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Motivated by this question on the SE philosophy board "Is there a one true set theory", I want to ask for a clarification:

I've seen e.g. ZF being expressed in Grothendieck set theory.

If I say I express a set theory $S$ within another different set theory $T$, does this mean I at best construct a model $\mathcal M_S$ in $T$ (of which there might be many different ones)? Or does it mean I am actually able to express the (modelless) syntactic structure of $S$ within $T$?

If the first is true, can I even use $S$ to judge about the merely syntactic derivablity of statements true in $T$?

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Nikolaj-K
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If you want the syntax of $S$ expressed in terms of the syntax of $T$, then apparently you will be dealing with a subtheory of a theory expressed in $T$, and this may not take you very far. Typically a new set theory $S$ would have new features in its syntax, which is what makes it interesting compared to $T$.

A typical example would be extending ZF to a theory such as NBG with a new syntactic feature, a "class". To give a less standard example, Edward Nelson's IST extends the syntax of ZFC by adding a unary predicate "standard", and then the elements that are not "standard" can behave like infinitesimals and infinite numbers, all within the ordinary real line constructed for example using the usual Dedekind cuts. For more details, see my answer here.

In both of these examples, one first needs to show that models exist so one can proceed with the new syntactic features.

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Mikhail Katz
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