Talk:Constructible universe

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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)

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)

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)

I removed the 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)

[edit] A standard model?

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)

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)

[edit] Usual notion of constructive?

The article includes sentences such as

The usual notion of constructive would not allow that ...

I doubt that there is a usual notion of constructive, and I think the majority usage would be that constructive means proved without using the axiom of choice (which is not the sense intended by the article). There are several incompatible definitions of constructivity in various schools of constructive mathematics. It would benefit the article if the specific sense of constructive that is meant could be defined. CMummert 13:52, 24 July 2006 (UTC)

I think that "usual" here means "in the sense of Constructivism (mathematics)". Aleph4 16:04, 24 July 2006 (UTC)

I agree. Unfortunately, constructivism in mathematics is like conservatism in politics - it means different things to different people. There is no single movement that carries the mantle of constructivism for everyone else to see. Also, the statement in the article that consists of “constructive sets” is strange because although you can call a proof technique constructive, it is less obvious what it means for a set to be constructive. Here's a particular example: the truth set of first order arithmetic is in ; in what sense(s) is this set constructive? CMummert 18:07, 24 July 2006 (UTC)

I changed the article to include a definition of what I meant by "constructive". To wit, set being constructive means that it has a code (set theory) which is recursive. How do you know that the truth set of arithmetic is in ? If you are correct (which I doubt), then it refutes my identification of the set of constructive sets. JRSpriggs 03:58, 26 July 2006 (UTC)

Because the truth set of first order arithmetic is at Turing degree 0^(ω). After you have the set of natural numbers, each additional level of the L hierarchy gets you (at least) one more Turing jump, so after ω levels you have all the finite jumps. Then an instance of Σ₀ collection allows you to collect these into a single set from which 0^(ω) is computable.

I think it is easier than that to see that does not consist exclusively of sets with recursive codes. Given a recursive code for a set of natural numbers (finite von Neumann ordinals), the question of whether n is in the coded set is decidable from 0''. Thus if every set in had a recursive code then every set of natural numbers in that model would be recursive in 0'', which is impossible this collection of sets is closed under Turing jump. CMummert 13:16, 26 July 2006 (UTC)

You are right. I rewrote the second paragraph of the section on constructive sets. Please check it over again. JRSpriggs 11:21, 13 August 2006 (UTC)