Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 2nd 2015

    Wrote cumulative hierarchy, and edited some at ZFC (idea section, reference, related articles).

    • CommentRowNumber2.
    • CommentAuthorspitters
    • CommentTimeAug 2nd 2015

    Should this be connected to algebraic set theory? This is really an algebraic construction.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 2nd 2015

    Oh sure, I had meant to say something there but forgot; please feel free to jump in yourself (and thanks for the reminder).

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 2nd 2015

    It was only recently that I realised how much set theorists like the cumulative hierarchy, and this is one reason they (or at least some) find structural sets less than satisfying: it’s not obvious that, say, ETCS comes with anything like being able to say “where the sets came from”. I guess in AST it’s different? One probably needs a form of Replacement to construct even for an arbitrary individual set a ’membership tree’ in the structural style. For sets with injection to some P^n(N) there is no problem, I think.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 2nd 2015

    I have tried to say a few things about AST in cumulative hierarchy. This however is all from memory, as I am having difficulty tracking down my copy of Joyal-Moerdijk.

    David, I remember a recent MO question of yours in which I started a long comment thread on the meaning of “ontology” for ZF-ists, and it seems that yes, it’s all about the cumulative hierarchy – although I don’t recall anyone actually using that phrase! (For me the most helpful replies were from François Dorais.) There are still a few mysteries for me from that thread, which I may get back to at some point.

    I see that Thomas Holder has helpfully added a related articles section; however, some of those nLab articles could use a little more love. :-)

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeAug 3rd 2015

    One probably needs a form of Replacement to construct even for an arbitrary individual set a ’membership tree’ in the structural style.

    What do you mean? Do you mean under what conditions is an ETCS-set isomorphic to the set of nodes of a rigid wellfounded tree (or whatever)? I think the answer to that question is “always”, because ETCS includes AC so that every set can be well-ordered and is hence isomorphic to a von Neumann ordinal.

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 3rd 2015
    • (edited Aug 3rd 2015)

    @Mike whoops! I must have been thinking of more general toposes as models for set theory… (I’d looked at the relevant section of the Baby Elephant recently) :-S

    @Todd - yes, it was that discussion that made me realise this. The fact that ZFC sets are exactly subsets of some set constructed so far is still far more rigid than the case for a structural approach: one merely has a monomorphism XP α()X \hookrightarrow P^\alpha(\mathbb{N}). So I guess a material set theorist would ask: “where did the domain of that mono come from?” But working up in an isomorphism-invariant way we just have to bear with that, as it were. Perhaps thinking of monos as elements in the powerset P α+1()P^{\alpha+1}(\mathbb{N}) (say we’ve chosen an initial object 11 once and for all: one choice instead of many) we get a bit tighter…

    Also: there’s the cumulative hierarchy constructed in the HoTT Book.

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 3rd 2015
    • (edited Aug 3rd 2015)

    @Mike actually, I don’t think I do mean that. Yes, one gets an isomorphism to the nodes of a rigid wellfounded tree from a well-ordering, but there is a massive amount of freedom. In particular, one might as well always choose an initial ordinal. What I’m thinking of is whether given any set XX, one can show the existence of some ordinal α\alpha (with no choices, or at least none explicitly or implicitly well-ordering XX) such that XP α()X \leq P^\alpha(\mathbb{N}). Such a result would not rely on Choice in the same way. Whether only well-orderable XX, in the absence of choice, admit such an embedding (it obviously doesn’t a priori well-order XX), is an interesting (to me at least) problem.

    Compare the problem of constructing something like LL inside a topos: its definition in material set theory doesn’t require choice.

    Cf also A cumulative hierarchy of sets for constructive set theory

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeAug 3rd 2015

    @David, do you just want to work in ETCS-minus-AC? I wouldn’t think XP α()X\le P^\alpha(\mathbb{N}) would imply well-orderability of XX; doesn’t it follow in ZF from the axiom of foundation?

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 3rd 2015

    @Mike ETCS-AC is as good as any to start thinking about such a problem.

    I wouldn’t think XP α()X\le P^\alpha(\mathbb{N}) would imply well-orderability of XX

    No, I didn’t think so. The trick is finding an ordinal such that it works. And, of course, one may not be able to use αω\alpha \geq \omega, since P ω()P^\omega(\mathbb{N}) may not exist in the topos: some Replacement may be needed.

    You’re probably right about Foundation, it sounds familiar.

    And all this is only at the level of musing, I’m not actively pursuing a result.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)