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I am wondering why we need to have a metatheory in order to talk about a theory- why can't we just add self-referencing terms to the language of the formal system on which the theory itself is based, such that all of the statements we make in a metatheory can be made in its object theory?

For example, this answer says an approach while studying ZFC is to have the metatheory be ZFC + "there is a model of ZFC", and to have the object theory be ZFC. However, why can't we dispense with the metatheory entirely and just have the axioms of our formal system be those of ZFC with the additional axiom "there is a model of these axioms"?

Even if the particular scheme above doesn't work to dispense with the metatheory, I would like to know why other approaches using metatheories are so common, instead of including self-referencing language into a theory.

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  • $\begingroup$ One general thought: paradoxes like the liar show that we can't simultaneously have a full-blown theory of reference (including self-reference) and full-blown theory of truth. If our language is powerful enough to permit full self-reference (or is extended to do so), then it won't itself have an adequate truth predicate. However in a meta language we can still talk about truth in the object language. $\endgroup$
    – blargoner
    Commented Jun 21 at 21:47
  • $\begingroup$ @Blargoner is this a strict implication (self-reference implies contradiction) or is it just that it has the potentiality to do so, which urges us to stay away from it from fear of this potentiality? $\endgroup$
    – Princess Mia
    Commented Jun 21 at 21:50
  • $\begingroup$ What would "there is a model of these axioms" look like as a sentence of the original object language of ZFC? Or if the idea is indeed to extend that original object language, what's the difference between what you are suggesting and having a (formalised) metalanguage for ZFC? $\endgroup$
    – Peter Smith
    Commented Jun 22 at 8:36

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The object theory is just the theory you want to study. The metatheory is just the theory you work in while you study the object theory.

Nothing stops you from studying ZFC while you yourself work in ZFC. This would keep the object theory and the metatheory identical. The only problem is that ZFC happens not to prove any interesting conclusions about itself. In particular, ZFC can't even prove that ZFC is consistent (equivalently, that it has a model)!

This constitutes an instance of Gödel's 2nd incompleteness theorem, which says that a theory that can prove its own consistency is either

  1. extremely weak (fails to reason even about simple arithmetic); or
  2. inconsistent (proves everything and its opposite too); or
  3. not recursively axiomatizable (you can't tell what its axioms actually are)

and so quite useless in practice.

ZFC proves that a theory is consistent precisely if it has a model. Therefore, a "self-referential" theory $T$, which extends ZFC set theory with the assertion that there is a model of $T$, is able to prove its own consistency. By Gödel's 2nd incompleteness theorem, the theory $T$ is therefore inconsistent.

And this is why we don't work in such a theory $T$.

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