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

Kreinovich, Vladik vladik at utep.edu
Sat Oct 11 00:36:06 EDT 2008

> -----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. 

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. 

> 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. 

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