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So I'm going to say no. At least to the "decide any question about any set theoretical statement" part. There are well known set theoretical problems which are independent of set theoretic axioms. Any sort set theoretic thing you construct will still operate within those axioms. Coming up with an extended or generalized or tweaked definition of Turing ...

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There is no notation for the smallest inaccessible cardinal. In fact, there is no notation "agreed" for any large cardinal property. One may abbreviate things like the tree property, or so, but those are usually properties that can occur at small cardinals. We would usually say something like, Let $\kappa$ be the least inaccessible cardinal... The ...

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Taking an ultraproduct of well orders by a sigma complete ultrafilter results in a well order, whereas an incomplete filter results in a linear order which is not well founded. The proof is the same as the ZFC case. The reason we don't really talk about this is model theory is that the consistency difference of a measurable cardinal assumption is quite ...

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