Talk:Saturated model

This article is rated Start-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects:

Mathematics Low‑priority This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Low This article has been rated as Low-priority on the project's priority scale.

This article is substantially duplicated in one or more external publications. Since these publication(s) copied Wikipedia, rather than the reverse, please do not flag this article as a copyright violation of the following source: Cyrus F. Nourani (2014), A Functorial Model Theory, Apple Academic Press

This article is in need of better explanations for the (informed) layman. It also uses the same notation S() for both sets of types and for the successor function. Kuratowski's Ghost 02:54, 20 Mar 2005 (UTC)

This seems to have been corrected. - Gauge 21:31, 10 July 2005 (UTC)Reply

I think the article should define "countably saturated." Is this just the case where |M|=omega? It's also unfortunate that complete type is used prominently in the definition, but there's no article. --Bcrowell 16:32, 14 January 2006 (UTC)Reply

"The model M is called countably saturated if it is Aleph_1-saturated": this should be Aleph_0, shouldn't it?! 82.124.30.249 (talk) 20:10, 15 January 2009 (UTC)Reply

Unfortunately not. It's an odd usage, but a standard one, and it's not up to Wikipedia to reform mathematical terminology. Algebraist 20:18, 15 January 2009 (UTC)Reply

Where did you get this definition? According to Chang & Keisler (p.100) a countably saturated model M is a countable model which is ω-saturated, i.e.

complete types realized over a finite subset of the universe of M. --Stefan.vatev (talk) 11:14, 25 February 2009 (UTC)Reply

I am not sure what is the standard definition, or if there even is one. I think most people use ω-saturated for the more useful notion, perhaps exactly because this avoids any confusion. I did find the odd definition of countable saturation as ω₁-saturation in the extended abstract of Michael Morley's groundbreaking paper. It sounds like this could have been the original sources, and as if this could have been intended as an ad hoc definition. However, with a Google Books search I found a book from 1997 that also used this definition. (I never heard of the authors before.)

I think it's best to mention that both definitions exist, and not to use the term on Wikipedia. --Hans Adler (talk) 11:15, 26 February 2009 (UTC)Reply

This article is unnecessarily technical.

• To begin with, it needs an introductory section that gives an easy-to-understand (i.e. in prose without unnecessary mathematical jargon (which includes the mathematicians' way of phrasing, e.g. the M in "a saturated model M is ...")) overview of the subject. This is for people like me, who are only looking for an overview of the subject and do have sufficient background knowledge to be able (at least in principle) to make sense of the topic at all.
• Secondly, it should add a few sections that provide in-depth information without the mathematical techno-babble (which can be found in the section "Definition"). This techno-babble only serves to drive the reader who is not interested in rigorous mathematical descriptions crazy. This addition is meant for readers who are ready for a more in-depth understanding of the subject, but who do not want to be bothered by rigorous mathematical phrasing (and who may even be scared off by such descriptions even though well informed). [This is, of course, not to say that we should get rid of a rigorous mathematical description, it only should be properly placed in the whole]
  

Wikipedia:Make technical articles accessible describes this in a more general way. --JorisvS (talk) 00:39, 3 January 2010 (UTC)Reply

The article can probably be improved somewhat from the point of view of accessibility, but a realistic sense of what is achievable is needed. You said in your edit summary this is way too technical for your-average reader to get. I'm afraid there is no way at all to make the concept of saturated models accessible to "your-average reader". The goal should be to enlarge the number of readers to whom it's accessible, not to make it accessible to people who never heard of model theory. --Trovatore (talk) 00:53, 3 January 2010 (UTC)Reply

Did you read the description I gave here? I get a feeling you didn't really... Because I'm not talking about people simply lack the background knowledge to be able to make sense of the subject.--JorisvS (talk) 10:54, 3 January 2010 (UTC)Reply

I also put in a context tag, as the lead section does nothing to indicate to stray readers that they are on the wrong page. In particular the issue that this article is about saturated models within model theory as opposed to saturated models in general (see the note on the top of that page) is insufficiently stated. --JorisvS (talk) 13:28, 3 January 2010 (UTC)Reply

As far as I know there's no such thing as "saturated models in general". The term saturated model is specific to model theory. The opening sentence says In mathematical logic, and particularly its subfield model theory.... How much more do you want to state it? Say it again, maybe? --Trovatore (talk) 18:56, 3 January 2010 (UTC)Reply

No, that's what I changed it into!;) Now, the first part of the sentence indeed seems fine.
If there is no saturated model in general (or whatever: any other usage of the term "saturated model"), then this article should also inform me about what I was looking for... I was looking for what it meant in relation to path models as used in Structural equation modeling. --JorisvS (talk) 19:30, 3 January 2010 (UTC)Reply

Ah, I wasn't aware there was such a thing, but a little Googling confirms that there is. That seems to have nothing at all to do with the subject of this article.

It looks like we probably need some sort of disambiguation. --Trovatore (talk) 19:37, 3 January 2010 (UTC)Reply

The article does not really explain what it means for a model to "realize types". This should be clarified. Tkuvho (talk) 11:20, 23 May 2010 (UTC)Reply

This article necessarily requires basic background in model theory; there is a whole article on types, which should be read first. It's a bit like reading the article on Noetherian Rings and demanding an explanation of ideals. Math is inherently hierarchical, that's why we wikify. Rodya mirov (talk) 01:37, 8 April 2011 (UTC)Reply

We are in full agreement as to the hierarchical nature of mathematics. You will find few treatments of noetherian rings that do not mention ideals, but Goldblatt presents a detailed treatment of saturation without mentioning "type" once. If your wish is to "wikify", do consider the fact that wiki policy is to make the lede of an article accessible to as broad an audience as possible. Tkuvho (talk) 07:05, 8 April 2011 (UTC)Reply

WP can't be a math journal. There must be a common language introduction and explanation, with no special meanings for common words used. A more technical section with special meanings can follow, but without the others leading in before, it's not encyclopedic. Attleboro (talk) 17:24, 15 August 2013 (UTC)Reply

WP is not a journal, but it is to some extent a scientific encyclopedia (that is, the corpus of articles on scientific topics constitute a scientific encyclopedia), and for some highly technical articles, some pre-knowledge must be assumed or you don't get anywhere at all.

Your issue seems to be with a different notion of saturated model, which is hatnoted at the top of this article, but maybe that's not enough. I don't really know which, if either, notion is the "primary topic" for the search term — it's quite plausible that saturated model should be a disambig page. I put a note at WT:WPM asking for thoughts. --Trovatore (talk) 19:06, 15 August 2013 (UTC)Reply

Thanks for creating that note. I'm used to a saturated model being one for which no more components can be added, given the limitations of the data. A saturated experimental design produces such a dataset, given a model form. A supersaturated experimental design produces a dataset insufficient to estimate all components of a given model, relegating the experimenter to a subset model of the original one. Is there a place here for that? I have a sneaking suspicion there may be a way to get from the formal definition in the article to (what's for me) the more common usage, or vice versa, but it's beyond me at the moment. Attleboro (talk) 20:31, 15 August 2013 (UTC)Reply

This is an "unavoidably technical" article. It is never going to be possible to make it accessible to readers without at least some specialized knowledge. That doesn't mean it's impossible to make it more accessible to more readers, but the {{technical}} template is inappropriate without a discussion of which points can be improved. --Trovatore (talk) 05:06, 12 April 2019 (UTC)Reply

I'm not sure if the following excerpt from the motivation section contains an error (note in particular the word "ω-saturated" in the end):

"a specific increasing sequence c_n having a bound can be expressed as realizing the type {x > c_n : n ∈ ω}, which uses countably many parameters. If the sequence is not definable, this fact about the structure cannot be described using the base language, so a weakly saturated structure may not bound the sequence, while an ω-saturated structure will."

Doesn't ω-saturated mean that all complete types over a finite parameter set are realized? Since here, infinitely parameters are used (one parameter c_n for each natural number n), shouldn't it say ${\displaystyle \aleph _{1}}$-saturated instead of ω-saturated? (I'm not an expert, just wondering.) Zaunlen (talk) 17:24, 24 September 2019 (UTC)Reply

You are right (for instance, (Q,<) is omega-saturated but does not bound the natural numbers.) I have corrected it. Felix QW (talk) 13:11, 13 July 2021 (UTC)Reply

While I have never heard anyone use the term "countably saturated" as a synonym for ${\displaystyle \aleph _{1}}$-saturated, Chang and Keisler (Model theory, 3rd Edition 1990, p. 100) use it to mean countable and ${\displaystyle \omega }$-saturated. I am not sure how helpful it is to have three usages of "countably saturated" in an article on saturated models, though. Felix QW (talk) 16:04, 13 July 2021 (UTC)Reply

What strikes me as weird is the use of "${\displaystyle \aleph _{1}}$-saturated" to mean what you would think you would call "${\displaystyle \aleph _{0}}$-saturated", but I guess this terminology is unfortunately standard. But the best we can do is report the usages that have currency. See for example this preprint as an example. --Trovatore (talk) 16:26, 13 July 2021 (UTC)Reply

Thanks, that's interesting! What I always found even less intuitive than ${\displaystyle \kappa }$-saturated is ${\displaystyle \kappa }$-presented, which is a substitute for cardinality, but means cardinality less than ${\displaystyle \kappa }$. I think that itself was actually designed to be consistent with ${\displaystyle \kappa }$-saturated... Felix QW (talk) 17:58, 13 July 2021 (UTC)Reply

I couldn't actually find any reference that uses countably saturated as meaning aleph_0-saturated; if someone finds one, feel free to add it again. Felix QW (talk) 19:18, 13 July 2021 (UTC)Reply

While sifting through sources on the lookout for meanings of countably saturated, I encountered a rather blatant backwards copy from Wikipedia: Page 206 of "A Functorial Model Theory" by Cyrus F. Nourani (published since 2021 by Taylor and Francis, and since 2014 by Apple Academic Press) is essentially a carbon copy of this article in its historical form. I am not sure what to do about this? Felix QW (talk) 19:36, 13 July 2021 (UTC)Reply