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Axioms of Symmetry: Throwing Darts at the Real Number Line
The Journal of Symbolic Logic
Vol. 51, No. 1 (Mar., 1986), pp. 190-200
Published by: Association for Symbolic Logic
Stable URL: http://www.jstor.org/stable/2273955
Page Count: 11
You can always find the topics here!Topics: Null set, Intuition, Axioms, Philosophical axioms, Cardinality, Real lines, Lebesgue measures, Real numbers, Symmetry, Counterexamples
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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 .) 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 Sierpinski 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.
The Journal of Symbolic Logic © 1986 Association for Symbolic Logic