[FOM] RV: Is Godel's Theorem surprising?
ignacio
natochdag at elsitio.net.uy
Sat Dec 9 05:41:09 EST 2006
Harvey Friedman wrote:
"ALSO: There are two senses of surprise for a theorem. One is that one is
surprised by the fact that the theorem is true. The other is that one is
surprised by the fact that anyone was able to prove that the theorem is
true. There were probably many people who weren't too surprised that it is
true, but who were shocked that anyone was able to prove such a thing."
In fact, the shock of Gödel's demonstration was not the result, it was the
achievement: the notion of circularity and inconsistency was in the air
throughout all the western world from the Warsaw circle in Poland to Pierce
in USA since the turn of the century, it could hardly come as a shock to
anybody, but the fact that it could be proved rigorously was thought
amazing. It is a silly dramatization of the historical events to think that
Hilbert's program was given a blow by the incompleteness proofs: Hilbert's
passion and vision was concentrated in the results that could be got by
reducing to a formal system any given demonstration: it was not a desire to
exhaust mathematics nor to reduce it to "a game with symbols"(something he
never said by the way): proof of this is the fact that Gentzen continued to
expand and refine the methods of formalism immediately afterwards of Gödel's
results, hardly being shocked by them, and Godel never lost interest in
Hilbert's methods throughout all his early period.
Harvey Friedman wrote:
"In fact, following a general line that I have discussed on the FOM fairly
recently, one can simply ask if PA is complete for sentences that are not
very long in primitive notation. This is of course a very difficult
problem."
This difficulty is described as early as 1917 by Weyl in his book on the
continuum problem, where he points out the impossibility of substituting the
natural numbers for functions: Weyl's particular interest here was the
foundation of real analysis, but it already shows how, from a Brouwerian
intuitive point of view, inconsistency and circularity was not a new and
surprising thing.
I.N.
-----Mensaje original-----
De: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] En nombre de
Harvey Friedman
Enviado el: Viernes, 08 de Diciembre de 2006 07:58 a.m.
Para: fom
Asunto: Re: [FOM] Is Godel's Theorem surprising?
On 12/7/06 8:12 AM, "Charles Silver" <silver_1 at mindspring.com> 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
> complete?
> 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?
>
I assume you are talking exclusively about Godel's First Theorem. Not
Godel's Second Theorem.
One reason it was regarded as surprising is that, up to that time, every
single example of an arithmetic theorem was easily seen to be provable in
PA. At that time, there was no idea that arbitrary arithmetic statements
might differ fundamentally from the arithmetic statements that came up in
the course of mathematics.
Of course, it is still true that PA may be complete for all arithmetical
sentences that obey certain intellectual criteria - criteria which are
normally left informal. I have no doubt that a lot of people will be very
surprised by examples of statements independent of PA that meet certain such
informal criteria. Much more surprised if PA can be improved to ZFC.
In fact, following a general line that I have discussed on the FOM fairly
recently, one can simply ask if PA is complete for sentences that are not
very long in primitive notation. This is of course a very difficult problem.
ALSO: There are two senses of surprise for a theorem. One is that one is
surprised by the fact that the theorem is true. The other is that one is
surprised by the fact that anyone was able to prove that the theorem is
true. There were probably many people who weren't too surprised that it is
true, but who were shocked that anyone was able to prove such a thing.
Harvey
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