There are two "atomic" sets that exist because an axiom says it does. One axiom for the empty set and one axiom for infinite sets. The other few axioms input a set, or multiple sets, and predicates, and output a set that will exist under specific circumstances. So we have two "seeds" and more than half a dozen inference-rule-like axioms. A thing, then, is a set if it is either an atomic set, namely Ø or a set equipollent to the set of natural numbers, or the result of applying finitely many other axioms to get "new" sets. If a property of sets is true of the atomic sets (i.e., the sets in the set theory for which there is an unqualified existential statement like "there is a set with the property that some formalism of 'is empty'.") and closed under the more than half dozen other axioms, then it is true of all sets.
At any rate, I think everyone might have been mistaken about how many axioms I am willing to drop (i.e., not adding its negation to the list, simply removing it from the list of axioms) in order to try to break incompleteness.
I blame myself for my lack of articulative quality in my original question. I am looking for the precise moment in a formal system when incompleteness is broken. If a FS can "model" a variant of ZFC or a variant of ZF, it can "model" PA. PA is what Godel used to prove incompleteness.
How about if we just had the empty set axiom (not even its more general brother the comprehension axiom) and the pair set axiom? Could that system model PA? Well I guess it would because we would still have something infinite going on, roughly speaking, and when that happens incompleteness creeps near.