Well, you're actually making metaphysical claims more than mathematical or physical ones. One of two assumptions, depending on how literally the sentiment that the universe is a formal system is intended, is being elided here:
1. That the universe _is_ a formal system, rather than being describable in the language of some formal system. It's not evident what the universe being a formal system even means, or how it squares with basic intuition regarding e.g. the fact that physical systems have state.
2. That, dropping the physical system <=> formal system equivalence and given some real system R consisting of some fundamental entities whose behaviors can be described in full in the language of some formal system S, (borrowing a useful construct from Lucas' anti-mechanism argument, even though I don't buy that argument) no machine can be constructed in R which computes theorems of some formal system S' in which all true statements of S are provable, meaning that no state of the system R can be said to contain a description of S', and that S' is therefore not describable by any arrangement of the entities in R (assuming some reasonable predicate over states of R that is true for a state when some arrangement of a subset of the entities in that state describes S'). Intuitively, this doesn't seem to hold up: by analogy, I can describe a universal Turing machine with a computer equipped with only finite memory. You could then attempt to go down the road of claiming that, even if a description of S' is possible in R, that a mind within R would not be capable of formulating that description, but then you're heaping on an even larger tangle of assumptions, unknowns, and things you have to define if you're going to argue the case rigorously.
The point being that confidence about _any_ hypothesis about the nature of reality made on the basis of Gödel's incompleteness theorems is not epistemologically warranted.
1. That the universe _is_ a formal system, rather than being describable in the language of some formal system. It's not evident what the universe being a formal system even means, or how it squares with basic intuition regarding e.g. the fact that physical systems have state.
2. That, dropping the physical system <=> formal system equivalence and given some real system R consisting of some fundamental entities whose behaviors can be described in full in the language of some formal system S, (borrowing a useful construct from Lucas' anti-mechanism argument, even though I don't buy that argument) no machine can be constructed in R which computes theorems of some formal system S' in which all true statements of S are provable, meaning that no state of the system R can be said to contain a description of S', and that S' is therefore not describable by any arrangement of the entities in R (assuming some reasonable predicate over states of R that is true for a state when some arrangement of a subset of the entities in that state describes S'). Intuitively, this doesn't seem to hold up: by analogy, I can describe a universal Turing machine with a computer equipped with only finite memory. You could then attempt to go down the road of claiming that, even if a description of S' is possible in R, that a mind within R would not be capable of formulating that description, but then you're heaping on an even larger tangle of assumptions, unknowns, and things you have to define if you're going to argue the case rigorously.
The point being that confidence about _any_ hypothesis about the nature of reality made on the basis of Gödel's incompleteness theorems is not epistemologically warranted.