Intuitionism

Spread (intuitionism)

In intuitionistic mathematics, a species is a collection (similar to a classical set 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 (1918B), and was used to define the real numbers (also called the continuum). 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 (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 tree with numerical vertex labels. (Wikipedia).

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Related pages

Decidability (logic) | Upper and lower bounds | Choice sequence | Set (mathematics) | Vertex (graph theory) | Michael Dummett | L. E. J. Brouwer | Continuum (set theory) | Real number | Intuitionism | Bitstream | Sequence | Tree (graph theory)