For sake of argument we will be presuming below that ZFC is consistent. While it is true that ZFC is model-complete in your sense (which holds for all first-or

For sake of argument we will be presuming below that ZFC is consistent. While it is true that ZFC is model-complete in your sense (which holds for all first-order theories as you note), ZFC has many non-standard models that come with with non-standard sets of natural numbers within them. There are Σ₁-sentences that are false in models with standard sets of natural numbers but are true in models with non-standard sets of numbers. Tarski's definition of truth, which I'm using for the definition of (in)completeness in my comment, excludes non-standard models because it requires, for example, for a Σ₁-sentence ∃x. P(x) to be considered true, that there actually exist a numeral (e.g. a closed term such as 42), 'n' where P(n) is true (recalling P is a decidable predicate the truth or falsity of P(n) can be determined (in principle) by calculation). When a Σ₁-sentence is only true in models with non-standard natural numbers, such numerals that witness P do not exist. Some people argue that non-standard natural numbers are just as good as standard natural numbers, and you can try that argument I suppose. However, if I'm hiring someone do formal analysis of some algorithm and they "prove" that it is terminating by showing it terminates in some non-standard number of iterations, then I'm just going to fire them because such a result isn't helpful for reasoning about programs. There is a material difference between standard and non-standard natural numbers. The standard natural numbers are formed in a natural way from the terms that make up the formal language of the first-order logic under consideration (i.e. the terms that are numerals), but the non-standard natural numbers cannot since they include "numbers" that have no associated numeral. Also note this way of characterizing the standard natural numbers won't work for characterizing "standard sets", because we do not expect all sets to come with canonical concrete terms denoting them, like we do expect for natural numbers.