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Transfinite Zippers

In my previous post, I described an approach for extending multi-stage programming (MSP) to infinity and beyond. We start with a simple computing “machine” M[0] such as the lambda calculus. Each machine M[a] has a successor machine M[a+1] which is capable of building and reasoning about programs from M[a], yielding towers of nested machines. From there, we note that MSP itself can’t be implemented in any such finite M[n], but must exist in some limit machine M[ω], yielding transfinite towers of machines M[a] for each ordinal a.