If we can't solve every mathematical problem, there's gonna be things in science that aren't solvable either.

Not at all, math and science are very different things. Math is a fixed system of rules that we constructed. Within these rules, there are possible statements which cannot be proven or disproved using only those same rules.

Science is different, we don't know the rules but we observe, measure, and make predictions. It's not possible to "solve" physics but that's because we can't make infinitely accurate measurements, there's nothing systemic to prevent us from making a complete theory.

Halting Problem

Goes for me into the same bucket as proofs of god.

I never said we'd be able to understand or prove everything, just that there is some logic underpinning reality. It might be that some things are fundamentally unknowable, but that doesn't mean it doesn't exist to be known, just that we'll never know it.

I also don't get what the halting problem proves about reality. It might be possible that infinities or unresolvable results are real, so long as we can still exist. The cosmological principle proves that we have to live in a reality that it is possible for us to exist in, otherwise we wouldn't be here to observe it. So long as the infinities or uncomputable problems don't prevent our existence, it might represent reality. If the equation doesn't allow us to exist, then it doesn't represent reality.