How the continuum hypothesis could have been a fundamental axiom (2407.02463v2)

Abstract: I describe a simple historical thought experiment showing how we might have come to view the continuum hypothesis as a fundamental axiom, one necessary for mathematics, indispensable even for calculus.

• The paper explores a hypothetical historical scenario where the Continuum Hypothesis could have become a fundamental axiom in mathematics.
• It suggests that a different early development of calculus, potentially integrating a categorical structure of hyperreal numbers, could have made CH indispensable.
• Hamkins' work reflects on the historical contingency of mathematical foundations and the nature of axiomatic systems and necessity.

Analysis of "How the Continuum Hypothesis Could Have Been a Fundamental Axiom" by Joel David Hamkins

Joel David Hamkins' exploration of the potential for the Continuum Hypothesis (CH) to have been regarded as a fundamental axiom reconstructs a version of history where the CH is integral to mathematical foundations. The paper is both a thought experiment and a reflection on the nature of mathematical axioms and structures.

Core Argument

Hamkins posits that slight alternate developments in the historical evolution of mathematics could have led to a scenario where the CH is indispensable to mathematical theory. The central thesis involves reimagining the early discovery and formalization of infinitesimals by Newton and Leibniz, suggesting they might have posited a distinct and structured field of hyperreal numbers. This field would extend the real numbers to include infinitesimals, thus anchoring CH as a necessary principle for its coherent construction.

Historical Reevaluation

The paper reconstructs the mathematical narrative by suggesting that if early calculus included a clearer foundational layering involving hyperreal numbers, calculus itself might have evolved with CH as an implicit requirement. By integrating hyperreal numbers into the foundation of mathematics, the exploration suggests that the axiomatization of set theory might have included CH from the outset. The proposition hinges on the existence of a categorical characterization of the hyperreal numbers, linking CH to aspects of set theory crucial for the uniqueness and robustness of mathematical structures akin to how Peano characterized natural numbers.

Theoretical Framework and Implications

Hamkins discusses several mathematical components:

• Hyperreal Numbers: He argues that the hyperreal number system could serve as both an extension and clarification of the concept of infinitesimals, with countable saturation and the transfer principle being analogous to the completeness of the reals.
• Categorical Characterization: The idea that mathematical structures are fundamentally understood through their unique characterizations up to isomorphism raises the potential inevitability of CH for hyperreal characterization were it historically necessary.
• Forcing and Independence: The observation that CH is independent of ZFC axioms does not inherently negate its potential fundamental role, as reimagined historical acceptance might parallel the acceptance of the Axiom of Choice.

Reasoning Against Historical Fixedness

Hamkins emphasizes historical contingency in mathematical truth, suggesting that what we jointly consider foundational could have been conceived differently had preceding thoughts been otherwise oriented. The reliance on CH for a coherent infinitesimals theory illustrates a standpoint of mathematical practice where dependent axioms become self-evident, thereby solidifying his claim on CH's historic potential.

Broader Theoretical Reflections

The implications extend to philosophical dialogues about foundationalism and the necessity of categorical descriptions. Hamkins' thought experiment conveys an alternate universe of set theory where CH might juxtapose current ideas favoring open, pluralistic approaches to foundational questions.

Future Perspective

Imagining a different historical development where CH is fundamental provokes deeper contemplation of what might constitute axiomatic inevitability. It concurrently engages with contemporary arguments in model theory and axiomatic set theory regarding the CH's current non-fundamental status.

In conclusion, while Hamkins' proposal is speculative, it serves as an intellectual prompt for investigating the philosophical underpinnings of mathematical axiom selection, reaffirming the non-fixed nature of mathematical intuition based on historical framing. His discussion invites reconsideration of the paths not taken and how such paths could redefine mathematical landscapes.