A Specker sequence. The nth digit of xk is 4 if n ≤ k and the nth Turing machine in a computable Gödel numbering halts on input n after k steps; otherwise it is 3.
In computability theory, a Specker sequence is a computable, monotonically increasing, bounded sequence of rational numbers whose supremum is not a computable real number. The first example of such a sequence was constructed by Ernst Specker (1949).
The existence of Specker sequences has consequences for computable analysis. The fact that such sequences exist means that the collection of all computable real numbers does not satisfy the least upper bound principle of real analysis, even when considering only computable sequences.
A common way to resolve this difficulty is to consider only sequences that are accompanied by a modulus of convergence; no Specker sequence has a computable modulus of convergence.
More generally, a Specker sequence is called a recursive counterexample to the least upper bound principle, i.e. a construction that shows that this theorem is false when restricted to computable reals.
The least upper bound principle has also been analyzed in the program of reverse mathematics, where the exact strength of this principle has been determined. In the terminology of that program, the least upper bound principle is equivalent to ACA0 over RCA0. In fact, the proof of the forward implication, i.e. that the least upper bound principle implies ACA0, is readily obtained from the textbook proof (see Simpson, 1999) of the non-computability of the supremum in the least upper bound principle.
number. The first example of such a sequence was constructed by Ernst Specker (1949). The existence of Speckersequences has consequences for computable analysis...
Alexander Specker (born 1918), Swiss sports shooter Specker See, lake in Mecklenburg-Vorpommern, Germany Speckersequence, bounded sequence Baer–Specker group...
computable sequence of computable real numbers need not be a computable real number. A sequence with this property is known as a Speckersequence, as the...
Erdős. Specker received his Ph.D. in 1949 from ETH Zurich, where he remained throughout his professional career. Speckersequence Baer–Specker group Ernst...
machines cannot solve their own halting problem. Computation in the limit Speckersequence Ross–Littlewood paradox Hamkins, Joel (2002-12-03). "Infinite time...
every arithmetical number is computable. For example, the limit of a Speckersequence is an arithmetical number that is not computable. The definitions of...
all functions from the reals to the reals are provenly continuous. A Speckersequence exists and then Bolzano-Weierstrass fails. Categorical logic Hyland...
(since it is a non-computable number). The supremum limit of the Speckersequences (since they are non-computable numbers). The so-called Fredholm constants...
converge to a correct solution of the halting problem by evaluating a Speckersequence. Many hypercomputation proposals amount to alternative ways to read...
{\displaystyle g(x)} are both undefined, or they are both defined and equal. Speckersequence A. Mostowski, "Examples of sets definable by means of two and three...
mathematical analysis — these spaces become natural under the analogy. Speckersequence See Simpson, Alex K. (1998), Brim, Luboš; Gruska, Jozef; Zlatuška,...
0 ∨ 0 ≥ x {\displaystyle x\geq 0\lor 0\geq x} . The existence of a Speckersequence is proven from C T 0 {\displaystyle {\mathrm {CT} _{0}}} . Such phenomena...
the Baer–Specker group, that is, the group of all integer sequences, with termwise addition. For each natural number n, let en be the sequence with n-th...
2D flying specksequence / Horton's anime sequence 86 20th Century Fox / Blue Sky Studios 2008 Kung Fu Panda United States Opening sequence and ending...
ten are: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000, 10,000,000... (sequence A011557 in the OEIS) In decimal notation the nth power of ten is written...
unnecessary in this case). The case of the perfect set property was solved by Specker (1957), who showed (in ZF) that if every set of reals has the perfect set...
gradients localize the signal in space. By varying the parameters of the pulse sequence, different contrasts may be generated between tissues based on the relaxation...
Given name Surname Patrilineal/Matrilineal Affixes Nobiliary particle By sequence First name Middle name Last name By trait Diminutive Double-barrelled Epithet...
title sequence for future seasons. The opening title sequence was created by Kyle Cooper and his company Prologue. He also created the title sequence for...
origin of replication (also called the replication origin) is a particular sequence in a genome at which replication is initiated. Propagation of the genetic...
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