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It is well known that the axioms of Separation and Replacement are axiom schema, that is, they subsume an infinite number of first-order axioms for ZFC. My question is, how large an infinity is this, or, perhaps better, how large an infinity should this be. One can naively imagine this infinity to be as large as Ord, the proper class of all ordinals. If it were, could ZFC then be considered 'complete' contrary to the Godel incompleteness theorems? Could one argue that ZFC could not have Ord many instances of Separation and Replacement?

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  • $\begingroup$ There are only countably many formulas, so the axiom schemes are countable. $\endgroup$ – André Nicolas Oct 2 '13 at 19:56
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    $\begingroup$ As a point of information, the plural of schema is schemata. $\endgroup$ – Brian M. Scott Oct 2 '13 at 19:59
  • $\begingroup$ So if you could choose countably many of your favorite instances of Separation and Replacement (e.g. CH or not-CH,PD or not-PD as you wish given the validity of the set-theoretic multiverse) would ZFC then be complete, contrary to the Godel incompleteness theorems? If not, why not? $\endgroup$ – Thomas Benjamin Oct 3 '13 at 20:57
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The language of set theory has only one symbol, $\in$, except variables and connectives and quantifiers (one can include equality, but it isn't necessary, and many old treatments of $\sf ZFC$ do not include equality).

This means that the set of formulas and sentences in this language is countable. Countable where? In the meta-theory, be it $\sf PA$ or some other arithmetic theory sufficient to develop logic; or even a larger universe of $\sf ZFC$.

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To add to Asaf's answer, although each schema has only countably many instances in the sense of containing only countably many axioms, if we think of an "instance" of separation or replacement as being determined by a logical formula and a set parameter for that formula, then there would indeed be different "instances" for each ordinal. This is not the standard usage of the word, however.

Note also that comparing the size of schemas with the size of sets or classes is problematic because the former (as Asaf says) is described in the meta-language and the latter is described in the object language. The distinction becomes important when considering models of set theory, where the class of ordinals of a model may be a countable set (in the universe $V$) and also the model and the universe may not agree on what it means to be an axiom of ZFC.

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