[FOM] Is Godel's Theorem surprising?
robblin at thetip.org
Thu Dec 7 16:49:01 EST 2006
The expectation that PM should be complete comes from the notion that
PM is a variety of logic which seeks to have "all and only the logical
truths". It should be a Sound and Complete system of logic.
Completeness is as important as Soundness because if a system is not
complete, then there are logical truths which are not derivable in the
formalism and consequently, the formalism does not embody a complete
formalism of the logical truths.
To see why this is important, a parallel case in Physics is worth
considering. A given Theory of Physical Reality (say Newtonian
Mechanics) allows one to derive MANY but not ALL of the observed
phenomenal truths. Until ALL of the observed phenomenon are accounted
for by the physical theory, we regard it as incomplete and in need of
improvement because we know (or have a strong suspicion) that there are
factors which ought to be taken into account in our system which are
not being accounted for. This can result (as it did in the case of
Newtonian Mechanics) in a complete falsification of the original theory
in favor of a stronger theory. That is, without knowing that our
theory is Complete, it remains a possibility that it is in fact FALSE.
This is slightly more troubling in logic since it means (for incomplete
systems of logic) that they are not the whole truth and it is not
clear, for them, whether the statements that would contribute to making
them complete wouldn't also make them inconsistent or false.
This is the surprising aspect of Godel's theorem as applied to PM and
With Godel's theorem, we effectively KNOW that any system that is
complete and at least as strong as PM is not consistent.
This is surprising and unfortunate because a pragmatic person looking
at it, as it were, from the outside, might conclude that PM (and
related systems) therefore can't be the "initial segment" of a complete
theory of arithmetic because any completion of them is inconsistent.
On Dec 7, 2006, at 5:12 AM, 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?
> I'm not intending the above to be questions of mathematical fact.
> I'm just wondering what accounts for the shock so to speak of Godel's
> Theorem. One answer I've read is to the effect that everyone at
> the time thought PM was complete. But for me, that's not
> satisfactory. I'd like to know *why* they should have thought it was
> complete. Did they have *intuitions* for thinking it had to be
> I'm also wondering, though this is a separate point, whether today
> the theorem is not only not surprising, but perhaps even intuitively
> obvious. One unsatisfactory answer would be that incompleteness is
> now not surprising because we now know it holds. But do we now have
> distinctly different *intuitions*, aside from the proof itself
> (though of course the proof can't entirely be discounted), that
> establish, let's say the "obviousness" of the result?
> Charlie Silver
> FOM mailing list
> FOM at cs.nyu.edu
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