What is the etymology of model? The answer is of course pre-WWW, but the better part of an hour in the library searching both classic model theory and modal logic textbooks turned up nothing. Every book I touched, without exception, uses the word in the usual way - a structure consistent with some theory - but of course gives no justification for it.

I say 'of course' because, given the word's common meaning of 'a representation of something', phrases like 'model theory', 'canonical model', etc. are quite jarring.

To my great relief, a couple of books commented on this jarring nature (it's not only me!) but I am genuinely curious as to who was the first person to use model to mean 'consistent structure', and even more curious as to why they did so.


Thanks one and all for your comments! I've come to realize that I simply had the 'mathematical model' usage of the word etched into my brain, but the past couple of days spent reading 'model' as '(toy) model' have jolted me into agreeing that perhaps 'model' is a reasonable choice of term after all. I still feel (perhaps wrongly, I'm not an expert in model theory) that a word like cast or casting (as in a die-casting) would be better; it conveys a sense of fitting (satisfying) some mould (theory) while still being short and a little light hearted like 'model'. But that is just me, and thanks to the comments here, I think that 'model' is quite good enough.

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    $\begingroup$ Perhaps the German term 'Modell' came first, maybe deriving from the nineteenth century idea of models of non-Euclidean geometry, such as the Beltrami-Klein model. $\endgroup$ – David Corfield Mar 30 '12 at 11:10
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    $\begingroup$ I'm puzzled, since I've always thought of "model" as a very natural word to use for this, precisely because of the "representation of something" meaning. A first-order theory expresses an abstract idea, and a model is a concrete realization of that idea, so it's not so different from architectural models, mathematical models in the simulation sense, etc. It's definitely stretching the non-mathematical definition a little, but in a way that feels comfortable to me. However, that could just be me, and I don't know who used it first or what exactly they had in mind. $\endgroup$ – Henry Cohn Mar 30 '12 at 11:31
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    $\begingroup$ See also: mathoverflow.net/questions/42298/… $\endgroup$ – Qfwfq Mar 30 '12 at 12:53
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    $\begingroup$ Like Henry, I always thought that "model" was a very appropriate choice of word. I'll point out that it makes particular sense in the phrase "a model of ZFC" because, just like a model airplane, a model of set theory is smaller than the real thing. $\endgroup$ – David E Speyer Mar 30 '12 at 13:06
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    $\begingroup$ While I also think that "model" is a fairly natural term, there is a sense in which the usage in logic is backwards from the way the word "model" is used in applied mathematics. In the physical sciences, one normally thinks of the physical world as being concrete reality, and a "model" is an abstract mathematical axiomatization that captures some (but not all) essential features of reality. $\endgroup$ – Timothy Chow Mar 31 '12 at 23:53

I'm sure the origin of the term is quite complex. Indeed, as Henry and David have pointed out, it is a very natural choice in this context.

I think the first definition of 'model' (not the first use) in the exact sense currently used in model theory is due to Tarski in O pojȩciu wynikania logicznego (On the concept of following logically; English translation MR1951812; German translation from Polish by Tarski himself). This is the paper where the current definition of logical consequence first appears:

We say that the sentence $X$ follows logically from the sentences of the class $\mathfrak{K}$ if and only if every model of the class $\mathfrak{K}$ is at the same time a model of the sentence $X$. [Translation by M. Stroinska and D. Hitchcock.]

I don't know much about Tarski's choice of terms here, but the commentary to the English translation by M. Stroinska and D. Hitchcock could be enlightening.

It is interesting to note that Tarski published his paper in 1936, half a decade after Gödel's Die Vollständigkeit der Axiome des logischen Funktionenkalküls (The completeness of the axioms of the functional calculus of logic; MR1549799). So it appears that these ideas were already known to members of the Vienna Circle and affiliates.

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    $\begingroup$ Concerning the last paragraph, it's interesting to note that in the cited work of Gödel, he used the term "Erfüllungssystem" (approximately "satisfactory system") to refer to what nowadays one would just call precisely "model". $\endgroup$ – godelian Mar 30 '12 at 21:45
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    $\begingroup$ Looking through the papers in From Frege to Goedel it seems that "domain" was used instead of "model" in the late 1920s and early 1930s. $\endgroup$ – Carl Mummert Apr 12 '12 at 12:00
  • $\begingroup$ In a logic book by Swedish author Jesper Carlström that I recently run into, it is suggested that the first usage of the word comes from Felix Klein's discovery of "models" of non euclidean geometry. He does not cite references for this, though. $\endgroup$ – godelian Aug 3 '12 at 11:41

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