I asked this question in Philosophy.StackExchange whilst trying to get to grips on Badious declared philosophy on using mathematics as ontology. But was advised to ask it here because of the mathematical content.
Can arithmetic when codified by the first-order Peano Axioms recreate the transinfinite (cardinal) hierarchy of Set Theory (ZFC)?
I suspect not, simply because we have no formal means of creating a set - and so we cannot even take the first step to define cardinality.
How about 2nd-order Peano Axioms, I suspect here it can (but am not sure), as in the introduction of the previous article we have:
It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics.
But if you can only take subsets of the integers, then although you can construct the reals (by identifying them with certain subsets of the naturals), you won't be able to take subsets of them - which will stop one from building bigger sets. But on the other hand you can always generalise then to 3rd-order arithmetic and so on.
However, in the SEP entry on higher-order logic they say that:
'there is a sense in which the power-set operation is definable in second-order logic'
'This second-order expressibility of the power-set operation permits the simulation of higher-order logic within second order'.
So maybe you don't have to walk up the ladder of n-PA but only simulate it in 2-PA.