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.
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.