Stephen Kleene, R. E. Vesley, The foundations of intuitionistic mathematics,North-Holland (1965)

The Kleene-Vesley topos as such is discussed for instance in

Jaap van Oosten, page 279 of Realizability: an introduction to its categorical side, Studies in Logic and the Foundations of Mathematics, vol. 152, Elsevier, 2008 (preface pdf)

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