Are you sure you still have a paddle?
Now I have heard of the concept of completeness with reference to Goedel's incompleteness theorem. But I have never heard this concept applied to a physics theory. It is usually just applied to mathematics.
But just in case, you mean the same thing as Goedel by the words complete and incomplete, what Goedel proved is that every system of mathematics, except maybe (as I remember) for plane geometry and the like, certainly for every arithemetic system, all such systems are incomplete.
So when you say all physics theories are incomplete, that is true in the Goedel sense as the underlying math cannot predict emergeant solutions that may be consistent with the assumptions of the theory.
However, I suspect that you mean the a complete theory can treat all of reality. I'm not sure what the other fellow means when he says that GR is complete and GR predicts singularities and therefore singularities must exist. My take is that a solution that is inconsistent with the assumption of covariance cannot be valid, and singularities are not covariant.
It is interesting to me to think of Goedel incompleteness as say applied to General Relativity. It would suggest that there are bone fide solutions in the math space of GR that cannot be derived from assumptions of covariance, yet are consistent with that assumption. I am having difficulty imagining what they could be.
Normally one finds solutions in the domain of a theory from having solutions to an encompassing theory, a larger theory. Real and complex variables are such an example. In real variable theory, if you encounter an infinity, the is a limit on the domain of your possible solutions. But you mcan use complex variable and analytic continuation to mgo around the singularity and find a host of other solutions that are consistent with the assumptions of real variable theory.
Consider a larger theory that includes GR, for example a TOE, whic has GR as a special case. A TOE would be such a theory. Taking the LQG theory as our TOE, what is suggested there is that space is a quantum foam of elementary loops. But what we find here is that LQG selects particular solutions, such as the correct cosmological constant, in a sea of GR solutions.
It might be that Cahill's quantum foam inflow, which yields nonsymmetric terms in the equations of GR are connected to LQG. But that remains to be shown. But this would be an example of solutions that are covariant but not predicted by GR. Still they are no emergent solutions, even if space is an emergent property of LQG as I have seen in print.
Perhaps a better example is dark matter, which is not actually predicted by GR theory. But from a variety of diverse measurements, we know it must exist. GR cannot predict how much mass exists. The amount of mass is an input to solutions in GR theory. But from astronomical observations we know that the amount of mass cannot be infinite. Otherwise we would all be in a black hole. So when it is said that matter collapses to infinity, we should be sure to realize that the total amount of mass remains finite. Putting all the mass at one point is a valid substitution when the matter is spherically symmetric, except within the distribution of the matter. The planetary orbits can be determined by this approximation.