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Out of the numerous set theories (NBG, ZFC, etc) which is the current accepted one, or is it that each one has built upon the other?

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Depends what you mean by "accepted." Perhaps you're asking "When people say a theorem has been proven without specifying which set theory (if any) they're working in, what should I assume they're working in 'by default'?" So I'll try answering this.

The vast majority of known mathematical results can be formalized as statements about sets, in the language of ZFC. And furthermore, if a theorem is stated "unconditionally" (i.e., they didn't specify which axioms they used), you can assume they only used axioms of ZFC. Since ZFC and NBG agree on which sentences in the language of ZFC are provable, you can choose to formalize the claim in either. NBG is only really necessary for the occasional claim that needs proper classes to be formalized. Otherwise, there's a general preference to work in ZFC due its simpler ontology (which is convenient when studying models of the respective theories.

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  • $\begingroup$ At least for now, ya? In the past, ZFC didn't exist. In the future, who knows what logicians will consider as the accepted foundations? =) $\endgroup$ – user21820 Mar 24 '18 at 16:47

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