Choice sequence

﻿
Choice sequence

In intuitionistic mathematics, a choice sequence is a constructive formulation of a sequence. Since the Intuitionistic school of mathematics, as formulated by L. E. J. Brouwer, rejects the idea of a completed infinity, in order to use a sequence (which is, in classical mathematics, an infinite object), we must have a formulation of a finite, constructible object which can serve the same purpose as a sequence. Thus, Brouwer formulated the choice sequence, which is given as a construction, rather than an abstract, infinite object.

Lawlike and lawless sequences

A distinction is made between lawless and lawlike sequences. A lawlike sequence is one that can be described completely — it is a completed construction, that can be fully described. For example, the natural numbers $\mathbb N$ can be thought of as a lawlike sequence: the sequence can be fully constructively described by the unique element 0 and a successor function. Given this formulation, we know that the ith element in the sequence of natural numbers will be the number i − 1. Similarly, a function $f : \mathbb N \mapsto \mathbb N$ mapping from the natural numbers into the natural numbers effectively determines the value for any argument it takes, and thus describes a lawlike sequence.

A lawless (also, free) sequence, on the other hand, is one that is not predetermined. It is to be thought of as a procedure for generating values for the arguments 0, 1, 2, .... That is, a lawless sequence α is a procedure for generating α0, α1, ... (the elements of the sequence α) such that:

• At any given moment of construction of the sequence α, only an initial segment of the sequence is known, and no restrictions are placed on the future values of α; and
• One may specify, in advance, an initial segment $\langle \alpha_0, \alpha_1, \ldots, \alpha_k \rangle$ of α.

Note that the first point above is slightly misleading, as we may specify, for example, that the values in a sequence be drawn exclusively from the set of natural numbers—we can specify, a priori, the range of the sequence.

The canonical example of a lawless sequence is the series of rolls of a die. We specify which die to use and, optionally, specify in advance the values of the first k rolls (for $k\in \mathbb N$). Further, we restrict the values of the sequence to be in the set {1,2,3,4,5,6}. This specification comprises the procedure for generating the lawless sequence in question. At no point, then, is any particular future value of the sequence known.

Axiomatization

There are two axioms in particular that we expect to hold of choice sequences as described above. Let $\alpha\in n$ denote the relation "the sequence α begins with the initial sequence n" for choice sequence α and finite segment n (more specifically, n will probably be an integer encoding a finite initial sequence).

We expect the following, called the axiom of open data, to hold of all lawless sequences:

$A(\alpha) \rightarrow \exists n[\alpha\in n \,\land\, \forall\beta\in n[A(\beta)]]$

where A is a one-place predicate. The intuitive justification for this axiom is as follows: in intuionistic mathematics, verification that A holds of the sequence α is given as a procedure; at any point of execution of this procedure, we will have examined only a finite initial segment of the sequence. Intuitively, then, this axiom states that since, at any point of verifying that A holds of α, we will only have verified that A holds for a finite initial sequence of α; thus, it must be the case that A also holds for any lawless sequence β sharing this initial sequence. This is so because, at any point in the procedure of verifying A(α), for any such β sharing the initial prefix of α encoded by n that we have already examined, if we run the identical procedure on β, we will get the same result. The axiom can be generalized for any predicate taking an arbitrary number of arguments.

Another axiom is required for lawless sequences. The axiom of density, given by:

$\forall n \, \exists \alpha [\alpha\in n]$

states that, for any finite prefix (encoded by) n, there is some sequence α beginning with that prefix. We require this axiom so as not to have any "holes" in the set of choice sequences. This axiom is the reason we require that arbitrarily long finite initial sequences of lawless choice sequences can be specified in advance; without this requirement, the axiom of density is not necessarily guaranteed.

References

• Dummett, M. 1977. Elements of Intuitionism, Oxford University Press.
• Jacquette, Dale. 2002. A Companion to Philosophical Logic, Blackwell Publishing. p 517.
• Kreisel, Georg. 1958. A remark on free choice sequences and the topological completeness proofs, Journal of Symbolic Logic volume 23. p 269
• Troelstra, A.S. 1977. Choice Sequences. A Chapter of Intuitionistic Mathematics. Clarendon Press.
• Troelstra, A.S.. 1983. Analysing Choice Sequences, Journal of Philosophical Logic, 12:2 p. 197.
• Troelstra, A.S.; D. van Dalen. 1988. Constructivism in Mathematics: An Introduction. North Holland.

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Sequence alignment — In bioinformatics, a sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences.[1]… …   Wikipedia

• Choice of law — Conflict of laws Preliminiarie …   Wikipedia

• Sequence motif — In genetics, a sequence motif is a nucleotide or amino acid sequence pattern that is widespread and has, or is conjectured to have, a biological significance. For proteins, a sequence motif is distinguished from a structural motif, a motif formed …   Wikipedia

• sequence analysis — A series of questions about how social processes are ordered, either temporally or spatially, together with the techniques for answering these. Many areas of sociology are concerned with events or actions in their temporal context or with what we …   Dictionary of sociology

• Choice Countertop — PlayChoice 10 PlayChoice 10 est un est un système de jeux vidéo pour borne d arcade compatible JAMMA destiné aux salles d arcade, créé par la société japonaise Nintendo en 1983. Il fut distribué par divers entreprises comme Irem ou Capcom ou… …   Wikipédia en Français

• Axiom of choice — This article is about the mathematical concept. For the band named after it, see Axiom of Choice (band). In mathematics, the axiom of choice, or AC, is an axiom of set theory stating that for every family of nonempty sets there exists a family of …   Wikipedia

• Cauchy sequence — In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. To be more precise, by dropping enough (but still only a finite number of) terms from… …   Wikipedia

• Axiom of countable choice — The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory, similar to the axiom of choice. It states that any countable collection of non empty sets must have a choice function. Spelled out, this means… …   Wikipedia

• Teen Choice Awards 2004 — Die fünften Teen Choice Awards wurden am 8. August 2004 in Los Angeles (Kalifornien), unter der Moderation von Paris Hilton und Nicole Richie, in 38 Kategorien, vergeben. Inhaltsverzeichnis 1 Choice Movie Actor Drama/Action Adventure 2 Choice… …   Deutsch Wikipedia

• Teen Choice Award 2004 — Die fünften Teen Choice Awards wurden am 8. August 2004 in Los Angeles (Kalifornien), unter der Moderation von Paris Hilton und Nicole Richie, in 38 Kategorien, vergeben. Inhaltsverzeichnis 1 Choice Movie Actor Drama/Action Adventure 2 Choice… …   Deutsch Wikipedia