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As an afterthought to this question on sets in set theory, and more specifically to the observation that a (first-order) logic requires a meta-language to explain itself (i.e. there is already an implication sign in the definition of rules of such a logic, which differ from the sign symbol of the logic itself), I wonder:

Are there attempts to close the problem of regression of meta languages (and then meta languages for these and so on)?

My idea would be that it might be possible to close the loop by having a structure that is able to define/model a framework, which is able to state axioms of a logic, which exactly mirror the first one. At least for some theories.

In what sense can some "structures" define a framework which is strong enouth to express a copy of the original "structures"?

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  • $\begingroup$ Set theory is clearly "strong enough". It turns out even arithmetic is "strong enough", but that's not at all obvious. $\endgroup$
    – Zhen Lin
    Apr 29, 2012 at 15:20
  • $\begingroup$ @ZhenLin: Is this what is used to formulate provability of theorems in such a theory in terms of its own language? $\endgroup$
    – Nikolaj-K
    Apr 29, 2012 at 15:26
  • $\begingroup$ The [regression] tag is for statistics and such, definitely not for this question. $\endgroup$
    – Asaf Karagila ♦
    Apr 29, 2012 at 17:53

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In order to talk about something you need a language. If you are dealing with languages, then to distinguish the language you are talking in and the language you are dealing with, the former is called meta-language. It is possible for two languages to coincide. For example almost all mathematics can be carried out in set theory (using the language of first order set theory). First-order arithmetic is probably the minimum you need for doing mathematical logic. However, there are subtle limitations in confining oneself to a formal language, Tarski's undefinability theorem being the most obvious one.

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