[FOM] Godel's First Incompleteness Theorem as it possibly relatesto Physics

[Brian Hart]

On Sat, Oct 11, 2008 at 12:36 AM, Kreinovich, Vladik <vladik at utep.edu> wrote:
>
>
>> -----Original Message-----
>> From: Brian Hart
>
>> Why doesn't Godel's 1st Incompleteness Theorem imply the
>> incompleteness of any theory of physics T, assuming that T is
>> consistent and uses arithmetic?
>
> It does.
>
>> Shouldn't the constructors of the
>> Theory of Everything be alarmed?
>
> The designers of the Theory of Everything are interested in predicting
> all possible events -- i.e., in effect, all possible ATOMIC statements.
> Goedel's theorem shows that one cannot easily predict the truth values
> of the corresponding quantifier statements, but this is not of any
> serious concern to physicists.
>
> If a physical theory like Newton's gravity can predict the position of
> all the planets in any given future moment of time, physicists will be
> very very happy.

I don't think it can, though.  For example, isn't the n-body problem
within Newtonian gravity where n >= 3 intractable?  One of the first
observable improvements upon Newton's formulation of the theory of
gravity made by General Relativity, for example, was to correctly
account for the precession of the perihelion of Mercury.

>
> And the fact that this theory does not answer a quantifier-based
> question -- like "is true that for every moment t, there is a further
> moment s at which Moon is exactly 250,000 km from the position of the
> Earth at moment t" is not of a big worry.

However, in the future it would be of practical value to be able to
answer questions like the above with arbitrary accuracy using physical
theory.  Does this require only tweaking the form and/or the content
of a given physical theory?

>
>> I know this suggestion of
>> application of Godel's theorem was made decades ago but why didn't it
>> make a bigger impact?  Is it because it is wrong or were there some
>> sociological reasons for mainstream ignorance of it?
>
> The reason is that incompleteness in physics and incompleteness in logic
> are two different things. Incompleteness in physics usually means a much
> simpler thing: that we cannot predict a future event; in other words, it
> is mainly about ATOMIC statements. Incompleteness in logic means,
> crudely speaking, that we cannot check whether a given quantified
> (NON-ATOMIC) statement is true or false.

Any sub-theory, i.e. quantum mechanics, general relativity, of the ToE
is merely an approximation to it, only relevant within a given domain
of theoretical validity.  Therefore, presuming the ToE offers a
complete description of the cosmos, any sub-theory of it would always
be incomplete in a descriptive and therefore predictive sense in that
there will always be a set of undecidable statements it can physically
determine through dynamical evolution.

>
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