Islam Hassan said:
Apart from difficulty of falsification, have there been any physical theories for which the underlying mathematical development has proven undecidable in the sense of Gödel's incompleteness theorem?
You might want to look at a book by Torkel Franzén, called Godel's Theorem: An Incomplete Guide to Its Use and Abuse.
Physics makes use of various mathematical theories.
Some of these, such as elementary Euclidean geometry or the first-order theory of the reals, are theories to which Godel's theorem doesn't apply. (Tarski, "A decision method for elementary algebra and geometry.") They are provably self-consistent, and for any statement within these theories, there exists rules for determining in a finite number of steps whether that statement is true or false. Let's call this set of theories A.
Some other theories used in physics, such as Zermelo–Fraenkel set theory with the axiom of choice, are theories that Godel's theorem does apply to. Call this set of theories B. Godel's theorem guarantees that these theories contain undecidable statements (unless they are not self-consistent, which nobody expects is the case). However, there is every reason to believe that these undecidable statements are ones that have no interesting implications for any real-world application such as physics. Franzen discusses this.
There may be a lot of different theories in B, but if you name two of them at random, you will typically find that they are equiconsistent with one another. For example, the theory of Euclidean plane geometry (not just the first-order theory) is in B, and so is the theory of elliptic geometry. These two theories have been proved to be equiconsistent with each other, meaning that if there is a contradiction that can be reached from the axioms of elliptic geometry, then the same is true for Euclidean geometry. Since nobody expects that to be true in the case of Euclidean geometry, we don't lose any sleep over possible contradictions in elliptic geometry.
Please also keep in mind that physical theories are essentially never stated as formal axiomatic theories, so Godel's theorems don't apply to physical theories. Franzen discusses this. An exception is that there is a book on the formalization of part of the Principia: Jacques Fleuriot, A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton's Principia. This example is noteworthy because it's so rare for anyone to go to such lengths to pose a physical theory as a formal axiomatic theory.
A theory can make clearcut predictions based on given data, and yet there can be undecidable statements
about the theory. This doesn't mean that there is something wrong with the theory. This will be true for any interesting theory. For example, Conway's game of life can be thought of as a physical theory describing a certain simple universe. The theory always makes clearcut predictions about the evolution of its universe. However, there are undecidable statements
about the theory.
We had an extremely long thread about this kind of thing last year:
https://www.physicsforums.com/showthread.php?t=455641
-Ben
