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[Set Theory] I'm confused about the relation between the Peano Axioms and ZFC set theory

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I'm sure there are folks that can answer this question better than me and this is partly my interpretation. Please correct me where I'm wrong
But Peano Axioms were created before ZFC for the hopes of giving arithmetic a foundation which would give rise to an axiomatized foundation of the Real numbers and therefore, analysis.
ZFC was created some years later as a foundation for all of modern mathematics. Which means that yes, you are able to derive (first-order, I believe) Peano axioms within ZFC. You'll define Natural numbers with the help of the empty set as your 'zero', the successor 'function' (it's not strictly a function at first) S(x) = x U {x} and the axiom of infinity. Then everything else about Natural numbers will be proven using rules derived from ZFC.
I really liked how Jech & Hrbáček approach to it in "Introduction to Set Theory". If your knowledge is on par with an undergraduate (like myself), I think it's a good place to start without necessarily going in too much depth right from the get-go.

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