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Why are both the terms 'structure' and 'model' used in mathematical logic / model theory? Are they just holdovers from different subjects or is there a principled reason for having both?

For clarification, I'm not confused about any actual definitions or usages, just why both terms came to be used; I could, after all, survive perfectly well using exclusively one or the other with little chance of confusion.

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    $\begingroup$ See also math.stackexchange.com/q/371526/462 and math.stackexchange.com/q/406537/462 $\endgroup$ – Andrés E. Caicedo Jun 5 '13 at 23:08
  • $\begingroup$ For clarification, I've already seen all of the responses given thus far (as well as those linked above) and found them rather lacking as a reason for introducing two distinct terms. I'm not confused about any actual definitions or usages, just why both terms came to be used (I could, after all, survive perfectly well using only one or the other with little chance of confusion). $\endgroup$ – 01001101.turing Jun 6 '13 at 1:14
  • $\begingroup$ You should probably edit that comment into your question, or at least comment on the given answers. It's only by mistake that I stumbled onto your comment here. $\endgroup$ – Asaf Karagila Jun 6 '13 at 6:18
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Models are structures, and structures are models. But when we say "model" we mean that there is a particular theory which holds in the structure, and when we say "structure" we are mainly interested in an arbitrary interpretation of the language.

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A structure is a set with some interpretable symbols(constants, relations and functions) within a fixed language. You do not ask for more from a structure.

However...

A model (of a theory) is a structure which satisfies the axioms of the theory. It makes more "structural sense"...

Maybe an example brings more clarification: Consider the theory of groups. $\mathbb Z$ is a structure in $\mathcal{L}=\{e, \cdot, ^{-1}\}$ but not a model since it is not a group. On the other hand, $\mathbb R- \{0\}$ is an $\mathcal{L}$-structure and further a model as it is indeed a group.

This is what I more or less know within a model-theoretic view. Someone else may give an answer also considering a perspective of universal algebra.

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The term 'structure' is a replacement for the earlier term 'system' which was used by various authors (Weber, Hilbert, Dedekind) and meant something along the lines of "a set with added features". The change from 'system' to 'structure' occurred in the 1950's and seems to be owed to Abraham Robinson and Bourbaki. 'Model', on the other hand, appears in Tarski's early works (mid 1930's), and seems to have arisen entirely separately from 'system'. The use of both in modern model theory is, to the best of my knowledge, accidental with only minor intensional differences (as elaborated on in other answers) distinguishing them. There is no principled reason for having both.

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There is no reason for introducing two different terms. Apparently, somebody introduced one term, and somebody else introduced a different term, either because he didn't like the first guy's term, or he hadn't heard of it. Or maybe it was the same guy and he changed his mind, or forgot what he called it before. How would I know, I'm not a historian (nor a mathematician).

The point is, in mathematics there is no official body with the power to decide what the terminology should be. This is different from other sciences, such as astronomy, where some organization claims the power to decide what's a planet. In mathematics, each writer goes his own way, and anarchy prevails. (If you think "model" vs. "structure" is bad, look at the terminology of graph theory.) Eventually, after a few centuries, a consensus is reached. Obviously, model theory (and graph theory) are too young to have reached that point.

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    $\begingroup$ Really? There's no reason? I think that saying "model of the language" or "structure for the theory" are both strange. When I talk to someone else and I said $\cal L$-structure, they immediately know that it doesn't matter what sort of theory I am assuming on the structure. On the other hand, when I say "a model of $\sf ZFC$" they also know, immediately, that I am talking about a model which satisfies all the axioms of $\sf ZFC$. I don't know about you, and how many people you get to talk about these things, but I find these two terms to be quite useful. $\endgroup$ – Asaf Karagila Jun 6 '13 at 6:20
  • $\begingroup$ Certainly, I misuse the language a lot myself. It's easy to do that. Sometimes it's even convenient. However that doesn't mean that the two are really synonyms. In a meta-mathematical level semantic level there is a distinction. Much like there is a distinction between "fornication" and "sexual intercourse" and "coitus" and "sex". See how they all mean the same thing? Wunderbar. But why is it when I say "me and that girl are fornicating", whoever hears me thinks that the said girl is cheating, but when I say "me and that girl are having sex" they are less likely to think that? Magic, I say $\endgroup$ – Asaf Karagila Jun 9 '13 at 3:07
  • $\begingroup$ I recall reading somewhere that Cohen invented forcing. But we don't really use his notation, his assumptions, or his actual methods. We use the notation and methods of Shoenfield, Scott, Solovay and Vopenka. Oh, yes. Cohen wrote a small book about forcing too. Your argument is nothing but an appeal to authority. $\endgroup$ – Asaf Karagila Jun 9 '13 at 21:53

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