Talk:Constructible universe

message: While the definition of V and V_alpha is correct,
the definition of constructible universe given here 
is completely wrong.
The universe V is the union of all the 
V_alpha sets.  The constructible universe L is the union 
of a different hierarchy L_alpha, which is obtained by 
iterating something weaker than powerset.
: You contributions to the article would be most welcome if they help improve its accuracy. Please feel free to make any edits you deem useful. I don't know anything about the topic and the original author(s) may not take notice of your comments. thanks, Dori | Talk 21:09, Dec 17, 2003 (UTC)
: I'm the original author, and since I did it from memory I'm quite willing to believe I've got it wrong. Please do fix it. Onebyone 21:36, 17 Dec 2003 (UTC)
replies: text: This definition is wrong for L, but it's perfect for the Von Neumann universe V. So I just moved it there! (The intro, in contrast, really does belong here.) -- Toby Bartels 22:40, 28 Jan 2004 (UTC)

text: This new defintion copied from Goedel's constructible universe needs to be cleaned up, but I can't do it now. -- Toby Bartels 23:00, 28 Jan 2004 (UTC)

text: Let x represent the set of all finite non-reproducible sets. Let S(x;n) represent the set of finite non-reproducible sets taken over the field of real numbers. Then S(n;n) is not admittive of all possible universes of sets that do not include S(n;n) as a possible finite reproducible subset. Mamoun's Eternal Party Method innovation to Goedel's theorem was to point out that S(n; not n), taken over the field of real numbers leads to a non-ergodic finite state of dimensional indeterminacy that co-incides in all cases except S(n;n), interpreted as to content in terms of determinate finite states, with the resultant universe of reproducible sets. This likewise corresponds with the Monster Set but, remarkably, compactifies the set to include a self-referential mapping of the set configuration states such that the Monster Set - S(n;n) complements are mappings that are mutually onto and into; this is equivalent to a finite state indeterminacy in all forms except the union of determinate and indeterminate states, which maintains supersymmetric conservation parity of all dynamic states taken over the field of complex numbers. The immediate application to D-brane anti-deSitter spaces allows for a non-ergodic complete state manifold capable of describing any cyclically descriptive dynamic (1...n-1) when taken over the field of Monster Set classes
I looks like it does not make sense, and even if it does, it is not connected 
with Gödel's constructible universe.   Aleph4 19:24, 23 Jun 2004 (UTC)
thread_title: Comment by Dori 

message: What is a standard model of set theory?  I know what a transitive well founded model that contains all the ordinals is.  Is there a reference (in print) for using the word standard to refer to such a model? The article would benefit from additional clarity on this point.  I have also noticed this issue at the Kripke-Platek set theory article, where the phrase standard model apparently means well founded transitive model. 
I'm not interested in philosophy here; I'm just asking whether this use of the phrase standard model to mean well founded transitive model is common in the literature. CMummert 13:38, 24 July 2006 (UTC)
replies: text: As it says in Inner models, "A model of set theory is called 'standard' or 'transitive' when the base class is a transitive class of sets and the element relation of the model is the actual element relation restricted to the base class. A model of set theory is assumed to be standard unless it is explicitly stated that it is non-standard. Inner models are usually standard because their ordinals are actual ordinals.". JRSpriggs 03:50, 26 July 2006 (UTC)

text: This old Usenet post by Bob Solovay mentions Kunen's and Jech's books on set theory as possible references. 75.62.6.87 (talk) 12:20, 3 April 2009 (UTC)
thread_title:  '''A''' standard model? 

message: "(L,ε) is a substructure of (V,ε)" should possibly be replaced with "(L,∈) is a substructure of (V,∈)" but since I wasn't sure I didn't fix it. --Sigmundur (talk) 09:11, 19 January 2010 (UTC)
replies: text: You are correct, that would be the preferable notation.  One does occasionally see ε used for the elementhood relation, but ∈ is much more usual (and to your implied question, yes, elementhood is what's intended here). --Trovatore (talk) 09:15, 19 January 2010 (UTC)
replies: text: I noticed that and fixed it (before I saw this section of talk). JRSpriggs (talk) 20:57, 19 January 2010 (UTC)
replies: text: Ah, I just looked at this in Konqueror, and here I see the ∈ as a little box, indicating that it doesn't know that Unicode (or doesn't have it in its current font).  Likely that's why someone put ε in the first place. --Trovatore (talk) 21:02, 19 January 2010 (UTC)
replies: text: Yes, if I remember correctly, when I first wrote that section I was using Internet Explorer rather than Firefox. So ∈ did not appear to produce ∈. JRSpriggs (talk) 22:34, 19 January 2010 (UTC)
thread_title:  typography check, someone who is knowledgeable, please! 

message: I have removed the following section:
:All constructive sets are constructible, where a set being constructive means that it has a code which is recursive. On the other hand, constructible sets are not necessarily constructive. When constructing constructible sets one is given free use of the ordinal numbers and also of the hierarchy Lα. This may not be constructive because there are ordinals which are not the order type of a recursive well-ordering of the natural numbers. And generating Def(X) from X is not a constructive operation, especially quantifying over X when it is infinite. Nor is taking the union of Lβ at non-constructive limit ordinals.
:Every set in Lω is constructive, but some non-constructive sets appear as early as Lω+1. Every ordinal less than \omega_1^{\mathrm{CK}} is constructive. Every constructive set belongs to the admissible set L_{\omega_1^{\mathrm{CK}}}, which is a standard set model of Kripke–Platek set theory plus the axiom of infinity.
My objection is twofold.  First, it's not stated what sort of code has to be computable.  I could claim that 0# is constructive, because it has a computable code consisting of the numeral 0 and the sharp sign.  There's a link to code (set theory), but set theory uses all sorts of coding schemes; there's not enough clarity here.
More seriously, I do not believe this is especially standard usage.  I don't think there is any agreement as to what, if anything, constitutes a "constructive set". --Trovatore (talk) 20:11, 31 July 2010 (UTC)
thread_title: The difference between constructible and constructive

message: Using transfinite induction, it may be provable that V=L. Consider Vα for which we already proven Vα∈L. For beginning we can even use V0. Then, by axiom of power set, P(Vα)∈L. By definition of Von Neumann universe, P(Vα)=Vα+1. So Vα+1∈L. By axiom of pairing, we construct set {V0,V1,...} up to limit ordinal α. Then, by axiom of union, union of {V0,V1,...}∈L. So, for every successor and limit ordinal α, Vα∈L. So V is within L. From definition L is subset of V, so V=L, Q.E.D. Wojowu (talk) 19:00, 3 June 2012 (UTC)
replies: text: There is something in L that appears to be powerset of a given set A, but it is actually L∩P(A) rather than P(A). JRSpriggs (talk) 04:16, 4 June 2012 (UTC)
thread_title:  Power set implies V=L 

message: To JRSpriggs: On the subject of countability of the constructible universe....
The countability of the countable universe does not seem to be explicitly stated on the wiki page. But numerous books do state that it is a countable universe, In particular, X=\{x\in2^\omega;\,\hbox{Cons}(x)\} should be countable, where Cons(x) means that that x is constructible. Both you and the textbooks say that constructibility is a property which is expressible within ZF, i.e. not just in the metamathematics.
I can't quite square that with the Cantor diagonalization procedure. Suppose f:\omega\to 2^\omega is a function satisfying X\subseteq\hbox{Range}(f). Then define g\in2^\omega as the set of ordered pairs g=\{(i,1-f_i(i));\,i\in\omega\}. Then g:\omega\to 2=\{0,1\} is a function satisfying g\notin X. But surely g is constructible if X is constructible. So g is both constructible and not constructible. Where is the error in this argument? Is X not really constructible? Is the Cantor diagonalization procedure not a construction? Does countability not necessarily imply the existence of the map f? I must have missed out on something here.
I think that these questions go to the core of the constructibility issue. So I think they should be directly discussed on the wiki page. If I can be perplexed, then maybe other people can also.
--Alan U. Kennington (talk) 04:09, 16 December 2012 (UTC)
replies: text: The constructible universe is simply not countable.  In fact, it contains all the ordinals (so it is not a set, let alone a countable one).  On the other hand, the set X = \mathbb{R}^L = \mathbb{R}\cap L of constructible reals can be countable (it is consistent with ZFC that it be countable), but no function f surjecting ω onto X can be constructible (because X can be constructible in the universe, but not in L) and the function g you define also is not constructible.  (If this helps, the situation is quite analogous to recursive sets: there are countably many recursive sets of integers, but they cannot be "recursively counted".) --Gro-Tsen (talk) 12:13, 16 December 2012 (UTC)
replies: text: To Alan: As Gro-Tsen says, f and g are not constructible. When you said "surely g is constructible if X is constructible", that was a mistake. You would have to have f constructible to get g to be constructible, but it is not so. JRSpriggs (talk) 18:12, 16 December 2012 (UTC)

text: From Constructible universe#Constructible sets are definable from the ordinals, one can see that the cardinality of the sets in a transitive model Lα of ZFC+V=L is ℵ0+|α| rather than just ℵ0. In other words, a model containing an uncountable number of ordinals would have uncountably many elements because the formulas which define its elements must contain constant symbols for its ordinals. Although one can use formulas without parameters to define some of the ordinals, others cannot be so defined. L is minimal given its ordinals (so do not take the section title too literally). JRSpriggs (talk) 04:44, 17 December 2012 (UTC)
thread_title:  Cantor diagonalization procedure 

message: Reading through the article, there are only two references, one on the original Godel paper introducing the constructible universe and another on hyperarithmetical subsets. The rest of the article is essay-like in its near complete lack of citations and references. Given the carefully written exposition, it seems silly to slap a "needs citations" tag on the article. But it would be useful to indicate from which references the various results in the article were gleaned. Unfortunately, I don't have the expertise in this field to do so. Mark viking (talk) 06:52, 15 December 2012 (UTC)
replies: text: I added most of the content of the article, and based most of that on Ulrich Felgner's "Models of ZF-Set Theory". There are six sources listed in Constructible universe#References. These are used throughout the article. Requiring in-line citations for all statements in the article would make the article unreadable (especially for editors). JRSpriggs (talk) 07:06, 15 December 2012 (UTC)
replies: text:  Thanks for the information. I agree, citations for all statements would be too much and that degree of granularity would likely be unhelpful to the reader. Perhaps it would be useful to indicate in the notes or references section that the article is largely based on the Felgner reference? Mark viking (talk) 07:35, 15 December 2012 (UTC)
replies: text: Well, I used Felgner because that was what I had. It may not be as useful as some of the other sources to a reader who has easy access to them. JRSpriggs (talk) 12:09, 15 December 2012 (UTC)
thread_title:  Citations and references? 

message: It isn't unusual if articles are in both a category and one of its subcategories if the article and the subcategory have the same name, see WP:Eponymous category. I think it is natural (and helpful to the reader) to have the article constructible universe both in the category Set-theoretic universes and its subcategory  constructible universe. — Tobias Bergemann (talk) 19:40, 16 December 2012 (UTC)
thread_title:  Eponymous category 

message: Substituted at 01:55, 5 May 2016 (UTC)
thread_title: Assessment comment

message: They seem conversational, almost friendly. Like something in a pamphlet. Not a bad pamphlet, but not an encyclopedia. I thought about bringing them more in line with the rest of Wikipedia's style (something dry and brief, like Universe does), then noticed I don't grasp the concept well enough to summarize the chapters.
Someone smarter might want to take a serious go at making them look a bit like "Relative constructability" does. If not, no harm done, probably. InedibleHulk (talk) 22:47, 9 March 2017 (UTC)
thread_title:  These headers are weird