Traditional Cavalieri principles applied to the modern notion of area (Q1263582)

In a remarkable paper [``Axioms of symmetry: throwing darts at the real number line'', J. Symb. Logic 51, 190-200 (1986; Zbl 0619.03035)], \textit{C. Freiling} sets out an ingenious argument that seems to establish that the continuum hypothesis, CH, is false. He shows, in fact, that a weak consequence of his argument, which he calls \(A_{\aleph_ 0}\), is equivalent to the negation of the continuum hypothesis, \(\neg CH\). In addition, he formulates a stronger consequence of his argument, \(A_{null}\), on the basis of which he establishes such results as the negation of Martin's axiom and an extension of Fubini's theorem. This paper begins by suggesting that the crux of Freiling's argument is the application of a Cavalieri principle. It continues by formulating a measure-theoretic principle, \(A_{full}\)- which incorporates a Cavalieri principle - the intent of which is to capture Freiling's argument for \(A_{null}\). Next, it shows that Freiling's Fubini theorem implies \(A_{full}\), which implies \(A_{null}\), and, hence, that \(A_{null}\) is equivalent to \(A_{full}\). Finally, it discusses why Freiling's argument, though very attractive, might not be regarded as utterly convincing.