>>12929200No it's not. In set theory "every natural number has a successor" holds without the axiom of infinity. If n is a natural number (von Neumann definition), then the successor of n, n+1 is defined as the union of n and {n}. The set {n} exists by the pairing axioms, the set {n, {n}} exists by the pairing axioms, then the union axiom applied to the set {n, {n}} tells us that n union {n} exists, that is n+1 exists. Therefore we have a proof in set theory without infinity that every natural number has a successor.
What the axiom of infinity does instead is it tells us that we can collect all the natural numbers into one definite set. That is, there is a set that contains ALL the natural numbers.
This is the assertion that infinity can be completed, and is the main point of contention between finitists and infinitists. Finitists are mostly fine with every natural number having a successor, the trouble comes form collecting them all into one infinite definite object.
