[Paper Review] Is set theory indispensable?

Nik Weaver|ArXiv.org|May 11, 2009

Philosophy and Theoretical Science23 references3 citations

TL;DR

This paper argues that Zermelo-Fraenkel set theory with choice (ZFC) is not indispensable for mainstream mathematics or science, as all such mathematics can be formalized in weaker, predicative systems that are more philosophically defensible and less metaphysically committed. The key contribution is a critique of set-theoretic platonism and the Quine-Putnam indispensability argument, concluding that ZFC lacks a solid philosophical foundation and may even prove false number-theoretic statements.

ABSTRACT

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.

Motivation & Objective

• To challenge the philosophical justification of Zermelo-Fraenkel set theory with choice (ZFC) as the foundation of mathematics.
• To argue that the Quine-Putnam indispensability argument fails because science and mainstream mathematics do not actually require set theory.
• To demonstrate that virtually all mainstream mathematics, including scientifically applicable mathematics, can be formalized in predicative, number-theoretic systems.
• To show that set theory as currently practiced is better understood formally than platonistically, undermining its foundational status.
• To question the truth of number-theoretic statements provable in ZFC but not in predicative systems, suggesting ZFC may prove false arithmetic.

Proposed method

• Analyzing the proof-theoretic strength of predicative systems and comparing them to ZFC in terms of formalizing mainstream mathematics.
• Applying results from proof theory and reverse mathematics to show that most theorems requiring 'set-theoretic' principles can be formalized in weaker, predicative systems.
• Critiquing the iterative conception of sets as relying on physicalist metaphors that fail to justify uncountable or non-standard models.
• Examining the epistemological and ontological problems of set-theoretic platonism, especially the lack of a coherent account of how we can know abstract sets.
• Evaluating the consistency and truth of ZFC by assessing whether it has models with a standard ω, arguing that such models are unlikely without special justification.
• Using the Löwenheim-Skolem theorem to argue that uncountable structures cannot be meaningfully described without presupposing set theory, undermining claims of physical or intuitive accessibility.

Experimental results

Research questions

• RQ1 Is Zermelo-Fraenkel set theory with choice (ZFC) truly indispensable for mainstream mathematics or scientific applications?
• RQ2 Can the full strength of ZFC be replaced by weaker, predicative systems that are more philosophically defensible and still sufficient for formalizing core mathematics?
• RQ3 Does the Quine-Putnam indispensability argument successfully justify set-theoretic platonism, given that science and mathematics do not actually require set theory?
• RQ4 Are there compelling philosophical grounds to believe that ZFC proves only true number-theoretic statements, especially those not provable in predicative systems?
• RQ5 Can set theory be understood and practiced without platonism, and if so, does this undermine its claim to foundational status?

Key findings

• All mainstream mathematics, including that used in science, can be formalized in predicative systems such as the system CM, which are as elegant and usable as ZFC.
• ZFC likely proves false statements in first-order number theory, particularly those not provable in predicative systems, due to the high likelihood that ZFC lacks a model with a standard ω.
• The Quine-Putnam indispensability argument fails because science does not actually depend on set theory; the mathematics used in science is inherently number-theoretic and predicative.
• Set theory as currently practiced is largely formalist in nature and does not require platonism for justification, undermining the need for a metaphysical foundation.
• The iterative conception of sets relies on physicalist metaphors (e.g., forming power sets by manipulation) that are incoherent when applied to infinite or uncountable sets.
• The existence of undecidable questions like the continuum hypothesis in set theory, contrasted with the absence of such issues in number theory, contributes to widespread skepticism about the truth-value of set-theoretic statements.