One can consider a variant of the Dedekind-Peano axioms in which one replaces the assumption that every number has exactly one successor by the assumption that every number has at most one successor, leaving the other axioms the same (and in particular retaining the second-order induction axiom). The models of this theory are then the initial segments of the natural numbers. Call it a "size-neutral" version of the second-order Dedekind-Peano theory. My question is, to what extent can one prove analogues of standard theorems about the natural numbers in this theory?
For instance, what would serve as a size-neutral analogue of the proposition that the set of primes is infinite? Ideally it would be some proposition P with the property that if you assume that P is true, and you assume that every number has a successor, then the standard claim about the set of primes being infinite would follow without too much trouble (where "too much trouble" means "going back to the beginning and proving the infinitude of the primes in the usual way without invoking P at all").
I'm aware that even defining "prime" is problematic. To begin with, in a size-neutral theory, one cannot define addition or multiplication in the usual recursive way, or rather, if one does, the operations are not provably total; if one's model cuts off at n, then sums and products that exceed this bound are undefined. One could choose to remedy this with a definition that basically says "If a number has no successor, define it to be its own successor; now proceed using this new, enlarged notion of succession instead of the old one", and then sums and products that in ordinary arithmetic exceed n would now equal n -- though I'm not sure this is a good way to go.
This can't be a new line of thought, so I'd be happy with answers of the form "Go read X".