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+----- Thread: How the continuum hypothesis could have been a fundamental axiom (/showthread.php?tid=2059)
How the continuum hypothesis could have been a fundamental axiom - mklabgr - 09-21-2026
Joel David Hamkins’s paper How the continuum hypothesis could have been a fundamental axiom presents a historical thought experiment about the Continuum Hypothesis (CH), which states that there is no set whose cardinality lies strictly between that of the natural numbers and the real numbers. Rather than arguing that CH is actually true, Hamkins asks how mathematicians might have come to regard it as an essential axiom of mathematics. He imagines an alternative historical development in which mathematical analysis and calculus were constructed using a particular continuum of real numbers whose properties effectively forced CH to hold. In that scenario, CH could have appeared not as an optional set-theoretic principle discovered later, but as something fundamental to the mathematical.
The broader point is philosophical: what we regard as an obvious mathematical axiom can depend heavily on the historical development of mathematics and the structures we choose to study. In modern set theory, CH is independent of the standard ZFC axioms, meaning that both CH and its negation are compatible with ZFC assuming ZFC itself is consistent.
Hamkins uses his hypothetical history to illustrate that this independence does not automatically settle the question of whether CH should be regarded as mathematically meaningful, natural, or fundamental. The paper therefore connects the technical issue of CH with deeper questions about mathematical foundations, alternative mathematical universes, and how mathematical practice can shape our conception of what is considered an axiom.
ARTICLE [PDF]