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Before the applications in theoretical computer science, Turing machines were developed as a model of what the idealized mathematician could possibly compute in order to give a precise definition of "effectiveness", and thus giving a universal model of computation. The affirmation that Turing machines actually are such a universal model, at least roughly speaking, is the Church-Turing thesis.

It thus seems like a natural question to ask: "Is there a universal theory for all possible mathematical theories?" In other words, does there exist some theory in which we could interpret any theory that an idealized human mathematician could possibly devise? Such a theory would be an honest to goodness "foundation of mathematics", in the sense that, unlike ZFC, in which we can express any most theories that mathematicians tend to think about, we can express anything possibly devisable.

Call this theory $X$, then for example, $X$ admits an interpretation of ZFC, ZFC + any large cardinal you could think of, MLTT, MLTT with arbitrary type universes, etc...

I would imagine that like the Chruch-Turing thesis, we could not possibly rigorously prove such a proposition, but that we should be able to find some reasonable candidate for this "$X$" theory. Has this, or similar notions, been thought about in model theory?

Nathan BeDell
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    Doubtful, considering you'd be asking $X$ to express contradictory things. – Hayden Sep 17 '16 at 22:37
  • Going back to computability theory, there are many possible turing machines, but importantly, Turing machines take input and produce output. A universal Turing machine is a Turing machine for which, given the appropriate input, the action of arbitrary Turing machines may be simulated. Thus, I suppose in model theory, the analog would not be a single theory $X$, but a function $X(x_0,x_1,x_2, \dots)$ that given appropriate input (e.x. the inputs could be additional rules and axioms) can interpret arbitrary theories in the sense I described in the original post. – Nathan BeDell Sep 17 '16 at 22:55

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Since you mention universal Turing machines, perhaps this satisfies what you are looking for:

Any reasonable, in practice usable axiomatisation of mathematics has the property that proofs in that system can be expressed as finite strings of characters from a finite alphabet; and if $T$ is a string of characters, then there is an algorithm (a computable function) which determines whether or not that is a valid proof in the system. E.g., for ZFC, the alphabet consists of (say) alphanumeric characters, parentheses, logical symbols, the symbol $\in$ and maybe some separators. The computable function checks whether the proof consists of well formed formulas, and whether each one either follows from a previous one (depending on the proof system chosen) or is an axiom (of ZFC or in the proof system).

Now such an algorithm can be formalized within a "weak" theory like Peano arithmetic. Thus, if we have $\mathrm{ZFC} \vdash \phi$, then we can also prove $\mathrm{PA} \vdash ``\mathrm{ZFC} \vdash \phi"$. For instance, we have that $\mathrm{PA} \vdash ``\mathrm{ZFC} +\mathrm{IC} \vdash \mathrm{Con}(\mathrm{ZFC})"$. Hence, every system of mathematics that I referred to above as reasonable/usable can be internalized in Peano arithmetic (or another sufficiently strong) theory.

Mees de Vries
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  • Thanks for your answer! I think this was basically what I was looking for, but I'm curious if you could elaborate on what you mean by a "weak theory", and if you could provide me with a reference detailing the construction of such an algorithm. Does Robinson Arithmetic, for example, also suffice to formalize such an algorithm? – Nathan BeDell Mar 08 '17 at 16:16
  • @Sintrastes, I don't have the expertise to answer that confidently; at best I could make educated guesses. I suggest you post another question. – Mees de Vries Mar 08 '17 at 17:20