Spread (intuitionism)
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In intuitionistic mathematics, a ''species'' is a collection (similar to a classical
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
in that a species is determined by its members). A spread is a particular kind of species of infinite sequences defined via finite decidable properties. In modern terminology, a spread is an inhabited closed set of sequences. The notion of spread was first proposed by
L. E. J. Brouwer Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...
(1918B), and was used to define the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
(also called the
continuum Continuum may refer to: * Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes Mathematics * Continuum (set theory), the real line or the corresponding cardinal number ...
). As Brouwer's ideas were developed, the use of spreads became common in intuitionistic mathematics, especially when dealing with choice sequences and the foundations of
intuitionistic analysis In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. This contrasts with ''classical analysis'', which (in this context) simply means analysis done according to the (more com ...
(Dummett 77, Troelstra 77). Simple examples of spreads are: *the set of sequences of even numbers; *the set of sequences of the integers 1–6; *the set of sequences of valid terminal commands. Spreads are defined via a ''spread function'', which performs a ( decidable) "check" on finite sequences. The notion of a spread and its spread function are interchangeable in the literature; both are treated as one and the same. If all the ''finite initial parts'' of an infinite sequence satisfy a spread function's "check", then we can say that the infinite sequence is ''admissible to the spread''. Graph theoretically, one may think of a spread as a rooted,
directed Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''D ...
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with numerical
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labels.


Formal definition

This article uses "\langle" to denote the beginning of a sequence and "\rangle" to denote the end of a sequence. A spread function s is a function that maps finite sequences to either 0 .e. the finite sequence is ''admissible'' to the spreador 1 .e. the finite sequence is ''inadmissible'' to the spread and satisfies the following properties: *Given any finite sequence \langle x_1,x_2,\ldots,x_i\rangle either s(\langle x_1,x_2,\ldots,x_i\rangle)=0 or s(\langle x_1,x_2,\ldots,x_i\rangle)=1 (the property s "tests" for must be decidable). *Given the empty sequence (the sequence with no elements represented by \langle\rangle), s(\langle\rangle)=0 (the empty sequence is in every spread). *Given any finite sequence \langle x_1,x_2,\ldots,x_i\rangle such that s(\langle x_1,x_2,\ldots,x_i\rangle)=0 then there must exist some k such that s(\langle x_1, x_2, \ldots, x_i, k \rangle) = 0 (every finite sequence in the spread can be extended to another finite sequence in the spread by adding an extra element to the end of the sequence) Given an infinite sequence \langle x_1,x_2,\ldots\rangle, we say that the finite sequence \langle y_1,y_2,\ldots,y_i\rangle is an ''initial segment'' of \langle x_1,x_2,\ldots\rangle if and only if x_1=y_1 and x_2=y_2 and ... and x_i=y_i. We say that an infinite sequence \langle x_1,x_2,\ldots\rangle is admissible to a spread defined by spread function s if and only if every initial segment of \langle x_1,x_2,\ldots\rangle is admissible to s.


Fans

A special type of spread that is of particular interest in the intuitionistic foundations of mathematics is a ''finitary'' spread; also known as a fan. The main use of fans is in the fan theorem, a result used in the derivation of the uniform continuity theorem. Informally; a spread function s defines a fan if and only if given a finite sequence admissible to the spread, there are only finitely many possible values that we can add to the end of this sequence such that our new extended finite sequence is admissible to the spread. Alternatively, we can say that there is an
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element ...
on the value for each element of any sequence admissible to the spread. Formally; a spread function s defines a fan if and only if given any sequence admissible to the spread \langle x_1,x_2,\ldots,x_i\rangle, then there exists some k such that, given any j>k then s(\langle x_1,x_2,\ldots,x_i,j\rangle)=1 (i.e. given a sequence admissible to the fan, we have only finitely many possible extensions that are also admissible to the fan, and we know the maximal element we may append to our admissible sequence such that the extension remains admissible). Some examples of fans are: *the set of sequences of legal chess moves; *the set of infinite
binary sequences Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that ...
; *the set of sequences of letters.


Commonly used spreads/fans

This section provides the definition of two spreads commonly used in the literature.


The universal spread (the

continuum Continuum may refer to: * Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes Mathematics * Continuum (set theory), the real line or the corresponding cardinal number ...
)

Given any finite sequence \langle x_1,x_2,\ldots,x_i\rangle, we have s(\langle x_1,x_2,\ldots,x_i\rangle)=0. In other words, this is the spread containing all possible sequences. This spread is often used to represent the collection of all choice sequences.


The binary spread

Given any finite sequence \langle x_1,x_2,\ldots,x_i\rangle, if all of our elements (x_1,x_2,\ldots,x_i) are 0 or 1 then s(\langle x_1,x_2,\ldots,x_i\rangle)=0, otherwise s(\langle x_1,x_2,\ldots,x_i\rangle)=1. In other words, this is the spread containing all
binary sequences Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that ...
.


Dressed Spreads

A key use of spreads in the foundations of intuitionisitic analysis is the use of spreads of natural numbers (or integers) to represent
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
s. This is achieved via the concept of a dressed spread, which we outline below. A ''dressed spread'' is a pair of objects; a spread s, and some function f acting on finite sequences. An example of a dressed spread is the spread of integers such that s(\langle x_1,x_2,\ldots,x_i\rangle)=0 if and only if :\forall y_\, (x_i=2x_\pm 1 \lor\ x_i=0), and the function f(\langle x_1,x_2,\ldots,x_i\rangle) = x_i\cdot 2^ (the dressed spread representing the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
).


References

*
L. E. J. Brouwer Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...
''Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten. Erster Teil, Allgemeine Mengenlehre'', KNAW Verhandelingen, 5: 1–43 (1918A) *
Michael Dummett Sir Michael Anthony Eardley Dummett (27 June 1925 – 27 December 2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He wa ...
''Elements of Intuitionism'', Oxford University Press (1977) * A. S. Troelstra ''Choice Sequences: A Chapter of Intuitionistic Mathematics'', Clarendon Press (1977) {{reflist Intuitionism