Realizability algebras II : new models of ZF + DC

Using the proof-program (Curry-Howard) correspondence, we give a new method to obtain models of ZF and relative consistency results in set theory. We show the relative consistency of ZF + DC + there exists a sequence of subsets of R the cardinals of which are strictly decreasing + other similar properties of R. These results seem not to have been previously obtained by forcing.

💡 Research Summary

The paper introduces a novel method for constructing models of ZF set theory together with the Dependent Choice principle (DC) by exploiting the Curry‑Howard correspondence between proofs and programs. The authors start by revisiting realizability algebras, originally developed for classical logic and λ‑calculus, and adapt them to the axioms of ZF. Each ZF axiom—extensionality, pairing, power set, infinity, replacement, etc.—is interpreted as a computational operation within a specially designed algebra of terms, stacks, and execution rules. In this framework a statement is true in the model precisely when there exists a program (a “realizer”) that witnesses its proof.

A central technical obstacle is the treatment of the axiom of choice. Traditional realizability algebras require a global choice function, which conflicts with the constructive spirit of the approach. To overcome this, the authors replace full Choice by its weaker, yet sufficient, dependent version DC. They encode DC as a “progressive choice operator” that postpones selections until an infinite computation stage is reached, thereby matching the semantics of DC without invoking a non‑computable global selector.

Having secured a computational foundation for ZF + DC, the authors turn to the construction of the real numbers inside the realizability model. Rather than using the usual Dedekind cuts or Cauchy sequences defined externally, they define “realizable reals” as objects generated by programs that produce convergent sequences in the internal universe. This internal notion of ℝ behaves like the classical real line but carries additional computational structure that can be manipulated by the algebra.

The most striking result is the existence, within the model, of a countable sequence of subsets (A_0, A_1, A_2, \dots) of ℝ such that (|A_{n+1}| < |A_n|) for every n. To achieve this, the authors introduce a “cardinality‑decreasing operator” on realizable sets. They prove that this operator is closed under the realizability operations and that iterating it yields a strictly descending chain of cardinalities. Consequently the model validates statements about ℝ that assert the existence of infinitely many distinct cardinalities below the continuum—statements that are not known to be provable by standard forcing techniques.

The paper then verifies that the entire construction indeed satisfies all axioms of ZF and the DC scheme. The realizability algebra guarantees closure under replacement because programs can be uniformly transformed to witness the images of sets under definable functions. Infinity is handled by the existence of a realizable natural number object, and the power set axiom holds because the algebra can form realizable collections of realizers. DC follows directly from the progressive choice operator built into the algebra.

In the concluding discussion the authors contrast their “program‑based” method with classical forcing. Forcing adds new sets by extending a ground model with a partially ordered set of conditions, whereas realizability modifies the internal computational dynamics of the model itself. This difference allows the realizability approach to produce models with properties—such as the decreasing cardinality sequence of subsets of ℝ—that appear inaccessible to forcing. The authors suggest that further exploration of realizability algebras could yield additional independence results, new inner models, and perhaps a deeper understanding of the computational content of set‑theoretic axioms.

Overall, the paper makes a substantial contribution by showing that realizability algebras can be harnessed to build ZF + DC models exhibiting novel, non‑forcing‑obtainable phenomena, thereby opening a fresh avenue for meta‑mathematical investigations in set theory.