1), since there are only countably many computations but uncountably many oracles. So a computer with a random Turing oracle can compute things that a Turing machine cannot.

The group G of symmetries of a realization V of an abstract apeirogon P is generated by two reflections, the product of which translates each vertex of P to the next. The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and possibly trivial reflection. Generally, the moduli space of realizations of an abstract polytope is a convex cone of infinite dimension. The realization cone of the abstract apeirogon has uncountably infinite algebraic dimension and cannot be closed in the Euclidean topology.

The infinite dihedral group G of symmetries of a realization V of an abstract apeirogon P is generated by two reflections, the product of which translates each vertex of P to the next. The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and a possibly trivial reflection. Generally, the moduli space of realizations of an abstract polytope is a convex cone of infinite dimension. The realization cone of the abstract apeirogon has uncountably infinite algebraic dimension and cannot be closed in the Euclidean topology.

Four-dimensional manifolds are the most unusual: they are not geometrizable (as in lower dimensions), and surgery works topologically, but not differentiably. Since topologically, 4-manifolds are classified by surgery, the differentiable classification question is phrased in terms of "differentiable structures": "which (topological) 4-manifolds admit a differentiable structure, and on those that do, how many differentiable structures are there?" Four-manifolds often admit many unusual differentiable structures, most strikingly the uncountably infinitely many exotic differentiable structures on R4. Similarly, differentiable 4-manifolds is the only remaining open case of the generalized Poincaré conjecture.

In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence depends on the axiom of choice. In 1970, Robert Solovay constructed a model of Zermelo–Fraenkel set theory without the axiom of choice where all sets of real numbers are Lebesgue measurable, assuming the existence of an inaccessible cardinal (see Solovay model).

That means that if one were to partition the rationals into two non-empty sets A and B where A contains all rationals less than some irrational number (π, say) and B all rationals greater than it, then A has no largest member and B has no smallest member. The field of real numbers, by contrast, is both infinitely divisible and gapless. Any linearly ordered set that is infinitely divisible and gapless, and has more than one member, is uncountably infinite. For a proof, see Cantor's first uncountability proof.

With rational coordinates and the actual Euclidean metric, Euclidean TSP is known to be in the Counting Hierarchy, a subclass of PSPACE. With arbitrary real coordinates, Euclidean TSP cannot be in such classes, since there are uncountably many possible inputs. However, Euclidean TSP is probably the easiest version for approximation. For example, the minimum spanning tree of the graph associated with an instance of the Euclidean TSP is a Euclidean minimum spanning tree, and so can be computed in expected O (n log n) time for n points (considerably less than the number of edges).

In computability theory, an undecidable problem is a type of computational problem that requires a yes/no answer, but where there cannot possibly be any computer program that always gives the correct answer; that is, any possible program would sometimes give the wrong answer or run forever without giving any answer. More formally, an undecidable problem is a problem whose language is not a recursive set; see the article Decidable language. There are uncountably many undecidable problems, so the list below is necessarily incomplete. Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i.e.

Clearly the number of distinct subsets that can be constructed this way is as . Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (or informally, the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum).

For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals for all integers ; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line.

They can also be finite, countably infinite, or uncountably infinite. For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}, commonly written {H, T}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}, commonly written {HH, HT, TH, TT}. If the sample space is unordered, it becomes {{head,head}, {head,tail}, {tail,tail}}. For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).

Although Lindström had only partially developed the hierarchy of quantifiers which now bear his name, it was enough for him to observe that some nice properties of first-order logic are lost when it is extended with certain generalized quantifiers. For example, adding a "there exist finitely many" quantifier results in a loss of compactness, whereas adding a "there exist uncountably many" quantifier to first-order logic results in a logic no longer satisfying the Löwenheim–Skolem theorem. In 1969 Lindström proved a much stronger result now known as Lindström's theorem, which intuitively states that first-order logic is the "strongest" logic having both properties.

FN is the product of countably many copies of F. By Zorn's lemma, FN has a basis (there is no obvious basis). There are uncountably infinite elements in the basis. Since the dimensions are different, FN is not isomorphic to F∞. It is worth noting that FN is (isomorphic to) the dual space of F∞, because a linear map T from F∞ to F is determined uniquely by its values T(ei) on the basis elements of F∞, and these values can be arbitrary. Thus one sees that a vector space need not be isomorphic to its double dual if it is infinite dimensional, in contrast to the finite dimensional case.

The classification of complex manifolds is much more subtle than that of differentiable manifolds. For example, while in dimensions other than four, a given topological manifold has at most finitely many smooth structures, a topological manifold supporting a complex structure can and often does support uncountably many complex structures. Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by the genus, are an important example of this phenomenon. The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a moduli space, the structure of which remains an area of active research.

Experiment shows that dropping the spheres in randomly will achieve a density of around 65% . However, a higher density can be achieved by carefully arranging the spheres as follows. Start with a layer of spheres in a hexagonal lattice, then put the next layer of spheres in the lowest points you can find above the first layer, and so on. At each step there are two choices of where to put the next layer, so this natural method of stacking the spheres creates an uncountably infinite number of equally dense packings, the best known of which are called cubic close packing and hexagonal close packing.

Additionally, consider for instance the unit circle S, and the action on S by a group G consisting of all rational rotations. Namely, these are rotations by angles which are rational multiples of π. Here G is countable while S is uncountable. 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. The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent.

An important step in the evolution of classical model theory occurred with the birth of stability theory (through Morley's theorem on uncountably categorical theories and Shelah's classification program), which developed a calculus of independence and rank based on syntactical conditions satisfied by theories. During the last several decades applied model theory has repeatedly merged with the more pure stability theory. The result of this synthesis is called geometric model theory in this article (which is taken to include o-minimality, for example, as well as classical geometric stability theory). An example of a proof from geometric model theory is Hrushovski's proof of the Mordell–Lang conjecture for function fields.

If a given manifold admits a geometric structure, then it admits one whose model is maximal. A 3-dimensional model geometry X is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on X. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called Thurston geometries. (There are also uncountably many model geometries without compact quotients.) There is some connection with the Bianchi groups: the 3-dimensional Lie groups. Most Thurston geometries can be realized as a left invariant metric on a Bianchi group.

Georg Cantor, 1870 Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite.. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article was published in 1874.

Infinitely many distinct tilings may be obtained from a single aperiodic set of tiles.A set of aperiodic prototiles can always form uncountably many different tilings, even up to isometry, as proven by Nikolaï Dolbilin in his 1995 paper The Countability of a Tiling Family and the Periodicity of a Tiling The best-known examples of an aperiodic set of tiles are the various Penrose tiles. The known aperiodic sets of prototiles are seen on the list of aperiodic sets of tiles. The underlying undecidability of the domino problem implies that there exists no systematic procedure for deciding whether a given set of tiles can tile the plane.

In particular, it satisfies a sort of least-upper-bound axiom that says, in effect: :Every nonempty internal set that has an internal upper bound has a least internal upper bound. Countability of the set of all internal numbers (in conjunction with the fact that those form a densely ordered set) implies that that set does not satisfy the full least-upper-bound axiom. Countability of the set of all internal sets implies that it is not the set of all subsets of the set of all internal numbers (since Cantor's theorem implies that the set of all subsets of a countably infinite set is an uncountably infinite set). This construction is closely related to Skolem's paradox.

The function type in programming languages does not correspond to the space of all set-theoretic functions. Given the countably infinite type of natural numbers as the domain and the booleans as range, then there are an uncountably infinite number (2ℵ0 = c) of set-theoretic functions between them. Clearly this space of functions is larger than the number of functions that can be defined in any programming language, as there exist only countably many programs (a program being a finite sequence of a finite number of symbols) and one of the set-theoretic functions effectively solves the halting problem. Denotational semantics concerns itself with finding more appropriate models (called domains) to model programming language concepts such as function types.

The existence of many noncomputable sets follows from the facts that there are only countably many Turing machines, and thus only countably many computable sets, but according to the Cantor's theorem, there are uncountably many sets of natural numbers. Although the halting problem is not computable, it is possible to simulate program execution and produce an infinite list of the programs that do halt. Thus the halting problem is an example of a recursively enumerable set, which is a set that can be enumerated by a Turing machine (other terms for recursively enumerable include computably enumerable and semidecidable). Equivalently, a set is recursively enumerable if and only if it is the range of some computable function.

There are also universality theorems for certain sets of well-known gates; such a universality theorem exists, for instance, for the pair consisting of the single qubit phase gate Uθ mentioned above (for a suitable value of θ), together with the 2-qubit CNOT gate WCNOT. However, the universality theorem for the quantum case is somewhat weaker than the one for the classical case; it asserts only that any reversible n-qubit circuit can be approximated arbitrarily well by circuits assembled from these two elementary gates. Note that there are uncountably many possible single qubit phase gates, one for every possible angle θ, so they cannot all be represented by a finite circuit constructed from {Uθ, WCNOT)}.

However, the language S(x) may not even be a recursive language, since there are uncountably many such x, but only countably many recursive languages. A function f on ordered pairs (x,y) is a selector for a set S if f(x,y) is equal to either x or y and if f(x,y) is in S whenever at least one of x, y is in S. A set is semi-recursive if it has a recursive selector, and is P-selective or semi-feasible if it is semi-recursive with a polynomial time selector. Semi- feasible sets have small circuits; they are in the extended low hierarchy; and cannot be NP-complete unless P=NP.

Disappointed, Bender sighs in sadness and claims that it was 'the greatest uncountably infinite bunch of guys I ever met'. In the scene where the Planet Express crew are, in Hermes' words, 'in the belly of the beast', Fry uses a reference of two characters from different books; Jonah, from the Old Testament/Tanakh; and Pinocchio, when he is swallowed by The Terrible Shark. Both characters that have been in the belly of a large whale/fish. The scene where Leela and the other crew members emerge from the whale's mouth is reminiscent of the final scene in Close Encounters of the Third Kind where the alien abductees emerge from the mothership.

In order to produce a computable real, a Turing machine must compute a total function, but the corresponding decision problem is in Turing degree 0′′. Consequently, there is no surjective computable function from the natural numbers to the computable reals, and Cantor's diagonal argument cannot be used constructively to demonstrate uncountably many of them. While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable. Here, for any given computable number x, the well ordering principle provides that there is a minimal element in S which corresponds to x, and therefore there exists a subset consisting of the minimal elements, on which the map is a bijection.

After ten years, Kleene and Post showed in 1954 that there are intermediate Turing degrees between those of the computable sets and the halting problem, but they failed to show that any of these degrees contains a recursively enumerable set. Very soon after this, Friedberg and Muchnik independently solved Post's problem by establishing the existence of recursively enumerable sets of intermediate degree. This groundbreaking result opened a wide study of the Turing degrees of the recursively enumerable sets which turned out to possess a very complicated and non-trivial structure. There are uncountably many sets that are not recursively enumerable, and the investigation of the Turing degrees of all sets is as central in recursion theory as the investigation of the recursively enumerable Turing degrees.

Each set in the countable sequence of sets (Si) = S1, S2, S3, ... contains a nonzero, and possibly infinite (or even uncountably infinite), number of elements. The axiom of countable choice allows us to arbitrarily select a single element from each set, forming a corresponding sequence of elements (xi) = x1, x2, x3, ... The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. I.e., given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, then there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N.

Description numbers play a key role in many undecidability proofs, such as the proof that the halting problem is undecidable. In the first place, the existence of this direct correspondence between natural numbers and Turing machines shows that the set of all Turing machines is denumerable, and since the set of all partial functions is uncountably infinite, there must certainly be many functions that cannot be computed by Turing machines. By making use of a technique similar to Cantor's diagonal argument, it is possible exhibit such an uncomputable function, for example, that the halting problem in particular is undecidable. First, let us denote by U(e, x) the action of the universal Turing machine given a description number e and input x, returning 0 if e is not the description number of a valid Turing machine.

There are uncountably many of these sets and also some recursively enumerable but noncomputable sets of this type. Later, Degtev established a hierarchy of recursively enumerable sets that are (1, n + 1)-recursive but not (1, n)-recursive. After a long phase of research by Russian scientists, this subject became repopularized in the west by Beigel's thesis on bounded queries, which linked frequency computation to the above-mentioned bounded reducibilities and other related notions. One of the major results was Kummer's Cardinality Theory which states that a set A is computable if and only if there is an n such that some algorithm enumerates for each tuple of n different numbers up to n many possible choices of the cardinality of this set of n numbers intersected with A; these choices must contain the true cardinality but leave out at least one false one.

For instance, there are uncountably many numbers whose decimal expansions (in base 3 or higher) do not contain the digit 1, and none of these numbers is normal. Champernowne's constant : 0.1234567891011121314151617181920212223242526272829..., obtained by concatenating the decimal representations of the natural numbers in order, is normal in base 10. Likewise, the different variants of Champernowne's constant (done by performing the same concatenation in other bases) are normal in their respective bases (for example, the base-2 Champernowne constant is normal in base 2), but they have not been proven to be normal in other bases. The Copeland–Erdős constant : 0.23571113171923293137414347535961677173798389..., obtained by concatenating the prime numbers in base 10, is normal in base 10, as proved by . More generally, the latter authors proved that the real number represented in base b by the concatenation : 0.f(1)f(2)f(3)..., where f(n) is the nth prime expressed in base b, is normal in base b.

If a base vertex is chosen in each connected component of G, then each end of G contains a unique ray starting from one of the base vertices, so the ends may be placed in one-to-one correspondence with these canonical rays. Every countable graph G has a spanning forest with the same set of ends as G.More precisely, in the original formulation of this result by in which ends are defined as equivalence classes of rays, every equivalence class of rays of G contains a unique nonempty equivalence class of rays of the spanning forest. In terms of havens, there is a one-to-one correspondence of havens of order ℵ0 between G and its spanning tree T for which \beta_T(X)\subset \beta_G(X) for every finite set X and every corresponding pair of havens βT and βG. However, there exist uncountably infinite graphs with only one end in which every spanning tree has infinitely many ends.

A Vitali set is a subset V of the interval [0, 1] of real numbers such that, for each real number r, there is exactly one number v \in V such that v-r is a rational number. Vitali sets exist because the rational numbers Q form a normal subgroup of the real numbers R under addition, and this allows the construction of the additive quotient group R/Q of these two groups which is the group formed by the cosets of the rational numbers as a subgroup of the real numbers under addition. This group R/Q consists of disjoint "shifted copies" of Q in the sense that each element of this quotient group is a set of the form for some r in R. The uncountably many elements of R/Q partition R, and each element is dense in R. Each element of R/Q intersects [0, 1], and the axiom of choice guarantees the existence of a subset of [0, 1] containing exactly one representative out of each element of R/Q. A set formed this way is called a Vitali set.