My question is motivated by the fact that (at lest from my biased viewpoint), Turing machines are more related to standard math, particularly PA. But set theory, in most of its versions, seem to go much farther than that. Specially if we consider the largest cardinal axioms. So my specific question, is: is it trivial to imagine a machine that is more naturally oriented to the kind of operations that people use in set theory? I mean, some algorithm that can decide any question about any set theoretical statement. Of course, they will be limited in practice by halting problems, but assuming we have oracles of any level for such kind of machine, will be trivial to imagine a machine that could "compute any set theoretical answer after a number of steps of arbitrarily high cardinality? (for instance V=L?). I image that a problem will appear when we try to "compute" answers for questions about large cardinals that cannot be reached from below; such as, does it exist a "rank-into-rank" cardinal?
2026-03-29 15:59:36.1774799976
Is there some natural equivalent of a Turing machine for set theory?
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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 machine doesn't change the axioms of math.