Is set theory indispensable?

Although Zermelo-Fraenkel set theory (ZFC) is generally accepted as the appropriate foundation for modern mathematics, proof theorists have known for decades that virtually all mainstream mathematics can actually be formalized in much weaker systems which are essentially number-theoretic in nature. Feferman has observed that this severely undercuts a famous argument of Quine and Putnam according to which set theoretic platonism is validated by the fact that mathematics is “indispensable” for some successful scientific theories (since in fact ZFC is not needed for the mathematics that is currently used in science). I extend this critique in three ways: (1) not only is it possible to formalize core mathematics in these weaker systems, they are in important ways better suited to the task than ZFC; (2) an improved analysis of the proof-theoretic strength of predicative theories shows that most if not all of the already rare examples of mainstream theorems whose proofs are currently thought to require metaphysically substantial set-theoretic principles actually do not; and (3) set theory itself, as it is actually practiced, is best understood in formalist, not platonic, terms, so that in a real sense set theory is not even indispensable for set theory. I also make the point that even if ZFC is consistent, there are good reasons to suspect that some number-theoretic assertions provable in ZFC may be false. This suggests that set theory should not be considered central to mathematics.

💡 Research Summary

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The paper “Is set theory indispensable?” challenges the widely‑held view that Zermelo‑Fraenkel set theory with Choice (ZFC) is the indispensable foundation of mathematics. Drawing on decades of proof‑theoretic research, the author argues that virtually all mainstream mathematics—especially the parts actually used in scientific applications—can be formalized in far weaker, essentially number‑theoretic systems such as RCA₀, ACA₀, and ATR₀. These subsystems require far fewer ontological commitments, avoid the unnecessary complexity of full ZFC, and are often more natural for the kinds of constructions that arise in analysis, differential equations, probability, and other applied fields.

The second major contribution is an updated analysis of the proof‑theoretic strength of predicative theories. While certain celebrated set‑theoretic statements (e.g., the Continuum Hypothesis, the existence of measurable cardinals) have traditionally been presented as requiring strong, non‑predicative axioms, recent work shows that most of the rare theorems that genuinely seem to need “large‑cardinal‑type” assumptions actually fall below the Feferman‑Schütte ordinal ε₀. In other words, they are provable in predicative systems that are strictly weaker than ZFC. This undermines the claim that ZFC is needed for any substantive piece of mainstream mathematics.

The third thrust is philosophical. The author contends that contemporary set theory is practiced in a formalist manner: mathematicians manipulate symbols according to the ZFC axioms without committing to a platonic realm of sets. The usual textbook description of sets as “collections of objects” is shown to be more pedagogical than substantive; in research, the focus is on formal derivations, not on any metaphysical ontology. Consequently, the argument that set theory is indispensable because it provides a “true” universe of sets collapses; set theory itself does not require platonism for its justification.

Finally, the paper raises a cautionary point about ZFC’s reliability as a source of arithmetic truth. Even if ZFC is consistent, Gödel’s incompleteness theorems imply that ZFC may prove Σ₁ statements about natural numbers that are false in the intended model. This possibility further weakens the case for treating ZFC as a central, trustworthy foundation for all of mathematics.

Putting these strands together, the author dismantles the Quine‑Putnam indispensability argument: (1) the mathematics needed for science does not depend on ZFC; (2) the few remaining “set‑theoretic” theorems that seem to need strong axioms are in fact predicatively reducible; and (3) set theory’s own practice is formalist, not platonist. Therefore, ZFC is not philosophically or mathematically indispensable, and mathematics can comfortably proceed on much weaker, more transparent foundations.