Does the Axiom schema of Replacement imply the Axiom of Infinity?

Question

The axiom of infinity says that the set of natural numbers exists, while the axiom of replacement says that if an object (a member of a set) exists, then all definable mappings of that object yield objects (e.g if 0 is an object, then 0+1 is also an object).
Doesn't this mean that the axiom of infinity is redundant since one can recursively prove the existence of the set of natural numbers using the successor function of the Peano axioms?

Answer 1

No, replacement doesn't imply infinity. What you've found is that if $S$ is a set so is $\{x\cup\{x\}|x\in S\}$, hence so is $x\cup\{x\}$. Each element of $\omega$ can be proven to be a set by this method, but we can't use this on its own to prove $\omega$ is a set, even though it's just the union of its elements.