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Axioms of symmetry: Throwing darts at the real number line. (English) Zbl 0619.03035
We will give a simple philosophical ”proof” of the negation of Cantor’s continuum hypothesis (CH). 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 rejeting the axiom of choice (AC) - we will point out along the way some extensions of our intuition which contradict AC.

MSC:
03E50 Continuum hypothesis and Martin’s axiom
03E25 Axiom of choice and related propositions
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References:
[1] Hypothèse de continu (1956)
[2] Handbook of set-theoretic topology pp 887– (1984) · doi:10.1016/B978-0-444-86580-9.50023-9
[3] Set theory and the continuum hypothesis (1966) · Zbl 0182.01301
[4] Measure and category (1971)
[5] DOI: 10.1016/0003-4843(70)90009-4 · Zbl 0222.02075 · doi:10.1016/0003-4843(70)90009-4
[6] Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques 14 pp 19– (1966)
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