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Hmm, no doubt this is naïve, but the axiom as formulated there seems a bit fishy to me. I mean if the proposition does not make use of size in any way, it is perfectly reasonable that it can be resized up and down. But if the proposition does make use of size, it seems difficult to believe it can be resized down. Take for instance the example I mentioned in another thread, that any category with all (large) colimits is a poset. How does that resize down?
Edit: I suppose that this example cannot be formulated as a proposition in the intended sense?
I imagine your edit is correct. You’d need to express your proposed counterexample in HoTT as a certain type in 𝒰i+1.
Indeed. I’d be interested to know in a bit more detail why the example I mentioned is not a proposition in this sense, though (I imagine it is clear to an expert, but it is not to me!). I think it is clear that it lives in a higher universe than whichever universe one takes to correspond to small sets in the usual sense.
There is no “how” to resizing; it’s an axiom which baldly asserts that all propositions can be resized down. “Any category with all large colimits is a poset” is indeed a proposition and thus, assuming propositional resizing, can be resized down. The axiom therefore produces a small proposition that is equivalent to the large one, but it doesn’t tell you “what” that proposition is other than “the result of resizing this other one”.
Thanks! What I was getting at was that I do not know how to think about it, because naïvely it seems that some propositions might not be able to be sized down. In the HoTT book, though, it says that the axiom can be proven under the assumption of excluded middle, i.e. non-constructively. In other words, it must make sense. Maybe I can put it another way. Even if Id(x,y) is inhabited for all x,y:A, Id(x,y) might be too large to live in a universe U. So it would naïvely seem that there are clearly more propositions in V⊃U than in U, unless V is equivalent in size to U, which is impossible. It seems like one would somehow have to be able to ’discard’ all except U’s worth of the terms of Id(x,y), and even with excluded middle that seems tricky to express.
Can you suggest why this is not the right way to think about it, and how one should rather think of it?
Edit: it might be enlightening to see the proof under the assumption of excluded middle.
Proof assuming LEM: given P:PropUi+1, either P≃1 or P≃0 (that’s what LEM means). But we have 1:PropUi and 0:PropUi, so in either case there is a Q:PropUi such that P≃Q.
It’s true that Id(x,y) may be too large to live in a universe U. That’s why the resizing axiom (as written here) states only that every proposition is some universe is equivalent (hence equal, assuming univalence) to a proposition in the smaller universe, not that it itself lives in a smaller one. A very “large” contractible type is nonetheless equivalent to a very small one. Voevodsky originally proposed a version of resizing as a rule stating that any proposition itself inhabits every universe:
P:PropUi+1P:PropUibut to my knowledge it is not known whether this is consistent.
Ah, thanks! I see much better now. As you say, the key seems to be that the notion of equivalence is ’poly-universal’, whereas I was thinking in terms more like the stricter version of the rule that you mention.
Also I was thinking in more semantical terms, like I think one can if the proposition is ’semantically size-independent’, whereas this is not the right way to think about it, at least in general, here.
Propositional resizing is about subobject posets of the terminal object of categories Set1 and Set2 which are well-pointed cartesian closed Heyting pretopoi with a natural numbers object, such that Set1 is a sub-(well-pointed cartesian closed Heyting pretopoi with a natural numbers object) of Set2.
Type theoretic models imply that one only needs the categories which are Heyting categories, rather than Heyting pretopos
Unless I am misunderstanding something, this page currently contains the astonishing claim that propositional resizing is inconsistent (https://ncatlab.org/nlab/show/propositional+resizing#properties). This is quite the claim as propositional resizing is implied by the law of excluded middle, so this page seems to me to imply that classical mathematics is inconsistent.
The page history shows that this page was created and exclusively edited by the Anonymous user who some time later was banned for making too many low quality edits.
If the page can be salvaged and you have the energy, then please do.
Or rather, I see now that “Anonymous” came back as “Anonymouse” and made the edit in question in revision 18.
This “Anonymouse” contributor seems to be abusing the nlab to include a lot of their own notes with personally developed non-standard terminology without citations. In particular, here they have invented their own terminology local propositional resizing and global propositional resizing. I would recommend banning them as they have been doing this for a long time and none of the type theory experts that hang out on the nforum (including myself) have the time to sift through all of their contributions
I will try to update this page, mostly by deleting things and adding some citations, but I think we should delete the local/global propositional resizing pages (which I don’t know how to do)
Thanks for getting involved, I have been struggling with this user for a long time. We need more community interaction on this issue, since banning accounts evidently only goes so far: The page history here suggests that “Anonymouse” is just the Anonymous user who I had already get banned in the past.
I have now cleared the two entries you mentioned.
Unfortunately, there are many other entries by this user.
Clearing an entry goes like this:
Go to edit the entry, delete all the text,
click “change page name” to “empty X+1” where “X” is found from looking at the list of all pages,
go back once more to the edit panel to also delete the redirect to the old page name that the software will have inserted
(this is important to make sure that the page be “orphaned”, in that no links point to it),
hit “submit”.
(This is the closest to deleting a page that one can do without admin access to the server. For the actual physical deletion of all “orphaned” pages there is a server command, which however has never been run yet, this being why the list of “empty X” pages keeps accumulating.)
Re #21: If banning “Anonymous” was effective, perhaps banning “Anonymouse” will work too?
An IP address or subnet ban may be even more efficient. I believe such a ban was considered for a different user in the past, but was ultimately unnecessary in that case.
And here is a list of all 180 pages created (as opposed to edited) by Anonymouse. Some of these seem to be harmless, e.g., pages for people and books like Reflections on the Foundations of Mathematics, and some appear to be about real concepts, e.g., transcendental extension and zero-dimensional ring.
?-modality A-function A-group A-monoid A-ring A-set AUC Alba Sendón Blanco Ansten Klev Archimedean field Archimedean ordered field setoid Brouwer’s continuity principle Cauchy quotient Chirantan Mukherjee Claudio Ternullo Connor Aberlé Coquand universe De Morgan laws Deborah Kant Deniz Sarikaya Dutch Categories and Types Seminar E. Hastings Moore ETCS plus epsilon Escardo-Simpson real number Fabius function Gustav Lejeune Dirichlet Hadamard product Heine-Cantor theorem Jaap Fabius James Hanson Jean-Philippe Bernardy Jesper Cockx John Major equality Juan Arias de Reyna Julia Semikina Kreisel-Lacombe-Shoenfield-Tseitin theorem Laura Fontanella Lifschitz realizability Merlin Carl Michèle Friend Ming Ng Mirna Džamonja Neil Barton Penelope Maddy Philip Welch PostBQP Reflections on the Foundations of Mathematics Stefania Centrone Sy-David Friedman Tr(ϕ³) Viviane Pons Vladimir Lifschitz Z-functor accessible topological space acute triangle acyclic type adjacency matrix admissible Archimedean ordered field affine function affirmative proposition algebra - contents algebraic element algebraically independent subset analytic LLPO analytic LPO analytic WLPO analytic principle of omniscience antithesis interpretation antithesis partial order apartness ring ascending chain condition on principal ideals asymptotic notation atomic domain axiom of punctual cohesion axiom of real cohesion axiom of sufficient cohesion bar induction bar bi-pointed set bounded total order cartesian product of A-sets classical set commutative operation congruence rule constructible number countably prime filter decidable injection decidable subsingleton definitional isomorphism dependent type theory with type variables embedding of types explicit conversion field setoid fractured homotopy type theory gaseous vector space geometric mathematics global propositional resizing globally univalent bicategory groupal setoid haecceity heterogeneous equality heterogeneous path type hierarchy of universes higher-order logic as a dependent type theory homotopical real numbers type impredicative dependent type theory impredicative polymorphism impredicative universe inequality ring interval type localization law of non-contradiction lesser limited principle of omniscience limited principle of omniscience linear algebra - contents local propositional resizing local ring setoid locally cartesian (infinity,1)-category locally finite type loop type modulus of continuity monoidal setoid mutual exclusivity mutually exclusive events null type obtuse triangle op modality ordered field setoid parametric dependent type theory paraunital algebra permutahedron pi-finite type postselection principal ideal ring punctured neighborhood rational dagger category real numbers type record type refutative proposition residually discrete local ring right triangle ring setoid semi-decidable proposition sequential colimit sequentially Cauchy complete Archimedean ordered field setoid set truncation sigma-complete Boolean algebra sigma-complete Heyting algebra sigma-frame of propositions sigma-locale sigma-pretopological space sober sigma-topological space spatial sigma-locale squash type strict inverse function strict linear order strict proposition strict singleton strong negation strongly predicative dependent type theory surfaceology tensor product of A-sets tight apartness ring transcendental element transcendental extension transcendental number truncation twisted arrow modality twisted arrow two-type identity type two type of affine propositions type of entire relations type of finite types type of strict propositions type theoretic axiom of choice unified topological space weak limited principle of omniscience weak type theory weighted graph zero-dimensional ring
If banning “Anonymous” was effective, perhaps banning “Anonymouse” will work too?
As I said above, the page history here further supports the suspicion that the banned “Anomymous” simply changed their user name to “Anonymouse” and went on. I imagine they see me alone bugging them and probably conclude not to take this seriously as long as there is no further community reaction. Some more social pressure will probably work wonders.
Therefore I am glad that maxsnew chimed in here.
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