An actual first-order formulation of ZFC?

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SUMMARY

The discussion centers on the first-order axiomatization of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Participants clarify that ZFC is indeed a first-order theory, often misrepresented in higher-order logics. They emphasize that while higher-order formulations exist, the first-order version can be found in numerous sources, including Wikipedia. Additionally, it is noted that first-order ZFC necessitates infinitely many axioms, whereas first-order NBG (von Neumann-Bernays-Gödel set theory) is equivalent and requires only finitely many axioms.

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  • Understanding of first-order logic
  • Familiarity with Zermelo-Fraenkel set theory (ZFC)
  • Knowledge of higher-order logics
  • Basic comprehension of axiom schemas
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  • Research first-order axiomatization of ZFC in detail
  • Explore the differences between first-order ZFC and first-order NBG
  • Study axiom schemas and their role in set theory
  • Examine higher-order logics and their implications for set theory
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Mathematicians, logicians, and students of set theory seeking a deeper understanding of first-order formulations and their applications in formal logic.

mpitluk
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Can someone point me to a first-order axiomatization of ZFC?

As I've mostly seen ZFC expressed in higher-order logics.
 
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ZFC is a first order theory, where have you been seeing second order formulations? I'm sure every one of the top Google results for "ZFC axioms" will give you a first order formulation. In particular, Wikipedia.
 
mpitluk said:
Can someone point me to a first-order axiomatization of ZFC?

As I've mostly seen ZFC expressed in higher-order logics.
The higher-order axioms are replaced with axiom schema. e.g. the axiom schema of subsets is collection of statements
{x in A | P(x)} is a set,​
one for every unary predicate P in the language of first-order set theory.First-order ZFC requires infinitely many axioms to specify. First-order NBG, however, is an 'equivalent' set theory in an important sense, but only requires finitely many axioms.
 

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