# Species

*in logic*

The intuitionistic analogue of the concept of a set; an exactly formulated criterion for isolating some of the objects out of an already defined population of objects under study. It is essential to note that the condition which defines the species is to be understood in the intuitionistic sense, so that, for example, the double negation of a condition is not necessarily equivalent to the condition itself. Operations are naturally defined over species in analogy to operations defined over sets, such as union, intersection and others, but since they are to be understood in the specific intuitionistic sense (cf. Intuitionism), the properties of these operations often differ from those of the corresponding classical operations. Thus, the statement that the complement of the complement of a species is identical with the species itself, does not apply in the intuitionistic theory of species.

In constructing the theory of species, the usual paradoxes are avoided by stipulating that the definition of the members of a species be independent of the definition of the species itself. Intuitionistic theories such as intuitionistic arithmetic or intuitionistic mathematical analysis may be constructed without using the concept of a species at all, but in more abstract domains of intuitionistic mathematics (proof theory, semantics, intuitionistic functional analysis) the development of the theory of species is an essential part of the theory.

#### References

[1] | A. Heyting, "Intuitionism: an introduction" , North-Holland (1970) |

[a1] | A.G. Dragalin, "Mathematical intuitionism. Introduction to proof theory" , Amer. Math. Soc. (1988) (Translated from Russian) |

[a2] | A.S. Troelstra, D. van Dalen, "Constructivism in mathematics, an introduction" , 1–2 , North-Holland (1989) |

**How to Cite This Entry:**

Species.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Species&oldid=55208