Then it might be that the standard natural numbers are the unique minimal element in this inclusion relationship.
Why would we care about the smallest model? Then, we'd end up doing weird things like rejecting the axiom of choice in order to end up with fewer sets. Set theorists often actually do the opposite.
Generally speaking, the model of Peano arithmetic will get smaller as the model of set theory gets larger.
And the point is not to prefer smaller or larger models; the point is to see if there is a unique definition of the natural numbers.
Building on the very bad Gödel anti-AI argument (computers's are formal and can't prove their own Gödel sentence, hence no AI), it occurred to me that you could make a strong case that humans could never recognise a human Gödel sentence. The argument goes like this:
Now, the more usual way of dealing with human Gödel sentences is to say that humans are inconsistent, but that the inconsistency doesn't blow up our reasoning system because we use something akin to relevance logic.
But, if we do assume humans are consistent (or can become consistent), then it does seem we will never knowingly encounter our own Gödel sentences. As to where this G could hide and we could never find it? My guess would be somewhere in the larger ordinals, up where our understanding starts to get flaky.