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- Why is the Axiom of Infinity necessary? 3 answers
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?