Preserving cardinals and weak forms of Zorn’s lemma in realizability models

Abstract

We develop a technique for representing and preserving cardinals in realizability models, and we apply this technique to define a realizability model of Zorn’s lemma restricted to an ordinal.

Realizing Realizability Results With Classical Constructions

“…If such H exists, we say that Ẋ is a measurable name. Finally, Ẋ is densely measurable 3 if for every H ∈ F there is K ⊆ H such that K measures Ẋ. Theorem 3.3. Let P, G , F be a mixable symmetric system admitting an absolute representative.…”

“…While corresponding with Krivine, he informed us that together with Laura Fontanella they proved some weak versions of the axiom of choice, specifically wellordered choice, in some of the models constructed by Krivine, but also choice from families indexed by some of the "paradoxical sets" added to the model. Another recent work, [3], by Fontanella and Guillaume Geoffroy is concerned with producing realizability models where DC κ holds for some uncountable ordinal κ.…”