Physics is not mathematics.
Physical laws are not axioms.
A physical theory is not the derivation of all possible hypotheses from a set of axioms.
Please repeat this like a mantra a hundred times a day for the next three weeks.
Instead a physical theory is patterned as a set of naive ontological assumptions about the approximate usefulness of a set of observational terms like "mass", "force", "gravity" etc.. "Laws of nature" are shorthand descriptions for a wide range of observations that these terms satisfy within a more or less well defined domain of application of the theory.
The theory is both expected and accepted to fail outside of this domain and some of the most interesting aspects of it is how it fails along the boundaries of this domain, where one or several different theories with different sets of observational assumptions and laws of nature take over. That failure boundary is where, at any given time, the active scientific work takes place.
One of the important questions of physics is the question how far one can push that boundary and what the relation between the terms on both sides of it are. Sometimes we don't know, yet, what lies behind the range of application of a theory, sometimes we have important hints from multiple theories of what that could be, but we haven't found the right language, yet, to express those ideas concisely.
I know that this must be awfully frustrating for someone who thinks about the world in terms of axiomatic set theory and existence proofs. The world just isn't like that and physics is not that neatly organized. That, however, is what makes physics endless intellectual fun!