A conditional set is a "surjective" sheaf over a complete Boolean algebra A satisfying the Assumption (P) which is necessary and sufficient for the Grothendieck sup topology on A to be generated by a basis of partitions. A class of conditional sets is comprised of all conditional sets on all relative algebras of a fixed algebra. The conditional inclusion is two-dimensional. It considers the partial order on relative algebras and the pointwise inclusion as in sheaf topoi. The conditional inclusion on the conditional power set has the structure of a complete Boolean algebra. This suffices to prove a conditional version of classical theorems like the Ultrafilter lemma, the Tychonoff, the Heine-Borel and the Banach-Alaoglu theorems. Conditional sets are related to Boolean-valued models. We are interested in understanding how far a class of conditional sets is a model of ZFC. An approach to answer this question is discussed. This talk is based on a joint work with Samuel Drapeau, Martin Karlizcek and Michael Kupper.