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atyy
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There's always a necessary tension between the two points of view. Even in mathematics - can there be true rigour? No, because just to define ZFC requires an undefined, intuitive meta-language. Presumably that meta-language was learned by an individual from a community. Nonetheless there is also the point of view that a proof is right or wrong regardless of what the community thinks.
http://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf (p31)
"Each axiom of ZFC other than Comprehension and Replacement forms (an abbeviation of) one sentence in the language of set theory. But the Comprehension Axiom is a rule in the metatheory for producing axioms; that is whenever you replace the ϕ in the Comprehension Scheme in Section I.2 by a logical formula, you get an axiom of ZFC; so really ZFC is an infinite list of axioms. Likewise, the Replacement Axiom is really an infinite scheme."
http://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf (p31)
"Each axiom of ZFC other than Comprehension and Replacement forms (an abbeviation of) one sentence in the language of set theory. But the Comprehension Axiom is a rule in the metatheory for producing axioms; that is whenever you replace the ϕ in the Comprehension Scheme in Section I.2 by a logical formula, you get an axiom of ZFC; so really ZFC is an infinite list of axioms. Likewise, the Replacement Axiom is really an infinite scheme."
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