Finiteness and decidability:I

Finiteness notions in fuzzy sets

“…Here, decidable means that the diagonal X → X × X is complemented. See [7] and [1] for details. Since for most notions of negation the only fuzzy sets which are decidable are crisp, this notion is not useful in a fuzzy context.…”

An induction principle and pigeonhole principles for K-finite sets

“…That is, we do not assume that x = y or x = y. In fact, by [1], such an assumption would allow us to work in a sub-universe (sub-topos) in which classical logic holds and would thus remove the whole point of working in intuitionistic logic.…”

“…This statement is a reformulation, in the absence of N, of the result of the first half of the proof given above. (See [1] for the connection between the sets n and finite sets B with decidable equality.) But this statement is not provable in intuitionistic type theory without the axiom of infinity.…”

“…Of the several concepts of finiteness that are equivalent in classical logic but not in intuitionistic logic, we shall use the one commonly called K-finiteness or Kuratowski-finiteness [1,9,10]. The definition and some comments on it are given in Section 2.…”

Objetos K-finitos y decidibilidad

“…De esta forma extendemos los resultados estudiados en [3] y [4]. La teoría que fundamenta estos estudios se puede consultar en [1], [2] y [5].…”