>From the point of view of consistency, there is no structure in >which the numbers can be interpreted that is more convincing than the >system of numbers. It was this that led Hilbert to the conception of >a syntactical consistency proof. (Or so I remember him as saying.) > >There a seems to have been no sense of an impossibility of a >syntactical consistency proof among the people working in proof >theory in 1929. This group included von Neumann, who was, as was >Landau, in Berlin at that time.
It would require more than consistency, of course, to establish basic results about numbers. There would need to be some proof of the soundness of the formal system, on the standard interpretation, before one could show even that 2+3=5, for instance.
But this follows just from the character of Hilbert's meta-mathematics; there was no need to wait on Goedel to point it out. So the sociology of those times was very extraordinary. Goedel's Theorems do not show, for example, that while some arithmetical truths are provable formalistically, others are not. In fact none are, since any derivation within the formal system must be supplemented with a demonstration of its soundness, on the standard arithmetical interpretation. The basic problem for Hilbert's programme was that it gave no account *at all* of what is true in a model of some formulae, being deliberately concerned with just the formulae themselves.
The common convention of not showing quotation marks around formulae, I think, is one possible reason why the group in 1929 (among many others) were confused. Within PM one might be able to derive some formula like 'Con(PM) -> G' (notice the quotes), and, knowing that the system was sound therefore be able to know that if Con(PM) then G (notice the lack of quotes, and the use of 'that', just then). But if we know that the system is sound then we know that it is consistent, so we could further deduce that G. Not everyone compares our capacity to know *that G* with the system's incapacity to derive 'G' - it is not the same thing which is knowable but not derivable.
Putnam suggested that we were in no better position than PM in being able to demonstrate its consistency (and hence did not really know that G). But if we were in no better position, then we would not even know that 2+3=5, either. At this point any one with any common sense should raise their hands, just like Moore did against the disbelievers in the External World.
There is certainly no 'proof', involving just a series of formulae, that Peano's Postulates are true on the standard interpretation. For *that Peano's Postulates are true on the standard interpretation* is not a sentence, and so, a fortiori, is not the last sentence of any rule-governed series of sentences. Neither, of course, can any arithmetical fact be in this position, since that 2+3=5, for instance, is equally not a sentence. The proof of this arithmetical fact has to be non-formal, at least at some stage, and can even proceed entirely in this way - see Wittgenstein's Remarks on the Foundations of Mathematics 2nd ed. 1978, p58f, for instance.
-- Barry Hartley Slater Honorary Senior Research Fellow Philosophy, M207 School of Humanities University of Western Australia 35 Stirling Highway Crawley WA 6009, Australia Ph: (08) 6488 1246 (W), 9386 4812 (H) Fax: (08) 6488 1057 Url: http://www.philosophy.uwa.edu.au/staff/slater