Relative consistency of a finite nonclassical theory incorporating ZF and category theory with ZF

Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker than ZF, that is finitely axiomatized, and that does not have a countable model (if it has a model at all, that is). Here we prove that T is relatively consistent with ZF. We conclude that this is an important step towards showing that T is an advancement in the foundations of mathematics.

💡 Research Summary

The paper addresses two long‑standing “pathological” features of Zermelo‑Fraenkel set theory (ZF): the need for an infinite axiom schema (separation, replacement, etc.) and the existence of countable models guaranteed by the downward Löwenheim‑Skolem theorem. The authors claim that these features are undesirable, especially from a foundational perspective that seeks a finitely axiomatized theory without countable models.

To remedy this, they refer to a recently introduced non‑classical first‑order theory T (Axioms 10(2): 119 (2021)). T’s language is categorical: its objects are sets, its arrows are functions, and a function is formally a “concatenation” of two sets (graph and domain). Functions are not themselves sets; a special class of “ur‑functions” has a singleton domain and singleton range. The theory consists of about twenty ordinary first‑order axioms (covering extensionality, existence of the empty set, pairing, union, power set, infinity, etc.) together with a single powerful non‑standard axiom called the Sum‑Function Axiom (SUM‑F).

SUM‑F states that for any non‑empty set X and any family of ur‑functions indexed by the elements of X, there exists a function F_X that is a surjection from X onto some set Y and that maps each element ξ∈X to the value given by the corresponding ur‑function f{ξ}(ξ). In formal notation the axiom uses a multiple universal quantifier (∀ f{ξ}) ξ∈X and a conjunctive operator ^ ξ∈X to compress what would otherwise be infinitely many ordinary quantifiers and conjunctions.

Because the language contains these non‑standard quantifiers, the authors supplement ordinary first‑order inference rules with six additional rules (R‑1 through R‑6). Roughly, R‑1 and R‑2 allow elimination of the multiple quantifier by substituting a constant for the indexed family of ur‑functions; R‑3 replaces an existential witness by a fresh constant; R‑4 and R‑5 introduce and eliminate the conjunctive operator; R‑6 re‑introduces the multiple universal quantifier from a formula that already contains the conjunctive operator. The paper claims that these rules are “intuitive” and enable derivations of standard sentences from SUM‑F and vice‑versa, but it provides no formal proof of soundness or completeness for the extended proof system.

The main technical result is a relative consistency proof: assuming ZF is consistent, the authors construct a model of T inside ZF. The construction proceeds by interpreting the categorical universe of T as a class of sets and functions definable in ZF. The standard axioms of T are shown to hold in this interpretation by straightforward translation. For the non‑standard SUM‑F, the authors argue that within ZF one can define, for any set X and any family of ur‑functions indexed by X, a function F_X that collects the images of the ur‑functions; this uses the axiom of choice (or a definable selection function) to pick the required family. The existence of the required surjection and target set Y is then obtained by standard ZF constructions (e.g., taking the image of X under the map ξ↦f{ξ}(ξ)). However, the paper does not give a detailed verification that the multiple‑quantifier syntax can be interpreted as a genuine first‑order statement in the ZF model, nor does it address potential circularities in using choice to satisfy a non‑standard axiom that itself seems to encode a form of choice.

Beyond the technical proof, the authors argue that T is “stronger” than ZF for two reasons. First, T proves all theorems of ZF while using only finitely many axioms, which they liken to a “lighter weight” lifting a heavier load. Second, T proves that no countable model exists, a property they claim is mathematically and physically desirable (e.g., for modeling continuous space in physics). The paper includes a philosophical discussion about the inadequacy of countable models for physical theories, citing Skolem’s “paradox” and historical remarks by Zermelo, von Neumann, and others.

Critically, the paper suffers from several shortcomings. The non‑standard syntax and inference rules are introduced without a rigorous semantic framework; there is no model‑theoretic definition of what a “concatenation” of sets is, nor a clear treatment of functions as objects distinct from sets. The soundness of the extended proof system is assumed rather than proved. The relative consistency construction is sketched at a high level, lacking a concrete definition of the family of ur‑functions and a verification that the multiple‑quantifier can be eliminated inside ZF. Moreover, the claim that T has no countable model follows directly from the SUM‑F axiom, but without a proof that SUM‑F is compatible with ZF’s axioms of infinity and choice, the argument remains informal.

In summary, the paper proposes an intriguing idea: a finitely axiomatized theory that subsumes ZF and avoids countable models by augmenting the language with a powerful sum‑function principle. While the motivation is clear and the high‑level strategy of a relative consistency proof is plausible, the lack of precise definitions, rigorous metatheoretic analysis, and detailed model construction means that the central claim—T is relatively consistent with ZF and strictly stronger—remains unsubstantiated at the present stage. Further work would need to formalize the non‑standard syntax, prove the soundness of the extended inference rules, and give a fully detailed ZF‑internal model of T.