[talanum1 (OP)]

Yes. If Mathematics is incomplete, so is Physics. Incompleteness is the inability to prove within the system some true statement.

What could be an example of such a statement?

[Kryptid]

There are a lot of things about physics that we don't yet know, like how relativity meshes with quantum mechanics, what dark matter is, and why matter dominates antimatter.

[Bored chemist]

If Mathematics is incomplete, so is Physics.

No.
It may be that there are mathematical  hypotheses which can not be proved one way or another.
But they can not apply to physics because in physics, you would simply do the experiment and find out.

Mathematics is incomplete for complicated reasons.
Physics is incomplete for a simple one.
This guy puts it well at 1:50.

[puppypower]

Math is like a faithful horse that pulls the wagon to market. The good ole horse does not care what is in the wagon, but it will faithfully bring it to the place the driver requests. For example, the physics engines within computer games use math to make the game both realistic and more fun. The math horse in computer games, can be made to follow any premises, such as infinite lives. The faithful horse of math does not care, but will accommodate the driver. Math proof is only as good as the assumptions, since any premise can be used for a math model.

Physics is more like one of the drivers of the cart. The physicists can give the horse the wrong directions and horse will go there. For example, dark matter has never been seen in the lab to know it is real. However, it is assumed to be real, by the drivers of others horses, with those horses  of math not caring if these direct lab experiments failed. The other math horse can still appear to show the affect of dark matter where we cannot do experiments.

String theory was an interesting situation. It was like the driver of the horse fell off the cart and the horse was driving itself. The horse then decided to walk down to the oat field and feast. The idea of strings was not based on experiment, but was a premise that gave math freedom; linear functions. In the end,the horse needed a driver to limit the errands.

Another twist is the difference between pure and applied science. Both can become drivers of a science cart. Pure science tries to define nature as it is. Applied science uses pure science and then tries to make nature even better. Since both use science, this can lead to confusion in terms of the driver of the natural cart, since applied science can be mistaken for pure science, and vice versa. In such cases, experiments may bear out both assumptions, and the math will follow.

The example,  if we compare natural diamonds to synthetic diamonds, synthetic is sort of an improvement on natural since it takes less time to make high quality diamonds, synthetically. If one had the theory that diamonds that formed near volcanoes, happen fast, by taking advantage of natural hot press conditions, like in the lab. We can prove this with our applied lab experiments; production facility. Now we are ready for the math. 

This is also a modern version of science magic. This type science can satisfy the philosophy of science and math because it is still science, albeit but applied science, sold as pure. The worse offenders may be in the life sciences, where statistics is over used. This math horse sits in the cart and the driver pulls the wagon for the horse.

We run a statistical experimental protocol which tells the driven where to go;  driver waits for the horse to tell him what to do, next. Ole Betsy wants to swim in the lake. Depending on how the experiment pan out , the driver will than theorize what it takes to justify the horse swimming in the lake. When published, it reads like a conscious and logical sequence of driver and horse.

[Bored chemist]

these direct lab experiments failed.

I doubt that you will find any failed lab experiments regarding dark matter.

The example,  if we compare natural diamonds to synthetic diamonds, synthetic is sort of an improvement on natural since it takes less time to make high quality diamonds, synthetically. If one had the theory that diamonds that formed near volcanoes, happen fast, by taking advantage of natural hot press conditions, like in the lab. We can prove this with our applied lab experiments; production facility.

That was a bad analogy the last time you used it; it hasn't got better.

[Eternal Student]

Hi Talanum, hope you are well.
   You've been watching YouTube videos again, I think.   Godel's incompleteness theroem and something Veritasium released recently about a hole in the bottom of Math?

[Eternal Student]

Well, no replies from talanum for a couple of days.   I guess interest was lost.  That's a bit of shame, there's hardly ever anything Mathematical discussed on this forum. 

     Non-linear chaotic systems seem to furnish us with many practical examples for undecideability and incompleteness.  Weather prediction algorithms, for example.

    Any piece of Physics that uses computers and algorithms can suffer from the halting problem (which is an undecideability issue in disguise).