Talk:Model theory

This article is written in British English, which has its own spelling conventions (colour, travelled, centre, defence, artefact, analyse) and some terms that are used in it may be different or absent from other varieties of English. According to the relevant style guide, this should not be changed without broad consensus.

History, sources?[edit]

I'd expect an encyclopedia to give more about the history of model theory. Shouldn't seminal works by Tarski be among the references, at least? He's hardly mentioned. DanConnolly 18:21, 26 June 2007 (UTC)[reply]

comment of Logicnazi[edit]

Is not the last sentence of the first paragraph (i.e. what can be proven given a set of axioms) closer to proof theory?

Ughh, the completness part at least needs some work. What it means for a theory to be complete is quite differnt from the completness theorem. Logicnazi 12:11, 27 Aug 2004 (UTC)

Also the statement about a theory being maximally consistant set of sentences is just wrong. Only complete theories are maximal consistant set of sentences, e.g. the theory consisting of only pure truths of predicate calculus is closed under implication but hardly maximal (otherwise we could never add axioms!!) Logicnazi 12:13, 27 Aug 2004 (UTC)

Just so no one tries to re-add the statement it is simply NOT TRUE that a complete theory fully specifies a model. The Low-Skol theorems easily prove that complete theories will have models of differnt cardinalities. Logicnazi

Maximal consistent set[edit]

Anyone fancy creating this node and providing the necessary discussion here? I'm creating a link from Consistency proof, but I have more than enough to do around proof theory. If not, I'll get around to it eventually... ---- Charles Stewart 07:48, 22 Sep 2004 (UTC)

Category theory[edit]

Can someone add words that clarify the distinction between model theory and category theory? Is model theory supposed to be a broadened, extended, generalized category theory? Or was historically inspired by category theory, while ditching the weighty baggage of the concept of "class" and the cardinality of class? linas 16:04, 12 Mar 2005 (UTC)

I don't think they are related. MarSch 17:04, 19 Apr 2005 (UTC)

Category theory is more general than model theory. A topos, which is a type of category, can be understood as a model of a set theory or a logic. Archelon 00:56, 11 Jun 2005 (UTC)

See Intuitionistic_type_theory, specifically the section titled Categorical models of Type Theory. Perhaps something regarding the relation to topos theory merits inclusion in the article? Marc Harper 02:49, 6 December 2005 (UTC)[reply]

models of set theories[edit]

What is meant by "a model of a set theory"? Does it mean that you try to make a model in one set theory of the other set theory? MarSch 17:20, 19 Apr 2005 (UTC)

Yes. For example, countable models of set theory exist; that is, models of set theory with only a countable universe. Such models "think" they have uncountable sets, but since the underlying universe is countable in that case, they don't. Things like this can be confusing at first. - Gauge 04:54, 29 October 2005 (UTC)[reply]

Very confused![edit]

If "a theory is defined as a set of sentences which is consistent", then "a theory has a model if and only if it is consistent" seems very confusing. By way of illustration, "a 'set of sentences which is consistent' has a model iff it is consistent", looks very much like tautology to me. The irony of that appearing in this article is not lost on me, but this article needs a more precise and expository rewrite.

The theory of an L-structure A over a language L is defined to be the set of L-formulae that are satisfied by A. In contrast, a theory over a language L is a set of sentences that is closed under deduction. Given a set of sentences S, you can close it up to get the theory of S, denoted Th(S). This is the smallest set of sentences containing S that is closed under deduction. Consistency is not required of a theory in order for it to be a theory, but it will only have a model if it is consistent. - Gauge 04:54, 29 October 2005 (UTC)[reply]

To Do List[edit]

I removed this "to do" list from the article, so I'm sticking it here.

TODO - Vaught's test. Extensions, Embeddings and Diagrams. To give a flavor, mentioning the hyperreals and/or the extension of the concepts of basis and dimension to strongly minimal theories would be good. (All of these need substantial filling out)

Josh Cherry 04:15, 21 Jun 2005 (UTC)

Definition of model theory[edit]

I don't think this article gives a very good sense, currently, of what model theory is. The way I would put it is, in most of mathematics you specify a structure and try to discover its theory (that is, what statements are true in the structure). In model theory, you turn this around: You specify the theory (the set of, usually, first-order statements) and look for properties of structures that satisfy it. Model theory, in other words, lives in the gap between elementary equivalence and isomorphism. The intro to Category:Model theory needs similar attention.

There seems to be a serious lack of articles on even the basic concepts of model theory (types, saturation, omitting types, homogeneity). Compactness at least exists on WP. --Trovatore 23:56, 26 November 2005 (UTC)[reply]

Actually, there is a saturated model article. I've added a bunch of the rest to Wikipedia:Requested articles/mathematics. --Trovatore 00:07, 27 November 2005 (UTC)[reply]

Infinitesimals[edit]

Might we discuss briefly (and provide links) how infinitesimals and nonstandard analysis can be developed from model theory? Or is it discussed somewhere, and I missed it? Thanks. MathStatWoman 18:04, 22 January 2006 (UTC)[reply]

p.s. ok, found link to hyperreals. MathStatWoman 18:07, 22 January 2006 (UTC)[reply]

Logical Constant[edit]

JA: The way I read it, constant means a symbol with a (relatively) fixed logical interpretation, that is, a logical constant like "and", "or", etc. So maybe some clarification of that is called for. Jon Awbrey 17:30, 6 August 2006 (UTC)[reply]

I don't recall hearing "logical constant" with that meaning (I would think a logical constant would be "true" or "false", or possibly a name for some other truth value in a multivalued logic). Where have you encountered this meaning? Do you have a ref? --Trovatore 20:19, 6 August 2006 (UTC)[reply]

JA: I'm not saying that it's my favorite usage, but it's pretty standard. Don't know who started talking that way — the distinction is already clear in Frege and Peano, but the vagaries of translation may smudge it there. Pretty sure that it's in Whitehead and Russell somewhere, as Gödel is basically just gistifying "the system obtained by superimposing on the Peano axioms the logic of PM" when he writes the following:

The basic signs of the system P are the following:
I. Constants: "~" (not), "∨" (or), "Π" (for all), "0" (nought), "f" (the successor of), "(", ")" (brackets). ... (Gödel 1931/1992, p. 42).
Kurt Gödel (1931), "On Formally Undecidable Propositions of Principia Mathematica and Related Systems", B. Meltzer (trans.), R.B. Braithwaite (intro.), Basic Books, New York, NY, 1962. Reprinted, Dover Publications, Mineola, NY, 1992.

JA: Jon Awbrey 21:32, 6 August 2006 (UTC)[reply]

Wow. Well, that's a reference, for sure. I don't think it's used much these days, though. I suspect the terminology most used these days is due to Tarski rather than those earlier workers, but I don't know that for sure. --Trovatore 04:12, 7 August 2006 (UTC)[reply]

JA: I'm pretty sure that Tarski, Quine, etc. all use the term that way, though Tarski somewhat famously commented that he thought the distinction between logical signs and extralogical signs might be arbitrary and thus a parameter of the formal system chosen. But if it's not clear then it needs to be explained somewhere. Jon Awbrey 04:26, 7 August 2006 (UTC)[reply]

Logical sign sounds a good deal different from logical constant. I really would be pretty surprised if he used the precise term "logical constant" that way, assuming (as I think) he was the one who introduced the now-standard notions of non-logical symbols consisting of constant symbols, function symbols, and relation symbols. But I haven't read any of his original work, that I recall, so I'm certainly willing to be proved wrong. --Trovatore 04:49, 7 August 2006 (UTC)[reply]

JA: For example:

Among the signs comprising the expressions of this language I distinguish two kinds, constants and variables. I introduce only four constants: the negation sign 'N', the sign of logical sum (disjunction) 'A', the universal quantifier 'Π', and finally the inclusion sign 'I'. (Tarski, 1935/1983, p. 168).

Tarski, A. (1935), "Der Wahrheitsbegriff in den formalisierten Sprachen", Studia Philosophica 1, pp. 261–405. Translated as "The Concept of Truth in Formalized Languages", in Tarksi (1983), pp. 152–278.
Tarski, A. (1983), Logic, Semantics, Metamathematics: Papers from 1923 to 1938, J.H. Woodger (trans.), Oxford University Press, Oxford, UK, 1956. 2nd edition, John Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983.

JA: Jon Awbrey 05:40, 7 August 2006 (UTC)[reply]

Article rating[edit]

The article is somewhat short, and could mention important ideas such as types, quantifier elimination, etc. These could be in summary style. There is no discussion of the history of the subject or of the current trends. The discussion on Godel's incompleteness theorem seems a little out of place here; a one-sentence clarification might be enough. CMummert 14:29, 25 October 2006 (UTC)[reply]

I'd go even further. The lead section is extremely misleading and does not at all reflect the content of what is ordinarily called "model theory". The two independence results given as examples (AC and CH) are ordinarily considered part of set theory, not model theory, though they use some elementary model-theoretic techniques. The lead needs a complete rewrite, preferably with input by a real live model theorist if we can find one. Are there any on WP? --Trovatore 15:52, 25 October 2006 (UTC)[reply]

The problem with rating these articles is that there is no grade between Start and B-class but many articles are in between the standards for them. I agree that this article is barely a B-class article, and that it needs significant work before it can be regraded as B+ or A class. After reading the rating gudelines again, I changed the rating to Start-class.

I also noticed the second para in the lead section, but I decided it was OK because it only claims that those are the "most famous results", which is probably correct because model theory is not well known outside of mathematical logic. If the rest of the article were stronger, that paragraph in the lead might help a lay reader to get into the spirit of the main article. CMummert 16:02, 25 October 2006 (UTC)[reply]

Shellgirl's recent edits[edit]

I have a few concerns about some of these additions. I will be very brief for now because I'm supposed to be working, but I didn't want her to do a lot of work that might be disputed later. So here is a short list; I'll look in more detail later.

The claim that model theory is based on the axiomatic method. This is not necessarily so; it can be treated as formally or informally as any other area of mathematics.
The identification of the Gödel incompleteness theorems as part of model theory. They are more typically thought of as proof theory or recursion theory, though certainly they have model-theoretic implications. --Trovatore 00:10, 6 April 2007 (UTC)[reply]

Jessica responds Hi!

First point. Although I am fond of the phrase "axiomatic method" (I think because I saw a talk given by Udi Hrushovski where he justified the new geometric tilt to model theory by saying something like, the axiomatic method and geometry have been interacting usefully since Euclid ) I understand your concern: I certainly don't want people to get the impression that model theory involves reams of formally correct first order proofs.

Second point. I agree. As much as I love and adore Gödel, and although I do consider his work with the constructible universe to be the work of a model theorist, his big results are clearly of most interest for foundations of math, set theory, proof theory, computability theory, and computer science (especially with the twist involving Kolmogorov complexity). Okay, the completeness theorem is arguably the most important theorem of model theory. But somehow it pales in comparison to the incompleteness theorem and the work on L. In my defense, I wasn't the one who put the incompleteness theorem in the first paragraph, I just reworded it.

Please let me know what you think of the new additions. I plan to make model theory an A class article. But it'll take a couple years and/or a bunch of model theorists jumping on board, especially since a lot of the work will involve making other pages.

Shellgirl 04:49, 6 April 2007 (UTC)[reply]

So first of all I want to say I'm delighted that someone who actually specializes in model theory is finally taking a look at this article. I'll just give some impressions in no particular order:

The lede is far too dense and technical. Wikipedia lede sections are supposed to summarize the whole article, but be accessible to as wide an audience as the subject matter permits. All of that information should go in the article, but most of it further down; at the top level we want to synthesize the content and give as broad an audience as possible (that would probably be professional mathematicians in general, for this sort of article) an idea of why it's important.
Really, you think the work on L is model theory? To me it's set theory. Of course the borderlines are fuzzy, but there isn't too much in Gödel's work on L that involves, say, types, which I've always thought of as more or less the defining feature of model theory.
I still don't really get how the axiomatic method is relevant. You could look at the model theory of models of some non-axiomatizable theory; say, true arithmetic. From my outsider's perspective, very roughly, if you fix the theory (even a non-axiomatizable one) and vary the models, then you're doing model theory, whereas if you fix a (set or class) model of set theory (to be, say, the complete V, or L, or L(R) ) and try to figure out its theory, then you're doing set theory. Does that sound like a reasonable rough demarcation to you?
Can't comment on the "geometric tilt"; I'm completely ignorant on that score.

But to sum up, great to have you on board! We can really use your help; I don't think there's a model theorist per se editing regularly. It would be useful for you to get familiar with the "house style", which is informally but fairly strictly adhered to. There are official manuals at WP:MOS and WP:MSM that you should at least scan, and other conventions that maybe aren't written down in a single place, but sometimes discussed at Wikipedia talk:WikiProject Mathematics. You might also like to add yourself to Wikipedia:WikiProject Mathematics/Participants. --Trovatore 07:03, 6 April 2007 (UTC)[reply]

I agree with you about the first paragraph. I'll take your advice about learning more about the house style, as well as basic social graces on wikipedia. Now that I've "been bold" I'll try to integrate myself.

I view skolemization as a method of model theory -- it is similar to Henkin's proof of the completeness theorem, which builds the model out of the syntax. Also, the Mostovski collapse is a model theoretic technique. As you say, the distinction is fuzzy. Also, I would prefer to leave a distinction like this to a working set theorist.
I'm off playing ultimate frisbee for the weekend. More changes next week.

Jessica Millar 12:23, 6 April 2007 (UTC)[reply]

underconstruction[edit]

I removed (commented out actually) the underconstruction template as I cannot see any edits to the article since 2 May 2007. Zero sharp 14:26, 9 May 2007 (UTC)[reply]

butbutbut... please restore it if you are actually, actively working on the article -- which I think is a fine idea! Zero sharp 14:32, 9 May 2007 (UTC)[reply]

I am working on it when I have time. There is lots of good stuff here, but the organisation needs a lot of work. I would be interested in comments about the organisation, because I think it is not right at the moment, but have not got a clear idea yet on what would be good. I see the following as a possibiility:

Initial motivation: elementary and pseudo elementary classes, decidability, QE, foundational issues: compactness and LS-theorems
Then maybe the basics of the classical theory: robinsons ideas, diagrammes, model completeness EF games, Horn sentences, definable sets and types and so on.
Imaginaries and interpretations.
Model theoretic constructions: omitting types theorem, Fraisse, ultraproducts, saturated, big, homogeneous structures, prime models. Hrushovskis construction.
Morleys theorem, Shelahs clasification theory and on to geometric stability theory.
o-minimality and related (definable completeness, weak o-minimality, d-minimality, thorn independence
recent important results and programmes (relating to each secion: eg. decidability of R_exp, some recent "pure model theory", valued fields - stable domination, maybe put Hrushovskis construction and the new Zilber analytic structures here, groups of finite morely rank, the implications for real geometry of o-minimality.

As regards computable model theory, I know nothing about it. I would be interested to know some more. As it regards this page, it is really close enough to the rest of (infinite, first order finitary) model theory, to warrent a section here? I look forward to your comments. Thehalfone 12:40, 22 June 2007 (UTC)[reply]

excellent! I worry a bit about leaving the article with empty sections... I guess at the least we could put whatever the appropriate stub/expand template is. I myself, just an amateur/layman/dilletante am very interested in Model Theory and would love to contribute to the article any way I can. Thanks Zero sharp 14:39, 22 June 2007 (UTC)[reply]

Hi Thehalfone,

The program above looks great, but not achievable in a single article, certainly not in the central article for a whole subject. What I would do in this article is try to give the basic thrust of what model theory does and hopes to accomplish, at a relatively high level. The central notions that need to be treated, I'd say, are those of model, isomorphism, elementary equivalence (and especially the fact that this is not the same as isomorphism), and types.

Then most of the above concepts should be farmed out to other articles where they can be treated in more detail. You might have roughly one subsection of the main article, per bullet point in your program above, where you mumble some generalities and point the reader to the detailed articles on the subject. This is what I've tried to do in the determinacy article; you might look at that and see what aspects of the organizational structure you think mught work here (leaving out, of course, those you think wouldn't be so good). --Trovatore 22:03, 22 June 2007 (UTC)[reply]

A couple of grammatical questions[edit]

pseudo elementary class or pseudo-elementary class?

the latter according to Hodges

non logical symbols or non-logical symbols?

I suppose the latter to be consistent

"any binary functions" or "binary functions"?

Theorys or Theories?

again the latter according to Hodges. I guess it was my edit that had all these problems. I will change it now.

Being a noob on Model theory, I don't want to make edits without getting a second opinion. 84.184.251.34 18:39, 26 June 2007 (UTC)[reply]

Thanks! Thehalfone 10:00, 27 June 2007 (UTC)[reply]

Pseudo-elementary -> elementary[edit]

The article defined "pseudo-elementary" classes as what is usually (Hodges, Chang-Keisler) called "elementary" classes. I corrected this and added a link to the relevant article, which, by the way, originally had a ~~wrong~~ non-optimal definition of elementary classes (requiring axiomatisability by a single sentence rather than a theory). Since Wikipedia has for some time been the number 1 source for (incorrect) information on pseudo-elementary classes on the web I have also covered them in the elementary classes article. If anybody has issues with the changes, please don't hesitate to contact me. --Hans Adler 16:19/20:01, 12 November 2007 (UTC)[reply]

Removed paragraph from introduction[edit]

I have removed the following paragraph below from the introduction. After my changes to the first paragraph it became a non sequitur, and anyway a bullet list of abstract examples is perhaps not ideal for an introduction.

For example:

One can classify structures depending on which sentences are true in them. This is generally a coarser classification than isomorphism classes.
One can classify sets of sentences depending on properties of classes of structures which satisfy them.
One has methods for finding or constructing structures satisfying a given set of sentences.
Given a structure, one can consider the sets definable within it via logical formulas. One can ask whether these sets have a "good" geometry.

--Hans Adler (talk) 14:38, 17 November 2007 (UTC)[reply]

Draft of a rewritten article[edit]

To address the obvious structural problems with this article as well as the question of duplication between the main article and those to which it refers, I came up with the following plan:

Give a quick overview of everything that is important in model theory.
Whereever possible define necessary notions intuitively by giving an example of their use rather than a definition. (Subarticles will invariably have more detail. With this approach we can actually profit from this.)
Organize by subfields rather than by methods, notions and theorems. Comparison between subfields can be done here better than anywhere else.
In spite of 3., tell a linear story.

I have tried it out in userspace, and it seems to work surprisingly well. Please have a look at my current (still very incomplete) draft here and comment on its talk page, or even better start editing in the draft if you like. --Hans Adler (talk) 15:49, 20 November 2007 (UTC)[reply]

I have now restructured the article and added my new sections (to avoid a longer fork in userspace). Some older material became redundant, since many notions are now defined in the sections on universal algebra, finite model theory and first-order logic. I have also removed many empty sections. I am going to write a section on classical model theory that will contain a large part of what is now in "Other notions". After that I will probably start with what I am really interested in (classification theory / stability theory). --Hans Adler (talk) 22:30, 28 November 2007 (UTC)[reply]

Defining Model Theory[edit]

As a model theorist, I'm still not quite satisfied with the opening paragraphs (though they're a lot better than they used to be). I think mainly I'd like to see more acknowledgement of the central role of the study of definable sets in contemporary model theory, to the extent that many researchers who call themselves model theorists spend more time thinking about categories of definable sets rather than classes of models.

As for how to define "model theory," the best one-sentence answer I've seen is from Hodges' Shorter Model Theory: "Model theory is about the classification of mathematical structures, maps, and sets by means of logical formulas." (Though I realize that strictly speaking this does not include some of the current research on abstract elementary classes.) I think this is a lot more informative than the somewhat cryptic formula "model theory = universal algebra + logic."

Other thoughts on this?

Skolemizer (talk) 09:15, 4 January 2008 (UTC)[reply]

This is a good point to raise. The opening paragraphs are always going to be difficult I think partly because model thoery has developed and evolved so quickly (and looks set to evolve a great deal more). It is difficult to try to summarise the work of say Robinson together with some modern applied model theory. I do think that more stress could be given to the study of definable sets (particularly if one conisders, for example, o-minimality: this field become widely accepted and used amongst real geometers who are still sometimes hesitant about working in models other than R). We have at the moment much more information about "classical" model theory here, and perhaps this has been approoached in the correct order. As we introduce more on modern model theory and applications, we can also think about improving the opening.

By the way, well done to all who have edited the article in the last few months! I have been rather busy and not been on Wikipedia. I was very pleasantly surprised to see how much the article has improved. Thehalfone (talk) 10:22, 6 March 2008 (UTC)[reply]

Unfortunate deletion of crucial examples[edit]

I am an applied mathematician taking an axiomatic and structural approach to physics. I am finding this approach is proving answers to long standing paradoxes in the subject. My view is that Model Theory has much to offer applied mathematics.

However, I find the presentation style of texts on mathematical logic to be rather inaccessible. I was very pleased to find an earlier edition of this article very helpful due to examples given. The excerpt I have pasted below was especially helpful in giving me an introductory understanding of structure and undecidability in mathematics that tends not to be given in books.

Unfortunately this text was deleted at 21:51 on 28 November 2007. The style of the article since is probably more structured but is in a language that suits pure mathematicians.

I believe the article could be significantly improved by the addition of examples throughout, or alternatively a section written with applied mathematicians in mind. I wonder if the following section could be reinstated.

Preliminaries[edit]

Languages and structures[edit]

The syntactical object we need is a language. This consists of some logical symbols (plus a binary relation symbol for equality of elements), a list of non-logical symbols known as the signature, and grammatical rules which govern the formation of formulae and sentences.

Let $L$ be a language, and $M$ a set. Then we can make $M$ into an $L$ -structure by giving an interpretation to each of the non-logical symbols of $L$ . The grammatical rules of $L$ are designed so that one can then give each formula and sentence of $L$ a meaning on $M$ . The class of $L$ -structures together with, for each structure, the interpretations of the symbols, formulae and sentences are the semantical objects which correspond to the language.

Examples.

Consider the first order language with non-logical symbols $\{\times ,+,-,0,1\}$ , where the grammar is arranged so that $\times$ and $+$ are binary operation symbols, $-$ is a unary operation symbol and $0$ and $1$ are both constant symbols.

Then if $M$ is a set, $f_{1},f_{2}:M^{2}\to M$ are any binary functions, $f_{3}$ is any unary function, and $m_{0},m_{1}$ are elements of $M$ then we can make $M$ an $L$ -structure by interpreting $\times$ by $f_{1}$ , $+$ by $f_{2}$ , $-$ by $f_{3}$ , $0$ by $m_{0}$ and $1$ by $m_{1}$ .

For example we can take the set of real numbers and interpret the symbols of $L$ by their usual meanings in the real numbers. If we ask a question such as "∃y (y × y = 1 + 1)" in this language, then it is clear that the sentence is true for the reals - there is such a real number y, namely the square root of 2.

One can also make the rational numbers into a structure (with the standard meanings for the symbols on the rationals). Then the sentence considered above is false for the rationals. A similar proposition, "∃y (y × y = − 1)", is false in the reals, but is true in the complex numbers, where i × i = − 1.

Steve Faulkner, 19 may 2008

This article is a nightmare[edit]

because, it does not give a definition -- informal or not -- anywhere near the beginning of the article. (And for all I can tell, anywhere.)

Let me guess -- this article was written by some guru of mathematical logic who has no idea of how to speak to mere mortals. Am I right, or am I right?

Here is the first sentence:

"In mathematics, model theory is the study of (classes of) mathematical structures such as groups, fields, graphs, or even models of set theory, using tools from mathematical logic. Model theory has close ties to algebra and universal algebra."

That is all well and good, but what is lacking is the definition of WHAT A MATHEMATICAL MODEL (in model theory, a branch of mathematical logic) IS. Even an informal one.

That is the NUMBER ONE REASON to have an encyclopedia article on such a subject -- to explain what a model is to people who don't already know what a model is.

Here is the so-called "Introduction" in its entirety:

"Model theory recognises and is intimately concerned with a duality: It examines semantical elements by means of syntactical elements of a corresponding language. To quote the first page of Chang and Keisler (1990): universal algebra + logic = model theory. In a similar way to proof theory, model theory is situated in an area of interdisciplinarity between mathematics, philosophy, and computer science. The most important professional organization in the field of model theory is the Association for Symbolic Logic. An incomplete and somewhat arbitrary subdivision of model theory is into classical model theory, model theory applied to groups and fields, and geometric model theory. A missing subdivision is computable model theory, but this can arguably be viewed as an independent subfield of logic. Examples of early theorems from classical model theory include Gödel's completeness theorem, the upward and downward Löwenheim–Skolem theorems, Vaught's two cardinal theorem, Scott's isomorphism theorem, the omitting types theorem, and the Ryll-Nardzewski theorem. Examples of early results from model theory applied to fields are Tarski's elimination of quantifiers for real closed fields, Ax's theorem on pseudo-finite fields, and Robinson's development of nonstandard analysis. An important step in the evolution of classical model theory occurred with the birth of stability theory (through Morley's theorem on uncountably categorical theories and Shelah's classification program), which developed a calculus of independence and rank based on syntactical conditions satisfied by theories. During the last several decades applied model theory has repeatedly merged with the more pure stability theory. The result of this synthesis is called geometric model theory in this article (which is taken to include o-minimality, for example, as well as classical geometric stability theory). An example of a theorem from geometric model theory is Hrushovski's proof of the Mordell–Lang conjecture for function fields. The ambition of geometric model theory is to provide a geography of mathematics by embarking on a detailed study of definable sets in various mathematical structures, aided by the substantial tools developed in the study of pure model theory."

Did you actually read it? Can you actually read it? It is "a tale . . . signifying nothing," especially to those who don't yet know what a model is.

I would gladly give up all the advanced material in the article in exchange for ONE SINGLE CLEAR EXPLANATION OF WHAT A MODEL IS.

OK, granted, some examples of models are given right off the bat: "groups, rings, fields, graphs, and even models of set theory".

That's real helpful, giving as an example of a model a "model of set theory" that uses the very same sense of the word model to explain it to people who don't know what it means yet. And giving no clue to what groups, rings, and fields have to do with modeldom.

HAS ANYONE CONSIDERED THE POSSIBILITY OF MENTIONING THE RELATION OF A MATHEMATICAL MODEL TO AN AXIOM SYSTEM? LIKE MAYBE SOMEPLACE IN THE FIRST 1000 WORDS?????

(Just a suggestion.)Daqu (talk) 05:33, 15 July 2009 (UTC)[reply]

I am sure you have cooled down a bit since writing this, so perhaps now you can also contribute some constructive comments. The idea of defining what a model is somewhere in the beginning is certainly a good start in this direction. I.e., it is a constructive proposal.

I am not sure it's a good proposal, though, since this article is about model theory, not about the model relation. In model theory the word "model" is often used as a synonym for structure (mathematical logic). In this sense it has nothing to do with axioms, and in fact a large parts of model theory are independent of logic.

The example "models of set theory" is not as directly tautological as it sounds, by the way. A model of set theory is intuitively something that satisfies the axioms of set theory, i.e. "model" here refers to the model relation. And any such thing happens to be a structure, i.e. a model in the other sense. I guess if it wasn't for another, unrelated terminological ambiguity, we would be calling model theory "structure theory".

I am afraid your long rant here had some similarities with a [hypothetical one] on Talk:Whale, complaining that the first paragraph doesn't state clearly that whales are fish. Hans Adler 19:51, 15 July 2009 (UTC) (edited 15:16, 16 July 2009 (UTC) after Daqu's complaint on my talk page)[reply]

Thank you so much, Mr. Adler, for inserting the phrase "[a hypothetical one]" in your sentence above -- but that is just a polite way of admitting that your reference to my supposed comment about whales is just a lie.

I am fully aware, thank you, that the subject of the article is model theory, not merely models per se. But what would you think of an article about group theory that omitted any reference to the definition of a group? About relativity theory that contained no reference to the definition of relativity? About algebraic number theory that didn't bother referring to the definition of an algebraic number? (Please don't answer these questions, Mr. Adler; they are rhetorical.)Daqu (talk) 15:28, 17 July 2009 (UTC)[reply]

Sorry for responding anyway, but I actually think that such articles can be fine:

Have you looked at Group theory? It turns out that it does not define what a group is. We have the article Group (mathematics) for that.
Relativity theory says it's named after the principle of relativity, but never says what that is. That's done in the separate article.

Similarly, the article Model theory defines what a model is, but very late, at Model theory#First-order logic. That's because we have the articles Structure (mathematical logic) and First-order logic for defining what the two kinds of model are (i.e. model as a structure and model as a model of a theory/of axioms). It's hard enough to write a semi-decent article on the topic without artificial constraints. Starting with explaining a term that needs as much explanation as "model" does is such an unnatural constraint. Hans Adler 19:37, 17 July 2009 (UTC)[reply]

I think that Daqu's complaint is fair, but he should see that there's not a whole lot of energy around to fix the problem. The mathematical logic article is adequately accessible to nonspecialists; the set theory and proof theory articles are written in the right style, but both are woefully incomplete. Tis article and the recursion theory article are more complete, but are too hard. I defend the value of having highly technical articles, but we should be more effectively using summary style to separate the parts of topics that are widely accessible from the parts that can't be.

SHOUTING is usually not effective rhetorically, but sometimes it does work well in communicating one's degree of frustration.

I'll follow up with some ideas on improving the article. I think the summary of model theory in the math. logic article would make a better start for this article than what we have. That concepts are defined elsewhere isn't a good reason not to summarise them here, given that this is one of our main math. logic articles. — Charles Stewart (talk) 10:40, 17 July 2009 (UTC)[reply]

I agree that Daqu has a point to some extent. While it is true that "large parts of model theory are independent of logic", it is also true that an awful lot of model theory is about logic, and this includes the motivation why model theory was born in the first place. The article definitely needs to start with an explanation of first-order formulas and the satisfaction relation, instead of misleading sections like "Universal algebra" (which completely ignores the fact that most of model theory concerns relational structures, not algebraic structures) or "Finite model theory" (which is quite out of place, it should come much later in the article, both for didactic reasons and WP:UNDUE). — Emil J. 12:18, 17 July 2009 (UTC)[reply]

Apologies for what was perceived by some as the textual equivalent of shouting, but in fact that is not how I intended it. The reason for the various capitalized and/or boldface words and phrases is this: So many, many times my comments on various Talk pages have been responded to by users who did not read what I wrote, that I felt it important to emphasize what my main points are. In the future I will make an effort to do so a bit less emphatically.Daqu (talk) 15:28, 17 July 2009 (UTC)[reply]

EmilJ, I think the article you have in mind is quite different from the ideal article that I have in mind – and obviously both are quite different from the current state, although I am not too unhappy with the organisation of the first few sections. The question for me is, how much overlap to first-order logic do we want? It would be very convenient to have a POV fork of First-order logic that presents the topic quickly and elegantly in the way that model theorists usually do it. But 1) it's against policy, and 2) in that case we would still need another article that actually discusses the various fields of model theory.

What I tried to do here is organise the article in such a way that every section describes a field of model theory, but that there is still a sense of natural progression as you advance through the article. That's why it starts with two of the more tangential fields: They need less prerequisites, and on the shallow level on which they are treated in the article they even form the basis for first-order logic. Hans Adler 19:51, 17 July 2009 (UTC)[reply]

I'm not suggesting to make a fork (POV or otherwise) of the first-order logic article, but to have a concise summary which would alert the reader of the basic concepts. The "first-order logic" section we have is not bad, though it could be better organized so that the reader can easily find links to the proper definitions of the relevant concepts (e.g., the current version suggests that the reader will find the definition of satisfaction in the T-schema article, which is actually useless crap). However, it should come first. I find the idea that finite model theory "needs less prerequisites, and on the shallow level on which it is treated in the article it even forms the basis for first-order logic" completely absurd, finite model theory is no less easy to introduce that general model theory, in fact it's quite the opposite, as it also involves descriptive and computational complexity theory (which connection is however not even mentioned in the section here, begging the question of why to consider it separately in the first place). As for universal algebra, it is an important field related to model theory, but I think that presenting it as a basis for first-order logic, and even deferring to it the definition of a structure is not just wrong (as universal algebra does not treat relational structures), but also counterproductive. The fields are not really much related apart from the shallow most basic level.

I suggest to move the "first-order logic" section up after the "introduction", incorporate in it a suitable concise definition of a structure, and tie up some loose ends as alluded to above. The "universal algebra" section can be shortened. The "finite model theory" section should be moved down, say, below "categoricity", and rewritten so that it actually has something to do with what is done in finite model theory. The big question is what to do with the "introduction"; it sounds reasonable to me, but Daqu's post strongly suggests that it is incomprehensible to an outsider; I'm not sure what can be done about it. — Emil J. 15:31, 20 July 2009 (UTC)[reply]

Comment from a non-mathematician: This article does need a definition of what a model itself is, because Model (logic) redirects here. I'm just a user with only a modest (i.e. freshman calculus, 30 years ago) math background. I got here because I want to understand relational database, which links to relational model, where the Overview section refers to a "finite (logical) model" and links to Model (logic), which I clicked on. All I want to know is, what's a model? I read this article as far as I could before my head started to swim, and I still have no idea what a model is. It would be great if somebody could add a brief explanation of that, in terms educated non-mathematicians could understand. Or else create a separate article instead of redirecting here (or to the mathematical logic article mentioned above by Charles Stewart, which is also no help at all). Thanks! 76.204.31.165 (talk) 17:51, 15 August 2009 (UTC)[reply]

Thanks for the reminder. I agree that this is a serious problem. It's a bit tricky, because the term has two major variants distinguished by the way you use it (model for a language, model of a theory) and also a very general, almost philosophical definition (a mathematical object for which the sentences of the language have meaning / for which the sentences in the theory are true) with a very dominant special case (structure (mathematical logic) / model of a first-order theory). So conceivably the necessary explanations could be put into mathematical logic, first-order logic, structure (mathematical logic) or into its own dedicated article at model (logic). Or indeed here. It's not clear to me which is the best solution; I am not particularly happy with any of them.

As a quick fix I have added an explanation to the lead of this article. [1] It may not be entirely clear yet, but I think these notions are hard to explain without assuming some prior knowledge, and unfortunately much of most basic terminology is not even standardised. (In some fields the terminology is modernised, and then others don't follow.) Hans Adler 19:17, 15 August 2009 (UTC)[reply]

Wow, still quite difficult, but thanks for the attempt. I think for somebody like me — despite my PhD! — to understand it, concrete examples would really help. That alone might justify breaking it out into its own article. 76.204.31.165 (talk) 20:56, 15 August 2009 (UTC)[reply]

Good point about the examples. Unfortunately the lead isn't the best place for that. Would the following help?

An possible language for abelian groups has a constant symbol 0 and a binary operation symbol +. The natural numbers with the obvious interpretations of 0 and + form a structure, a model for this language. Since 1 does not have an inverse in this structure, the structure is not a model of the theory of abelian groups.

If it helps, perhaps I can find a place for something like this further down in the article, or cram it into the lead anyway. Hans Adler 21:38, 15 August 2009 (UTC)[reply]

Well, I won't say it doesn't help the article for its intended audience, but I'll say it doesn't help me so much. I think what you're trying to get across may simply be way too abstract for the background knowledge I bring to it. But as I read further along in Relational model I find I'm getting the picture pretty well, even without understanding the abstraction of "model". Perhaps there should be some article at an intermediate level of abstraction, that Relational model could link to in order to take readers of that article just a little further beyond, and then that article could link here (or to a dedicated Model (logic)). It just seems too great a leap has been taken. Anyway, thank you for your efforts!

By the way, this is definitely the deepest I've ever gotten into Wikipedia, and the first time I've ever added to a Talk page or done anything other than fix a comma here and there. I don't see any way to subscribe/follow a topic; I've just been returning periodically to check whether you've commented. Do I have to create an account in order to access such a subscribe/follow feature? (I haven't wanted to create one so far because I thought there'd be lots of rules to learn that I don't have time for — I suppose I've probably even broken one of those rules by asking this question here!) Thanks.76.204.31.165 (talk) 23:24, 15 August 2009 (UTC)[reply]

It's been a while since I checked out this article, but recently I was explaining some things to a friend about how Goedel's undecidable statement "There is no proof of me" works (and all I learned about that came from the book by Nagel & Newman), and I wanted for my own sake to know what a model is, even if it was only an informal idea.

I am honestly shocked and saddened to come back to this article almost a year later and find not even one iota of improvement in the problem(s) that I pointed out (OK, maybe too emphatically) when I opened this section of the Discussion page.

I have never engaged myself in the process of formally trying to get an article removed from Wikipedia before, but in the case of this article I feel it is extremely important to remove it from Wikipedia. It is absolutely the worst possible kind of article: It is written by people who haven't the vaguest idea of what an encyclodpedia is for, and they're "too busy" to do anything that might actually enlighten someone who doesn't already know what a model is, even a teeny tiny bit.

But it's time I learned the procedure for initiating the removal of an article, because this one is the best candidate for removal I've seen in five years.

This reflects poorly on mathematicians, and I am in favor of trying to improve the image of mathematicians in the eyes of the general public. Mathematicians have a reputation, not entirely undeserved, for being unable to communicate with anyone else, for being either unable or unwilling to lower their level of discourse to the point that even the intelligent, interested layperson can understand them.

I am looking forward to the complete absence of this article so that people who do have some idea of the purpose of an encyclopedia can replace those who obviously have none, in the rewriting, from scratch, of the article on Model Theory.

(Defending the absence of any definition of a formal model in this article by pointing out that the article on group theory fails to contain the definition of a group makes as much sense as saying that 1 must be the largest positive integer, because for any other positive integer n > 1, n cannot be the largest because n² > n.)Daqu (talk) 00:21, 16 June 2010 (UTC)[reply]

It'll be a snow keep — you'll be laughed out of AfD. First of all, AfD is not for cleanup. Even if it were, you haven't explained how you want to clean it up. So either explain how you want to fix it, or go away. No one is interested in nonspecific "I don't understand this" whinges. --Trovatore (talk) 01:42, 16 June 2010 (UTC)[reply]

It's almost amusing that someone who got his Ph.D. seven years ago in set theory (nine years after receiving his bachelors) but is either unwilling or unable to explain what a model is in this article -- and is now not working in mathematics -- should be taking such a superior attitude towards someone they haven't met but who has published papers in the Annals of Mathematics, who has had postdoctoral fellowships at Harvard and at the Institute for Advanced Study in Princeton, and who got his Ph.D. over 35 years ago (four years after receiving his bachelors) [I better stop here or someone might think I'm being immodest or something]. —Preceding unsigned comment added by Daqu (talk • contribs) 17:21, June 16, 2010

Very nice; compliments. So then you shouldn't have any problem understanding the content. Explain how you want to fix it. --Trovatore (talk) 18:04, 16 June 2010 (UTC)[reply]

Primes and Irreducibles[edit]

In the first order logic section, there are example formulae "phi" and "psi". The article claims that of the "ring" of natural numbers, the only elements satisfying "phi" are primes and the only elements satisfying "psi" are irreducibles. This seems pretty bad to me. To begin with, the natural numbers are not a ring at all -- the author may have meant just the "set" of natural numbers or perhaps the "ring" of integers, I'm not sure which. The term "irreducible" doesn't apply unless you're in the context of a ring, though. Secondly, the definitions as stated are incorrect. The elements satisfying "phi" are primes and UNITS (e.g. 1 and -1), and the elements satisfying "psi" are irreducibles and units. I'm not sure what a good solution to the problem is, since I don't know what the author's intention was in including these examples... —Preceding unsigned comment added by 71.198.191.61 (talk) 19:43, 21 October 2009 (UTC)[reply]

You are right about the units (and 0 satisfies the formulas, too). However, the article does not claim that natural numbers are a ring. It only specifies that we consider the structure of natural numbers in the language of rings, i.e., {×,+,−,0,1}. This is actually also incorrect, as the language of the structure of natural numbers is not supposed to include −, but that's a different problem. — Emil J. 12:33, 22 October 2009 (UTC)[reply]

Oh dear. I haven't checked, but I guess it was me who wrote this when rewriting large parts of the article. If so, I am pretty sure I had the integers in mind and simply forgot to exclude the units. How embarrassing that this survived so long!

The intention was just to give some easy but non-trivial examples of how we can encode actual mathematics in first-order logic. Hans Adler 13:13, 22 October 2009 (UTC)[reply]

Classes[edit]

I'm quite confused at the notion of class.

From my limited understanding they are supposed to be objects C in models of class theories like NBG, where there for every X the membership relation M(C,X) does not hold, i.e. classes can't be elements of something else. Then we start talking about the class of all models of NBG. It seems to me there’s a comprehension scheme here at work, i.e. the predicate “is a model of NBG”. I find this weird in two ways:

To have predicates we need a background universe, but the background universe of “is a model of NBG” seems to be the collection of absolutely everything, whatever that means.

We defined classes to be special objects in a specific model of NBG (I want to avoid saying “any model”), but by saying the class of all model it seems to me to be circular. Aren't we supposed to take models as primitive mathematical objects?

So what exactly is a class? Money is tight (talk) 00:19, 20 January 2010 (UTC)[reply]

Classes, in normal set-theoretic discourse, are not really objects at all. They're a convenient way of talking about predicates.

When you're talking about theories like NBG and KM that have an actual sort for classes, you have to adjust this slightly. The best way to think about it is that the intended interpretation for these theories is not really "all sets and classes", but rather "all sets up to a certain rank, and all collections of those sets". In the case of NBG, you might be reinterpreting "class", not as arbitrary collection of sets up to that rank, but only definable such collections. --Trovatore (talk) 02:12, 20 January 2010 (UTC)[reply]

Since Trovatore has answered one thing, I will answer the other. When people say "all models of NBG" or "all models of ZFC", they are speaking in a metatheory. If you take that metatheory to be ZFC itself, then "all models" means "all models that happen to be sets". Thus a model of NBG is simply a structure that is a set and satisfies the axioms of NBG. Of course ZFC cannot prove that there is a model of ZFC, but people often add an assumption that a model exists, since this is implied by the existence of an inaccessible cardinal or by the arithmetical axiom Con(ZFC).

In any case, the class of all models of a particular theory is a perfectly well-defined, definable class, for any first-order theory. It may happen that the class is empty, if the metatheory does not prove that there are actually models of the theory in question. But talking about all models of ZFC or NBG is not really any different than talking about all models of the theory of abelian groups.

Sometimes, people use the term "class model" to refer to a generalized sort of model of a theory in which the domain may be a proper class. Class models are much more difficult to treat in general when the metatheory is ZFC, but could be handled in a metatheory of MK. — Carl (CBM · talk) 03:22, 20 January 2010 (UTC)[reply]

Well, I don't really agree about the metatheory thing. That's how you would formalize it, sure. But if you're just talking about the things that exist, and are true, simpliciter, then you're not speaking "in a theory" at all. You're just stating things that happen to be true (or false, as the case may be). --Trovatore (talk) 03:25, 20 January 2010 (UTC)[reply]

I think this is the "no metatheory" thing again. In any case, when people say "model of ZFC" everyone knows they mean a set model, unless they say "class model of ZFC" or some derived notion such as "inner model". For example, we would agree that ZFC does not prove there is a model of ZFC (for then it would prove Con(ZFC) by the completeness theorem), but ZFC does prove there is a class model of ZFC (namely, {x | x = x}). — Carl (CBM · talk) 03:30, 20 January 2010 (UTC)[reply]

Thanks, both of your answers are very helpful. I still have a few inquiries though.

To Trovatore: So do you mean, suppose we have a fixed model of ZFC. Then by “the class of all sets” we just mean the predicate used to represent x=x? $\forall z[z\in x\Leftrightarrow z\in y]\land \forall z[x\in z\Leftrightarrow y\in z].$ And when we say the class of all sets that don’t belong to themselves we just mean the predicate not(M(x,x)). Of course, in ZFC these aren’t objects of our fixed universe/domain of discourse, but in NBG or MK they are objects, which are called proper classes.

To Carl: I never knew how to see things like groups (which is a special kind of structure), do we see them as literally sets (not just the underlying domain), or just primitive objects? When I say see them literally as sets, I mean this:

The natural numbers are taken to be primitive concepts, but we can code them as sets (von Neumann’s definition), and similarly we can code ordered triples as sets, so a model itself can be understood as a set (not just its underlying domain).

With that aside, I still have one last question related to both of your answers. In MacLane’s Categories for the working mathematician 2nd edition, he first defines meta-categories as a two sorted first order theory (I think). Then at the section Foundations page 22 he defines what a Grothendieck universe U means, and on page 23 he defines a class to be a subset of U. I think this is the context Trovatore is alluring to, when we have a fixed model U and then define classes (we can of course restrict the subsets of U to be subsets that are definable). But on page 24 he says

“Our foundation by means of one universe does provide, within set theory, an accurate way of discussing the category of all small sets (members of U) and all small groups (groups with small set as domain), but it does not provide sets to represent certain meta-categories, such as the meta-category of all sets or that of all groups”.

By all sets does he mean the cumulative hierarchy? Or, in realist terms, every set that can possibly exist? It definitely doesn’t seem to depend on any of the set theories, and in particular he didn’t say the class of all sets/groups. I think that since the time of Frege people had a conception of a platonic universe of sets, but since Frege’s theory was inconsistent people developed the idea of cumulative hierarchy as the universe of sets. I know this is a subject I can fully understand only through time, I just need you to tell me if my understanding is right or wrong (that's something everyone can understand :D). Money is tight (talk) 21:28, 20 January 2010 (UTC)[reply]

Your post is very long, so I will only try to respond to part of it. The difficulty with category theory is that, if one defines a category to be a set of objects and a set of morphisms, then the collection of all groups is not a category, because the collection of all groups is not a set. Hence the distinction between "small" categories and other categories. But you may find that MacLane expects a lot more background in set theory than you would think, so that he leaves a lot unsaid.

You are right that the cumulative hierarchy was developed as a way to recover a consistent idea of sets, after earlier set theories were shown to be inconsistent. — Carl (CBM · talk) 03:44, 21 January 2010 (UTC)[reply]

P.S. For general questions like this, you should ask at the math reference desk page instead of article talk pages. You'll get a better variety of responses there, and it saves talk pages for discussing the corresponding articles. — Carl (CBM · talk) 03:48, 21 January 2010 (UTC)[reply]

Example[edit]

Would it be useful to include an explicit discussion of an example at an early stage in this article? Its purpose would be to illustrate the basic relationship involving syntax and semantics in the context of some interesting models. Thus, one can start with suitable axioms for the natural numbers on the syntactic side. Going on to the semantic side, one could mention the usual counting numbers as a model, and point out that Skolem developed alternative models in the 30s (perhaps earlier?). This would give some substance to discussions of "interpretation". If this is not a good example, perhaps someone can suggest a better one? Tkuvho (talk) 14:45, 16 June 2010 (UTC)[reply]

Split of model (logic)[edit]

An editor recently created an outline in model (logic). I propose it be merged back into model theory until such time that something which makes sense to put in a separate article is said. — Arthur Rubin (talk) 00:30, 17 June 2010 (UTC)[reply]

Oppose merge -- I am opposed to the merge on the basis that there should be separate articles for an academic displine (or sub-field) and the concepts which are its objects of study. There are numerous places that link to model, with no appt article describing what a model is (including this person on the upper part of this talk page -- very passionate.) Greg Bard 00:36, 17 June 2010 (UTC)[reply]

Greg, his passion counts against him, not for him. This is a lesson you could well learn. --Trovatore (talk) 00:40, 17 June 2010 (UTC)[reply]

The correct target for model (logic) is structure (mathematical logic), not model theory. --Trovatore (talk) 00:38, 17 June 2010 (UTC)[reply]

Good point, redirecting. — Arthur Rubin (talk) 00:42, 17 June 2010 (UTC)[reply]

And removing merge tag. — Arthur Rubin (talk) 00:44, 17 June 2010 (UTC)[reply]

compactness theorem[edit]

The page currently states that "The Löwenheim–Skolem theorem and the compactness theorem received their respective general forms in 1936 and 1941 from Anatoly Maltsev." I was under the impression that the compactness theorem derives from Maltsev's 1936 paper Untersuchungen aus dem Gebiete der mathematischen Logik. [J] Rec. Math. Moscou (Matematicheskii Sbornik), 1 (43) (1936), 323--335. Tkuvho (talk) 15:01, 13 December 2011 (UTC)[reply]

That was my fault [2]. Regardless of whether it was true at the time, now that we have been saying it for a long time it will no doubt be true.

More seriously, I don't have access to Dawson's "The compactness of first-order logic: from Gödel to Lindström" from home, but I think that's what I had read when I made the edit. And indeed, a Google Books search for Maltsev + 1941 gives me the following in a snippet: "There the matter rested until, in his article in Russian Maltsev 1941 finally gave the first explicit formulation of the compactness theorem (or, as he called it, 'the general local theorem') for uncountable first-order languages." Hans Adler 16:18, 13 December 2011 (UTC)[reply]

My understanding is that one needs the compactness theorem to prove Löwenheim–Skolem. Robinson's book states on page 48 that the finiteness principle of the lower predicate calculus is due to Malcev (1936). He goes on to point out that the name "compactness theorem" was proposed by Tarski in 1952. I tend to rely more on Robinson than on snippets. Tkuvho (talk) 13:15, 21 December 2011 (UTC)[reply]

When I made that edit years ago, I was of course using the full article. I don't make such edits based on snippets. I used the snippet to confirm that that was my source. And it reminded me of what the article says: That, surprisingly, although all the ingredients for the compactness theorem were there and from a modern POV people were using it, it was so unexpected that they didn't see, let alone formulate, it for many years. Hans Adler 13:48, 21 December 2011 (UTC)[reply]

Apart from that issue, Robinson does appear to assert that Malcev proved the compactness theorem in that 1936 paper. I don't read German and am just curious about the content of that paper. Tkuvho (talk) 13:59, 21 December 2011 (UTC)[reply]

I still can't access Dawson right now, and I can't access Malcev 1936 either. However, on the second page of this paper, Vaught says that Malcev essentially extended the compactness theorem to the uncountable case in 1936. That's compatible with the first explicit formulation being in the 1941 paper. Hans Adler 14:12, 21 December 2011 (UTC)[reply]

If the uncountable case was only "essential" then the countable case was explicit already in 1936, to pursue your deduction a step further. Let me know if you find Malcev. Tkuvho (talk) 14:15, 21 December 2011 (UTC)[reply]

OK, I finally found the full text of the paper linked from here. (On my tiny netbook screen at first I only found the misleading PDF link further up.) Malcev 1936 does formulate the compactness theorem very much like we do today, for the uncountable case. But it's not immediately clear to me that his proof makes sense. Hans Adler 15:00, 21 December 2011 (UTC)[reply]

I have since learned that Malcev 1936 was his first paper ever. So maybe it did have a mistake, after all, which was corrected in 1941. (I still haven't seen the 1941 paper, so this is pure speculation.) I would personally have no problem with stating that the formulation in the general case appeared in the 1936 paper (rather than the 1941 one), although we should double-check with Malcev 1941 and with Dawson, before or after such a change. Hans Adler 23:57, 22 December 2011 (UTC)[reply]

The tried and true method of doing this would be to attribute the statement to Robinson without making a commitment. After all, we are only supposed to be verifiable? Tkuvho (talk) 08:28, 23 December 2011 (UTC)[reply]

I don't know what the history is here, but I can point out a general phenomenon. Modern authors often attribute theorems to older papers if the older papers have enough insight that the author of the original paper "could have" proved the result, if only they had meant to, or if only they had the right terminology. So it is not uncommon for the first person to explicitly state a particular theorem to not be the person who gets the attribution for it. The other option would be to cite the first person to state the theorem, even if "everyone knew" the result before then. In some cases it can be very difficult to see exactly why a modern author gives credit to a particular older author. — Carl (CBM · talk) 14:35, 21 December 2011 (UTC)[reply]

I would agree with this reasoning if for some reason there were a "default" assumption that Malcev did this in 1941. But is there in fact such a default? Does Dawson represent a consensus? I am just trying to figure out what the standard in the field is. Tkuvho (talk) 17:11, 21 December 2011 (UTC)[reply]

History of mathematics is a woefully neglected subject, so there is not going to be a large collection of sources about this. We could just say "Dawson (...) attributes the result to Malcev (...)" and leave it at that. — Carl (CBM · talk) 15:35, 23 December 2011 (UTC)[reply]

Vaught, Robert L.: Alfred Tarski's work in model theory. J. Symbolic Logic 51 (1986), no. 4, 869–882 similarly reports that Malcev (essentially) proved the uncountable case of the compactness theorem in 1936. Tkuvho (talk) 13:02, 26 December 2011 (UTC)[reply]

Another Math Related Article Wiklipedia[edit]

Too many idiots make contributions to articles they have little expertise in on Wikipedia. Those that have some knowledge, after often unable to describe mathematical ideas (and those in logic) in a manner that is both elegant and comprehensible.

The first sentence of too many articles on mathematical subjects start with an awful first sentence, that is verbose, contains truism or is factually incorrect. It is usually at this point I stop reading.

The writer of the first sentence clearly has no idea what model theory is. For starters model theory is a branch of logic, not mathematics. The second sentence has truism common to many wikipedia articles. Which states that "X theory studies Xs", as if anyone but an idiot needed to be told this. — Preceding unsigned comment added by 86.27.193.180 (talk • contribs) 22:53, 20 December 2011

Thank you for your help. Your criticism would be even more constructive if you could suggest an alternative to the first sentence.

The first sentence has been saying more or less the same things since the previous incorrect first sentence was corrected as part of this series of edits by User:Shellgirl, a model theorist. Later, in November 2007, I rewrote [3] the sentence in a form that has been stable over the last four years, except for minor changes. As a model theorist myself, I am slightly concerned by your observation that "[t]he writer of the first sentence clearly has no idea what model theory is" and would appreciate further information. In particular, your claim that "model theory is a branch of logic, not mathematics" is intriguing. It appears to suggest that there are significant non-mathematical areas of model theory, hitherto unknown to me. Logic being a field around the intersection of philosophy, mathematics and computer science, I guess that these would have to be either in philosophy or in computer science. (Given that the 1998 ACM Computing Classification System agrees with the AMS 2010 Mathematics Subject Classification in filing model theory under mathematical logic, I guess that this blind spot of mine lives in philosophy?) So, may I ask you to provide further information? In particular, I would be thrilled to read a book about model theory written by a model theorist who is not a mathematician. Any pointers would be greatly appreciated. Hans Adler 10:19, 21 December 2011 (UTC)[reply]

Would it be helpful to paraphrase the lede in terms of the syntactic/semantic dichotomy? Tkuvho (talk) 12:04, 21 December 2011 (UTC)[reply]

I am not sure how this would help, as the first paragraph is already very clear that model theory lives on the semantics side and only uses syntax as a tool. The first sentence of the Introduction section makes this fully explicit, and I think that's enough. Hans Adler 12:32, 21 December 2011 (UTC)[reply]

Could some of the "introduction" be moved usefully to the lede? Tkuvho (talk) 12:39, 21 December 2011 (UTC)[reply]

I would much rather reduce the redundancy in the lead than add more ways to say essentially the same thing. Hans Adler 12:53, 21 December 2011 (UTC)[reply]

When I got to "For starters model theory is a branch of logic, not mathematics." I stopped taking this complaint very seriously. — Carl (CBM · talk) 13:56, 21 December 2011 (UTC)[reply]

Hi Carl. Could you comment on the previous section (compactness)? I am curious what the historical fact is. Tkuvho (talk) 13:57, 21 December 2011 (UTC)[reply]

Algebraic geometry without fields[edit]

Can anyone explain what (if anything) this means? Thanks. 67.119.15.30 (talk) 03:19, 8 September 2012 (UTC)[reply]

I also think that this idea should be described in more detail in the article. The opening section quotes "model theory = algebraic geometry - fields", but this point of view is hardly elaborated on in the rest of the article. As far as I can tell it is only discussed in the description of "geometric model theory" in the section "Branches of model theory". I suggest that this quote should not be in the opening section unless it described in sufficient detail within the article, and otherwise we should move to the section "Branches of model theory". Itaibn (talk) 12:52, 15 August 2017 (UTC)[reply]

Its a much simpler idea than that. What's an algebraic variety? Its a place where some polynomial, or some intersection of polynomials are zero. Whats a model? Its a place where a sentence (or the intersection of a bunch of sentences) are true. There are other concepts that are similar/analogous: ideals, stuff similar to Hilbert's Nullstellensatz, duality between languages and the grammars that generate them, factorization into things that look like prime ideals, etc. I don't believe that the correspondence can be made very deep, but rather, its kind of cute, and its usually trotted out in the preface or in chapter one of books on model theory. 67.198.37.16 (talk) 05:15, 23 October 2020 (UTC)[reply]

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Expert attention[edit]

Recently, User:D.Lazard recently (5 Oct 2020) made several large edits to this article; in particular, erasing the introduction. I complained at his talk page, making assorted points:

The slogan "model theory = algebraic geometry - fields", can be found in in Wilifred Hodges book on model theory, in the preface or in chapter one. It strikes me as a rather clever observation, and so should not be removed. Other authors have made similar remarks. The erased intro gave a reference to one-such.

Model theory is interdisciplinary. If you look at satisfiability modulo theories aka SMT, the "theories" are exactly those things of model theory. SMT solvers are commonly used for computer chip fabrication and computer chip verification, and if you dig into engineering texts on quality control, they will typically have an intro chapter that recapitulates model theory in more-or-less exactly the same way that Hodges describes it in his books. I kid you not; you can do some google searches for SMT solvers and silicon verification testing, and you will find these texts. Yes, I found it mind-blowing when I stumbled over it in the early 2000's, but the trade rags actually claim that it was a bit of a break-through that upset the fortunes of more than a few companies in the verification industry. (This is distinct from the SAT-solver revolution, which revolutionized digital simulations; here, I believe that the models are for verifying analog circuit components, serial lines, timing diagrams, but I may be confused about that.)

Regarding the erased the stuff about semantic and syntactic duality; this appears in the chapter-one of books on proof theory, where they seem to have this obligatory sentence that says "if model theory is the sacred, then proof theory is the profane." The guys who write those books seem to like to say that, and, from what I can tell its partly about pecking order in math departments, and its partly about the syntactic vs semantic duality in all the various theorems.

The current article does seem to be ... ugly and in need of ... a (major!) rewrite ... I assume that there are other articles that explain basic terminology like "what is a model" and "what is a theory" and "what is a language", "what is a valuation", "what is an interpretation", etc. But I have not looked.

D.Lazard replied on his talk page, I cut-n-paste that reply below.

This discussion should be placed in the talk page of the article. Neverthess, it seems that you misinterpreted my comment in the tag. I never wrote "its not about model theory". I wrote "its not specifically about model theory". This means that a large part of the content is about theories that have been developed independently of model theory, that they are probably related to model theory, but the way they are presented do not explain their relations with model theory. D.Lazard (talk) 09:42, 23 October 2020 (UTC)

67.198.37.16 (talk) 20:17, 23 October 2020 (UTC)[reply]

I have moved inside 67.198.37.16's post the copy of my post on my personal talk page, and I have formatted it as a quotation.

67.198.37.16 has reduced the lead to a single sentence that defines model theory, but does not summarize the content of the article. They have also added to the body of the article the controversial sentences that I have removed from the lead. So, although the lead does not complies with MOS:LEAD, it is not worth to continue to discuss it before fixing the content of the body.

Nevertheless, the article still requires to be completely rewritten by an expert, because it gives to the non-specialist (I am one) the impression that the authors of this article claim that model theory contains as subfields universal algebra, semantics, most of mathematical logic, and even a part of algebraic geometry. This is clearly ridiculous. Moreover, reading carefully above 67.198.37.16's post, it seems that there are several distinct model theories that are not distinguished in this article. One is a tool for software verification (this is probably this tool that the IP has in mind when he refers to industry); one is a method of proof used in logic for proving that a theory is consistent if another theory is (Robibson's theorem is an example); another should be the model theory (I am not sure of its existence) that justifies the detailed section on universal algebra. D.Lazard (talk) 10:26, 24 October 2020 (UTC)[reply]

The model theory of hardware verification is identical to the model theory of mathematics; although it does not require/use/invoke any statements about ordinals & cardinals, since everything is finite. Wilfred Hodges "A shorter model theory" exemplifies that. If I recall correctly, he never mentions anything pertaining to set theory or to ordinals (e.g. inner and outer models, sigma-pi hierarchy, AC, CH, ZF, etc.), nor does he mention any engineering applications. 67.198.37.16 (talk) 13:17, 6 June 2021 (UTC)[reply]

I think for purposes of discussion (not necessarily in the article) we need to be clear on whether we're talking about (as it were) elementary model theory, or about model theory in the sense of what people who consider themselves model theorists do. The general stuff about structures and their semantics is table stakes for model theory, but it's table stakes for the rest of math logic as well, and I'm uncomfortable calling it "model theory" per se.

My understanding of model theory in the second sense is that it tends to be about the distinctions between different models of the same formal theory (the theory being usually complete and often decidable). This is the fundamental difference in point of view from model theory to, say, set theory or number theory, each of which is more about finding out the theory of a fixed intuitive structure, or proof theory, which is about studying the nature of the implications within a formal theory.

I think it would be good if the article could explain this distinction, assuming good sources can be found, but in any case in any discussion here, all participants need to be aware of what the others mean when they say "model theory". --Trovatore (talk) 22:35, 7 June 2021 (UTC)[reply]

I hadn't noticed that there was more than a half-year gap between D.Lazard's last comment and 67's reply. Some of the comments are a little more understandable with that in mind. There is no longer a "detailed section on universal algebra", but there used to be, last October. The first paragraph of the section on first-order logic,

Whereas universal algebra provides the semantics for a signature, logic provides the syntax. With terms, identities and quasi-identities, even universal algebra has some limited syntactic tools; first-order logic is the result of making quantification explicit and adding negation into the picture

now looks a bit orphaned and should probably be removed. However, I don't think any recent version of the article implied that universal algebra was a subfield of model theory; the point seemed to be more that model theory was a refinement or strengthening of universal algebra.

Without having made a detailed analysis of the changes, my general sense is that the article is significantly more focused and generally better than it was seven months ago. It could still use a lot of work, ideally with oversight from an expert (Hans Adler maybe?). Just a few points at random I'd like to see addressed, in no particular order:

The notion of "type" is arguably the real subject of model theory, or at least is second only to the notion of "model" itself. There is a brief section on types, but it appears late in the article, long after coverage of topics that you can't understand without understanding types.
Lots of things are mentioned without being defined or even glossed. One of the main preoccupations of a lot of model theorists at around the time I left academics was o-minimality, which is mentioned once in the article, with no hint given as to what it might be. Similarly for stability.
Categoricity is treated in a dissmissive way, essentially giving the idea that there's no such thing except in trivial cases, without mentioning that categoricity in a particular cardinality is a fundamental tool.
Saturation is not mentioned.
The language is self-referential; there are four mentions of "this page". That's lazy style. It can be tricky to work around, but it's almost always worth the effort.

Anyway, as I say I think there's been progress, but the guidance of a subject-matter expert would be useful in sorting some of these things out. --Trovatore (talk) 07:25, 8 June 2021 (UTC)[reply]

I am sorry, but I no longer edit Wikipedia. I was driven away years ago in connection with a mob pushing unreasonable numbers of unencyclopedic images into the Muhammad article just to spite Muslims and another mob pushing male genital mutilation by wikilawyering any mention of cultural motivations out of the circumcision article. When NewYorkLawyer (spelling?), a user I had previously highly respected, agreed to formally censor me for 'unprofessionalism' in an Arbcom decision related to the former problem, for no valid reason visible to me, this was the last straw. My comments on this page are the first after several years, and it may well be a few more years before I log in again. As far as I am concerned, this project is a zombie, as all the political parts have been taken over by advertising agencies and the like. Hans Adler 13:42, 8 September 2021 (UTC)[reply]

If I would know nothing about model theory, the article, in particular the introduction, is now in a worse state than before. Why were the (trivialized of course) characterizations like model theory = universal algebra + logic been deleted? I don't think this contributes to conveying the superstructure between those theories to the reader. Lambda C (talk) 02:30, 15 June 2021 (UTC)[reply]

The characterization of model theory as "universal algebra + logic" is not only not accepted by most specialists, but most published articles on model theory do not refer to universal algebra at all. So, having this controversial assertion in the lead is definitely misleading. Trivializing may be acceptable, but not if this provides wrong information to readers. D.Lazard (talk) 10:18, 15 June 2021 (UTC)[reply]

It's weird to claim that "model theory = universal algebra + logic" is "not accepted by most specialists". It's still as correct as the newer characterization "model theory = algebraic geometry - fields" is. They just operate on different levels. The former describes where the basic notions and original motivations of model theory come from. It's what is most relevant to beginners. The latter describes what the majority of today's model theoretic research is like and where it comes from.

The only real reason for removing "model theory = universal algebra + logic" that I can think of is that model theory is flourishing, logic is very much alive, but universal algebra has become a backwater field that most people never heard about because it has been superseded by model theory. But the point of referring to universal algebra here was not an assumption that people know what universal algebra is, or to advertise for it. It is a handy way of referring to the simple basics of model theory that are not explicitly related to logic. One could also say "model theory = semantics + syntax", but that's not as catchy and it's probably a new coinage. Hans Adler 13:35, 8 September 2021 (UTC)[reply]

I would be happy to help rewrite or improve the article if we can achieve some consensus on the skeleton outline. I obtained my doctorate in the field of non-elementary stability theory in 2018 and I am currently working at the intersection of (infinite) model theory, finite model theory and machine learning, so I do have an overview of the topic. However, although I consulted the relevant style manuals, I am a new editor, so I am not as 'bold' as I might be!

I would argue for instance that currently the three examples of applications given in the "Informal description" somewhat misrepresent the subject - The canonical proof of the independence of CH from ZFC uses forcing and inner models, both techniques that are counted to set theory proper. Similarly, while the relational model is certainly related to model theory (and in particular finite model theory is a standard part of the database theory arsenal), Erik Meijer's argument (unreferenced, so I am not sure if that is even what is referred to) for the duality of SQL and noSQL is purely category-theoretic. SMT solving does at least build heavily on Tarski's Quantifier Elimination in the real field, which is definitely considered model theory. Felix QW (talk) 10:04, 22 June 2021 (UTC)[reply]

I have not the competence for discussing most of the content of the previous post. However, "quantifier elimination in the real field" does not belongs to model theory, or, more exactly, does not belongs to model theory anymore, since the introduction of cylindrical algebraic decomposition by George E. Collins, which reduces this problem to algebra of polynomials and real algebraic geometry. Moreover, this change of paradigm allows practical implementation on computers, which is definitively impossible with Tarski's methods. I have read many papers (and written some) and books on quantifier elimination on the real, and I have never seen model theory mentioned there, except, maybe in historical sections. D.Lazard (talk) 11:22, 22 June 2021 (UTC)[reply]

While I would maintain that in general, quantifier elimination of logical theories belongs to the subject area of model theory and therefore the fact that the theory of the real field has quantifier elimination is part of model theory (and would typically be covered in a model theory lecture, including a model-theoretic proof), I concur that the actual implementation underlying computer science applications relies on alternative approaches from real algebraic geometry. I do think one can mention the Tarski-Seidenberg theorem as a model theory result on this page, but your argument does make the claim that SMT solvers directly apply model theory questionable (in absence of an actual reference).Felix QW (talk) 12:35, 22 June 2021 (UTC)[reply]

Definitions[edit]

What is the general consensus on typesetting definitions? I see that most of the definitions in this article have the concept to be defined in bold, while for instance the (featured) article on groups uses italic, as I would in a research paper. I couldn't find any guidance in the mathematics style manual. Felix QW (talk) 19:15, 29 June 2021 (UTC)[reply]

Bold should be used for the subject of the article, and when introducing synonyms for the subject, especially terms that would redirect to this article. It should not be used for defining other concepts. --Trovatore (talk) 16:54, 6 July 2021 (UTC)[reply]

British English Convention?[edit]

It seems to me (an American) that this article is written primarily in British English. Should this be tagged as the national variety of English for this article? SaturnFogg (talk) 23:01, 31 July 2021 (UTC)[reply]

I hope that I did this correctly by adding the template at the head of talk page. Felix QW (talk) 15:22, 29 August 2021 (UTC)[reply]

Still too specialized and technical[edit]

I read some complaints about this article's introduction's impenetrability for nonspecialists from around 2010. There have been some changes but the complaint is still valid. For instance, the introduction should give an intuition for the meaning of "model". I added a parenthetical remark to do that, which may not be literally correct and may be improvable, but it is supposed to give the idea of what we are talking about. The introduction still is not much use to a professional mathematician (me) whose specialty is not logic or model theory.

Asking, as in various replies from around 2010, nonexperts to tell the experts how to fix the article is not entirely fair. You experts are supposed to know the facts. On the other hand, the nonexperts may be able to come up with specific questions that guide the experts. Experts, mainly do not show off your knowledge in the introduction. Yes, it's hard, but try, and we will try to help you. Zaslav (talk) 22:44, 7 November 2021 (UTC)[reply]

Thank you for your alterations and comments! It is not easy to define what model theory is (see the discussion above that got me involved in the first place), and I am aware that metaphors and generalities are not particularly helpful to the general reader.

The fact that we have three historically minded and somewhat conflicting sections in "Varied Emphasis", "Branches" and "History" is not great either.

I will try to look into both issues as soon as I get some time! Felix QW (talk) 17:07, 8 November 2021 (UTC)[reply]

I might have had more time for the complaint in 2010 if it had been expressed less unpleasantly. Looking back over it, it seems that the first concern is that this article does not prominently define "model".

In my opinion, the proper article for that is not this one, but rather structure (mathematical logic), where we (should if we don't currently) go into detail on Tarski semantics for first-order logic over structures. A model is just a structure that satisfies a particular theory.

That said, it would be reasonable to put a brief recap somewhere in the opening section.

I would still like to see my points from June 2021 addressed. I could try to work on it myself, but I am not now nor have I ever been a model theorist. --Trovatore (talk) 19:01, 8 November 2021 (UTC)[reply]

I think I have gone some way to addressing the points you mentioned then. Types now have their place where I think they belong chronologically, saturation is at least defined, categoricity and stability have their own section, and o-minimality is defined in the subsection on minimality.

Since I am unhappy with the wikilinking to T-schema that someone put in recently to explain the satisfaction relation, I am on the look-out for an article that explains it. There is a basic explanation in Satisfiability, but maybe Structure (mathematical logic) would be the better place for it? Felix QW (talk) 10:03, 17 January 2022 (UTC)[reply]

Assessment[edit]

I have the feeling that this page now includes "most of the material needed for a complete article; all major aspects of the subject are at least mentioned." and so should be on its way to a B rating - what do others think? Felix QW (talk) 10:19, 17 January 2022 (UTC)[reply]

Very nice article[edit]

I hope it's not inapproriate for me to say: I very much appreciate the work done on this article, making the subject accessible to beginners^*!!!

^* Meaning: beginners in model theory, not necessarily beginners in mathematics.