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  • 16 - Axiomatics of Set Theory

  • N. H. Bingham, Imperial College London, Adam J. Ostaszewski, London School of Economics and Political Science
  • Book:
    Category and Measure
    Published online:
    14 January 2025
    Print publication:
    23 January 2025, pp 268-277

  • Summary

    We begin with the canonical status of the reals: this extends up to uniqueness to within isomorphism as a complete Archimedean ordered field, but not up to cardinality aspects. We discuss four ‘elephants in the room’ here (an elephant in the room is something obviously there but which no one wants to mention). The first elephant (from Gödel’s incompleteness theorem and the Continuum Hypothesis, CH): one cannot properly speak of the real line, but rather which real line one chooses to work with. The second is ‘which sets of reals can one use?’ (it depends on what axioms of set theory one assumes – in particular, the role of the Axiom of Choice, AC). The third is that there are sentences that are neither provable nor disprovable, and that no non-trivial axiom system is capable of proving its own consistency. Thus, we do not – cannot – know that mathematics itself is consistent. The fourth elephant is that even to define cardinals, the concept of cardinality needs AC.