Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

84 possible collapse of contemporary mathematics12. Theorem. 0 is an additionable number.proof. Let y be a counting number. By Axiom 10, we need to show thaty + 0 is a counting number. But y + 0 = y.13. Theorem. Ifx is an additionable number, so is Sx.proof Let x be an additionable number and let y be a counting number; byAxiom 10, we need to show that y + Sx is a counting number. But y + Sx =S(y + x). By Axiom 10, y + x is a counting number, so by Axiom 9, S(y + x) isindeed a counting number.14. Theorem. Ifx 1 and x 2 are additionable numbers, so is x 1 + x 2 .proof. Let x 1 and x 2 be additionable numbers and let y be a counting number.By Axiom 10, we need to show that y + (x 1 + x 2 ) is a counting number. Buty + (x 1 + x 2 ) = (y + x 1 ) + x 2 (the associative law for addition), and y + x 1 isa counting number by Axiom 10, so (y + x 1 ) + x 2 is indeed a counting number,also by Axiom 10.We can define an even stronger notion, multiplicable number, and show not onlythat multiplicable numbers are counting numbers but that the sum and product of twomultiplicable numbers is again a multiplicable number.15. Definition. x is a multiplicable number in case for all additionable numbers y, theproduct y · x is an additionable number.By the same kind of elementary reasoning, one can prove the following theorems:16. Theorem. Ifx is a multiplicable number, then x is an additionable number.17. Theorem. Ifx is a multiplicable number, then x is a counting number.18. Theorem. 0 is a multiplicable number.19. Theorem. Ifx is a multiplicable number, so is Sx.20. Theorem. Ifx 1 and x 2 are multiplicable numbers, so is x 1 + x 2 .21. Theorem. Ifx 1 and x 2 are multiplicable numbers, so is x 1 · x 2 .The proof of the last theorem uses the associativity of multiplication. The significanceof all this is that addition and multiplication are unproblematic. We have defineda new notion, that of a multiplicable number, that is stronger than the notion of countingnumber, and proved not only that multiplicable numbers have successors that are multiplicablenumbers, and hence counting numbers, but that the same is true for sums andproducts of multiplicable numbers. For any specific numeral SSS ...0wecanquicklyprove that it is a multiplicable number.But now we come to a halt. If we attempt to define “exponentiable number” in thesame spirit, we are unable to prove that if x 1 and x 2 are exponentiable numbers then sois x 1 ↑ x 2 . There is a radical difference between addition and multiplication, on the onehand, and exponentiation, superexponentiation, and so forth, on the other hand. Theobstacle is that exponentiation is not associative; for example, (2 ↑ 2) ↑ 3 = 4 ↑ 3 =64, whereas 2 ↑ (2 ↑ 3) = 2 ↑ 8 = 256. For any specific numeral SSS ...0wecanindeed prove that it is an exponentiable number, but we cannot prove that the world of