Talk:Model theory
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[edit] comment of Logicnazi
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
[edit] Maximal consistent set
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)
[edit] Category theory
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)
[edit] models of set theories
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)
[edit] Very confused!
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)
[edit] To Do List
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)
[edit] Definition of model theory
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)
- 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)
[edit] Infinitesimals
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)
p.s. ok, found link to hyperreals. MathStatWoman 18:07, 22 January 2006 (UTC)
[edit] Logical Constant
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)
- 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)
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)
- 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)
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)
- 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)
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)
[edit] Article rating
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)
- 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)
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)
