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

Antonino Drago drago at unina.it
Thu Oct 16 18:14:18 EDT 2008

Brian Hart asked:

> 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?

Part of Antonino Drago's reply went

"Third. In theoretical physics quantifiers play an important role only when
they are included in some principle: " *Any* body either in rest or in
uniform motion perseveres in its state
unless a force changes its motion" (Newton 1687). Of course, this statement
is incomplete in the sense that Newton did not predicated this statement
about *all* bodies if not in an idealised way, i.e. in a non experimental
way. Is physics this? Again, it is a conjecture in order to obtain results
fitting experimental data. But notice that theoretical physics is capable to
disregard this kind of principles which include quantifiers; the same
inertia principle was stated without quantifiers by Lazare Carnot (1803) as
follows: "Once a body is at rest it does not change its state; when in
motion it is not capable to change its speed and its direction". Hence
physics is capable to dismiss at all the quantifiers; quantifiers come occur
in a way of speaking that make use of more elaborated statements. In the
more adequate statements to the experimental reality, no quantifier does
exist. In such a case there is not reason to apply Goedel's theorem."

Keith Brian Johnson asked:

But can quantified statements in theoretical physics be dismissed so easily?
When I look at "Once a body is at rest it does not change its state; when in
motion it is not capable to change its speed and its direction," I see a
universally quantified statement, as "a body" seems to refer to any
arbitrarily chosen body, rather than to some specific body. Hence, it seems
to say something like, "For all x, if x is a body at rest, then it does not
change its state, and if it is in motion, then it does not change its speed
or direction, without a force's acting on it." Am I misunderstanding
Carnot's formulation?

Antonino Drago reply:
It is not so simple. It corresponds to fix a body in a given system of
reference in a give state by means of experimental tools which are not
included in the statement of the principle. It amounts to renounce to
predicate through the principle an universal statement (for all bodies in an
infinite class of physical situations).
Of course, the  subsequent development of the theory is different;
analytical the
Newtonian theory, aimed to establish operative rules or invariants Carnot's
one (see my paper in Am. J. Phys. 72 (3) 2004, 407-9.

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