I have been thinking about Gödel's incompleteness theorems and their ramifications for the whole of mathematics.
In this question I assume some fixed formal system F expressive enough for the theorems to go through.
The essence of the theorems seems to be that there are number-theoretic facts that are true but unprovable.
Now, let's talk about possible foundations of mathematics. Arithmetic of natural numbers is an integral part of mathematics, so the foundation should include arithmetic by definition. Also, the foundation has to have some ontology to it. For example, the ontology of ZFC includes pure sets only and nothing else.
Here's where things get interesting: every object of ZFC is a set, including numbers, so the incompleteness spreads not only to numbers, but to sets as well, giving rise to all of the undecidable problems seemingly completely unrelated to arithmetic.
Now, can we come up with a foundation which has the following ontology: numbers exist, and some other fundamental objects exist as well having no formal connection to said numbers?
Now, these numbers in this foundation never leave the "quarantine zone", with their unprovability problems never spreading like wildfire across the whole foundation, leaving the other fundamental objects "healthy".
Is such a thing feasible? Is the situation with ZFC inevitable for every conceivable foundation of mathematics?
Is my understanding flawed?