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Axioms of symmetry: Throwing darts at the real number line

  • Chris Freiling (a1)

We will give a simple philosophical “proof” of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpiński and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will in fact show why there must be an infinity of cardinalities between the integers and the reals. We will also show why Martin's Axiom must be false, and we will prove the extension of Fubini's Theorem for Lebesgue measure where joint measurability is not assumed. Following the philosophy—if you reject CH you are only two steps away from rejecting the axiom of choice (AC)—we will point out along the way some extensions of our intuition which contradict AC.

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[1]Cohen, , Set theory and the continuum hypothesis, Benjamin, New York, 1966.
[2]Martin, and Solovay, , Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), pp. 143178.
[3]Oxtoby, , Measure and category, Springer-Verlag, New York, 1971.
[4]Erdös, and Hajnal, , On a problem of B. Jónsson, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 14 (1966), pp. 1923.
[5]Kunen, , Random and Cohen reals, Handbook of set-theoretic topology (Kunen, and Vaughan, , editors), North-Holland, Amsterdam, 1984, pp. 887911.
[6]Sierpiński, , Hypothèse de continu, 2nd ed., Chelsea, New York, 1956.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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