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7 Paradoxes of Infinity 21 Let us precede the solution of the paradox with a comment that there is no doubt that the conclusion asserting equality of both radii is inadmissible. , the revolution of the larger circle only accompanied by the smaller one, both circles cover the distance of 2pR during a full revolution of the larger circle. In the second case, when the smaller circle revolves accompanied by the larger one, both circles cover the distance of 2pr after one full revolution of the smaller circle.
G(b, p+) is deducible from L+. (ii) The conclusion: Correct V(p++) ? G(a, p++) is deducible from L++. Proof Ad (i): 1. 2. 3. 4. 5. 6. 7. Vt (OtZb $ (GBt(b,p) and -ZBtb)) Ot+Zb $ (GBt+(b,p) and -ZBt+b) -GBt+(b,p) -Ot+Zb Correct V(p+) V(p+) ) Ot+Zb -Correct V(p+) P1a, D12, P5 1 P3, P7 2, 3 hypothesis hypothesis 4, 6, D13; 46 3 Paradoxes of Ambiguity Correct V(p+) and -Correct V(p+) 9. -(V(p+) ) Ot+Zb) 10. V(p+) ) -Ot+Zb 11. G(b, p+) 12. Correct V(p+) ? G(b, p+) 8. 0; 5–11. Ad (ii): 1. 2. 3. 4. 5.
Dotted lines symbolize simple operations of one-to-one projecting of the smaller circle on the larger circle, and thus also the larger circle on the smaller circle. Consequently, the mediation through sections of the same length is eliminated in this argument. The arguments we encounter in the problem of Aristotelian circles and all other proofs showing equipollence of a set with one of its proper subsets23 seem paradoxical as long as we believe that every set has such a property that elimination of one of its elements must invariably result in a decrease of the number of elements of that set.