[FOM] Is Godel's Theorem surprising?
Rob Arthan
rda at lemma-one.com
Sat Dec 9 07:48:57 EST 2006
On Thursday 07 Dec 2006 1:12 pm, Charles Silver wrote:
> Why should it be so surprising that PA is incomplete, and even (in a
> sense) incompletable?
> Or put the other way, why should we have thought PA (or, for Godel,
> the system of Principia Mathematica and related systems) would have
> to be complete? It has been alleged, for example, that at the time
> of Godel's proof John von Neumann had been working on proving
> *completeness* for PM or some related system? Why would von Neumann
> have thought *intuitively* that the system could be proved complete?
Yu may find some clues in the papers by von Neumann and the accompanying
commentary in van Heijenoort's collection of papers "from Frege to Goedel".
The commmentary says that von Neumann was looking for a "minimal model of set
theory that could be uniquely characterised". Perhaps, the intuition was that
that minimal model would come equipped with or maybe even be defined by a
complete logic.
>...
> I'm also wondering, though this is a separate point, whether today
> the theorem is not only not surprising, but perhaps even intuitively
> obvious.
But then there must be some intuitively obvious difference between a complete
system like the first-order theory of the reals and an incomplete system like
PA or ZF. What would that intuition be?
Regards,
Rob.
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