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".

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