Subcountability

In constructive mathematics, a collection is subcountable if there exists a partial surjection from the natural numbers onto it. The name derives from the intuitive sense that such a collection is "no bigger" than the counting numbers. The concept is trivial in classical set theory, where a set is subcountable if and only if it is finite or countably infinite. Constructively it is consistent to assert the subcountability of some uncountable collections such as the real numbers. Indeed, there are models of the constructive set theory CZF in which all sets are subcountable[1] and models of IZF in which all sets with apartness relations are subcountable.[2]

For example Cantor's diagonal argument establishes that the real numbers can not be countable but in the constructive interpretation they may still be subcountable.

References

  1. ↑ Rathjen, M. "Choice principles in constructive and classical set theories", Proceedings of the Logic Colloquium, 2002
  2. ↑ McCarty, J. "Subcountability under realizability", Notre Dame Journal of Formal Logic, Vol 27 no 2 April 1986


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