We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov’s principle but meshes well with some classical principles, including the lesser limited principle of omniscience and weak König’s lemma. We discuss some applications, as well as some previous results in the literature.
"Bounded modified realizability." J. Symbolic Logic 71 (1) 329 - 346, March 2006. https://doi.org/10.2178/jsl/1140641178