# [FOM] Godel's First Incompleteness Theorem as it possibly relates to Physics

Giuseppe Longo Giuseppe.Longo at ens.fr
Tue Oct 14 05:03:35 EDT 2008

```Poincare' 1890 Three Body Theorem may be seen as a form of 'undecidability
result'. Consider the formal set of nine (Newton-Laplace) equations
determining the evolution of three gravitational bodies on a plane and a
statement on the state of the physical system parametrized in a
sufficiently remote future. Then one provably cannot decide the later by
an analysis of the equations. Today, the time of unpredictability of the
evolution of a deterministic system has been quantified for the Solar
System (Laskar, 1994) (see the references in teh quotations below).
Poincare''s theorem though cannot be considered a technical predecessor of
Godel's (at most an epistemological one: Poincare' explicitly stressed in
several writings the importance of ''negative'' and ''unsolvability''
results). This is because the theorem doesn't concern a purely
mathematical statement: unpredictability pops out in the relation of a
physical process to the intended mathematical model. And (classical)
physical measure, by principle an interval (an approximation), is
essential to yield unpredictability of a deterministic system (a
fluctuation/perturbation below measure may produce an observable, even
major, change in time).
However, there is a deep link between deterministic unpredictability and
recursion theoretic undecidability. In deterministic chaotic systems one
can give various notions of randomness. A relevant and purely mathematical
one is Birkhoff ergodicity. It deals with infinite trajectories (thus, it
is given in and concerns mathematics only, even if it is inspired by
physics). Along the lines of previous
work, it has beeen recently shown by two PhD theses (by M. Hoyrup and C.
Rojas) under my suprevision, that this asymptotic form of randomness (a
mathematical extension of unpredictability in deterministic systems)
coincides with Martin-Loef randomness, in soundly effectivized measure
spaces for physical dynamics.
Now, ML-radomness is a "godelian" notion (extensively developped also by
Chaitin and others), as it is given in purely recursion theoretic terms
and it is a strong form of undecidability for infinite sequences.
In conclusion, the unpredictability of non-linear (even simple) dynamical
systems is known since long: no need for a Theory of Everything. Its
rigorous relation to undecidability (worked out in
great physical generality in the theses above), is a relevant but
delicate (asymptotic) result.

such as
G. Longo, T. Paul.  The Mathematics of Computing between Logic and
Physics.  Invited paper, "Computability in Context: Computation and Logic
in the Real World ", (Cooper, Sorbi eds) Imperial College Press/World
Scientific, 2008.

Hoyrup and Rojas theses may be found in
http://www.di.ens.fr/users/hoyrup/
http://www.di.ens.fr/~rojas/

Finally, the incompleteness paradigm may be a useful tool for knowledge
also in other disciplines:
G. Longo, P.-E. Tendero.  The differential method and the causal
incompleteness of Programming Theory in Molecular Biology.  In
Foundations of Science, n. 12, pp. 337-366, 2007.

Giuseppe Longo
http://www.di.ens.fr/users/longo/

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