The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

from which it follows that d is finite, since by proposition (143) of Grundgesetze {viz. dPb → aP*b → aP*=d}, any predecessor of a finite number is finite.

Theorem (FA).17 Suppose dP #[x: xP*=d]. Then #[x: xP*=d] is finite.

Proof. In FA, define h: [x: 0P*=x] → [x: xP*=d] by:

h(0) = d; h(n+1) = y if yPh(n) and
= h(n) if ¬∃y yP h(n).

The definition is OK since P is one-one.

Since in general yR*z ↔ z(R⋃)*y,18 ∀x(xP*=d ↔ d(P⋃)*=x), and so h is onto. Therefore [x: xP*=d] is countable, i.e. either finite or countably infinite. If the latter, then #[x: xP*=d] = א0 and by the supposition of the theorem, dP א0. But as we saw just after the proof of (4′), xP*= א0 ↔ x = א0. Since dP א0, dP*=א0, d = א0, and #[x: xP*=d] = 1, contra #[x: xP*=d] = א0. Therefore #[x: xP*=d] is finite.

Thus Frege could have proved (1) after all and thus appealed to induction 2 to prove (0′). Of course the technology borrowed from second-order arithmetic used in the proof just given, particularly the inductive definition of h, is considerably more elaborate than that needed to derive induction 3 from induction 2. The conjectural proof is unquestionably to be preferred to this new one on almost any conceivable grounds.

So. Frege erred in §§82-3 of Die Grundlagen, where an oversight marred the proof he outlined of the existence of the successor. Mistakes of that sort are hardly unusual, though, there are four or five ways the proof can be patched up, and Frege's way of repairing it cannot be improved on. But even if one ought not to make too much of Frege's mistake, there is lots to be made of his belief that (1) was true but unprovable in his system. One question that must have struck Frege is: If there are truths about numbers unprovable in the system, what becomes of the claim that the truths of arithmetic rest solely upon definitions and general logical laws? Another that may have occurred to him is: Can the notion of a truth of logic be explained otherwise than via the notion of provability?


APPENDIX 1. COUNTERPARTS IN GRUNDGESETZE
OF SOME PROPOSITIONS OF DIE GRUNDLAGEN
Proposition of this paper Proposition of Grundgesetze
Hume's Principle 32,49
∀x¬Fx ↔ 0 = #[x:Fx] 94,97
78.1 114
78.2 113
____________________
17
This result is due to the second author.
18
Ru is the converse of R.

-423-

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