Does there exist a characterization of Goedel's constructible universe $L$ in purely category-theoretic terms, or is constructibility an 'artifact' of material set theory? If, in fact, constructibility is an 'artifact' of material set theory, is there a category-theoretic 'analogue' for $L$?
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asked Feb 20, 2017 at 12:09
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2$\begingroup$ As far as I know, this is an open problem. $\endgroup$– Mike ShulmanFeb 22, 2017 at 3:39
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$\begingroup$ @MikeShulman: Has there been any progress made on this 'open problem'? Has there been any interest by category-theorists in developing such a characterization? $\endgroup$– Thomas BenjaminFeb 22, 2017 at 6:38
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$\begingroup$ If, in fact, my question is an open problem, Peter Koepke's work on ordinal turing machines (and the equivalent formulation of $\ast$-recursion on $Ord$ provided by Koepke and Koerwien in their paper, "The Theory of Sets of Ordinals") might provide a basis by which this open problem might be solved. $\endgroup$– Thomas BenjaminFeb 23, 2017 at 6:00
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$\begingroup$ I don't know of any significant progress. I expect people would be interested in such a characterization. $\endgroup$– Mike ShulmanFeb 23, 2017 at 11:59
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1$\begingroup$ Yes, of course. There are lots of ways of representing material sets in structural set theory, see for instance ncatlab.org/nlab/show/pure+set or arxiv.org/abs/1004.3802 and references cited therein. $\endgroup$– Mike ShulmanFeb 24, 2017 at 18:12
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1 Answer
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Consider the following statement from Lawvere's paper, "Cohesive Toposes and Cantor's 'lauter Einsen' (Philosophia Mathematicae (3) Vol. 2 (1994), pp. 5-15):
It might seem that G$\ddot o$del's $L$ would rest, if anything does, on von Neumann's a priori $\epsilon$-chains, but that was refuted twenty years ago by William Mitchell who showed how to construct it from a suitable given category, structured only by composition of maps; however, to my knowledge this has not been pursued since.
Though it might not be 'kosher' to answer a question with a question, why does Mitchell's construction not provide the necessary category-theoretic characterization of $L$ I seek (Mitchell's construction is contained in his papers, "Boolean Topoi and the Theory of Sets" (Journal of Pure and Applied Algebra 2(1972), 261-274) and "Categories of Boolean Topoi" (Journal of Pure and Applied Algebra 3 (1973) 193-201))?
answered Aug 2, 2017 at 16:37
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$\begingroup$ As far as I know, what's involved here is just that membership-based set theory can be interpreted in appropriate categories of sets, either by using the structure $(\text{TC}(\{x\}),\in)$ from your earlier comment or by using the trees obtained by unraveling this structure. Once this is done, you can just translate traditional set theory (including the definition of $L$) into category-theoretic language. $\endgroup$ Aug 2, 2017 at 16:46
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$\begingroup$ Technical addendum: The set theory that gets interpreted into the theory of well-pointed topoi with natural-numbers object is a version of Zermelo set theory (so without replacement) with separation only for formulas with all quantifiers bounded (but the bounds can include power sets, so it's not too weak). The development of $L$ in that theory requires a bit of caution. $\endgroup$ Aug 2, 2017 at 16:49
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$\begingroup$ As a standard comment: one can strengthen the definition of (well-pointed) elementary topos a little to get analogues of the replacement axiom, and Replacement after doing the above construction. Mike Shulman's paper on the stack semantics does this using the concept of autology, though there are other approaches, eg McLarty's. $\endgroup$– David Roberts ♦Aug 2, 2017 at 21:07
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1$\begingroup$ The caution that I had in mind comes from the fact that the usual construction of $L$ uses transfinite recursion on the ordinals and thus involves replacement. Without replacement, one needs to work with general well-ordered sets rather than von Neumann ordinals, and one needs to formulate the definition of $L$ in a way that doesn't use replacement to justify transfinite recursion. $\endgroup$ Aug 4, 2017 at 16:24
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1$\begingroup$ On your other question, about the weakest set theory needed for a particular theorem, I don't know enough about extremely weak set theories to give you useful information about that. Presumably, you don't need any form of power set, so it would be quite different from what topos theory gives you. $\endgroup$ Aug 4, 2017 at 16:25