 One way to interpret this equivalence is that we can take the existence of an initial algebra for  Fx  =  x+ 1  as an axiom of set theory, in place of the usual Axiom of Infinity.   That axiom says that there is an algebra for the singleton functor  Sx = {x}  on sets which contains ∅ as an element and whose structure is the inclusion. This principle is easier to state than the algebraic reformulation. It takes a bit of work to use the simpler standard formulation to derive the Recursion Principle, and this is one of the basic topics in any course on axiomatic set theory. 