The problems I find with your argument are that you are taking a partially true statement and extending it as if it fits the whole of science. Yes there are parts of physics theories that support direct verifiability. The double slit experiment with electrons is not one of them. After all, nobody has ever seen an electron, only indirect observation is possible. You can see the result on phosphor, you can measure the induced voltage, ... none of these are direct observations.

But with the double slit experiment we are not that far from direct observation. Let's take the theory that there is a black hole in the center of the milky way ... would you care to explain to me how I can observe that directly ?

So your statements are misleading. Some tiny parts of some physics theories are directly verifiable. Some are indirectly verifiable. Most current ones are not really verifiable at all. Astronomic theories are both especially important for physics, they are used as support arguments, and let's just say they are very good examples of indirect observations, and there are plenty of historical arguments for physics theories coming out of astronomy that turned out to be wrong. One of the initial proofs of relativity theory, the measurement of the movement of Saturn's moons was a wrong measurement. It lead to the right conclusion, but the measurement itself was incorrect, which makes it especially interesting to me.

But that's just peanuts. The real kicker is : some physics theories are known to be wrong (which was the real point of referring to black body radiation : it was a known hole in a theory for almost a century), but we don't know how they're wrong, and nobody's ready to dump the theories. E.g. does the standard model match the "real" structure of the universe ? Well, we know it doesn't, yet everybody's using the standard model. Everybody believes in electrons and positrons and protons and large antiprotons ... yet we know something doesn't match up because we can run experiments that don't match predicted outcomes of the standard model theories. Have you seen anybody claim that the standard model is wrong ? See anyone dumping it ? I haven't. Did I miss something ?

Some theories in physics are direct contradictions. Take relativity versus big bang theory for example. Care to explain how the speed of light limit fits in with inflation theory (which as far as I know is still part of the current state of the art) ?

TLDR: In physics we have different theories that don't really interact, not in the maths part anyway. In order to progress within physics we just assume that you can just pick and choose whatever matches the observed data best. Gravity resistance in neutron stars ? Oh that involves the Pauli exclusion principle ... that sounds cool but the Pauli exclusion principle comes from a theory that ... doesn't have gravity. How is this consistent ? Whatever you want to call it, I call foul on this reasoning.

But such arguments, while wrong to any logical mind, have one big redeeming quality : they work. That's how science really works : part of it is giving us the ability to make cell phones, cpus, ... what have you. Part of it is an interesting story to tell "the public", and to put on grant requests of course. Part of it is interest and what appears to make sense to human minds, however broken the logic is.

But all of it flies in the face of the idea that science, without pick-and-choose tactics, satisfies any reasonable standard scientists sometimes claim it does. Physics, taken as a whole, doesn't satisfy either falsifiability, verifiability, or predictive ability, and you'd be hard pressed to find even small portions of it that have no known, shall we say "bugs" : observations that don't match, math that doesn't quite work. Hell, the theories don't even satisfy mutual consistency.

I remember the first physics lesson I had. I remember it because of the math the professor used. You start with constructing an equation for the path of an electron flying through a magnetic field. Then there was five minues of "this is nearly zero" (professor scratches that part of the equation out). This matches that, not exactly, but under normal circumstances they're more or less equal (scratches two non-matching things from above and below the fraction line) ... I was shocked. But let's be fair here : while yes, there are better methods for this particular problem, but this is how physics works. Mathematical rigour destroys most physics methods.

The argument you're making about maths is similarly flawed. You're taking a single example, and extending it around the entire theory, which is not a valid reasoning.

The axiom of choice was one of the initial tries to fix incompleteness, it's a debacle because the attempt was flawed. Also maybe pertinent here : English is not my first language, so if debacle is not the right word, just pretend I used another. It was triumphantly held up as a solution to incompleteness, and was flat-out wrong.

As for Gentzen's proof, proving that one theory is consistent using another theory that may not be consistent itself, while a remarkable achievement, does not resolve the problem Godel raised. It does not pull us out of the mud.

And let's not be too strict here. E.g. I realize what you mean by referring to Cantor's diagonal argument, but technically it is not applicable to logical statements. You're right that I confused relativity and quantization when referring to black body radiation.