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I was just reading P.T. Johnstone's introduction to his book "Topos Theory", where he uses the term "elementary" many times to classify the nature of theorems and definitions, examples below. I certainly do not think it is used purely in the common sense of "simple / straightforward" but with a certain technical connotation in mind. I would suppose that if that is so, then it is one also intended by Lawvere in coining the terms "Elementary Theory of the Category of Sets", and "Elementary Topos" with Tierney. I can only discern a fairly woolly sense of a theorem or definition being first-order in nature from the examples I have seen so far.

Here are some examples of the usage of the term by P.T. Johnstone:

  1. " [Mitchells' Embedding Theorem] was soon followed by Lawvere's paper setting out a list of elementary axioms which, with the addition of the non-elementary axioms of completeness and local smallness, are sufficient to characterize the category of sets "

  2. " [Lawvere] observes that the presence of [a subobject classifier] in an arbitrary category enables us to reduce the Comprehension Axiom to an elementray statement about adjoint functors "

  3. " Giraud himself has proved a relative version of his theorem (by non-elementary means) for Grothendieck toposes, and W.Mitchell had formulated the correct elementary form "

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    $\begingroup$ Elementary here is meant in the sense of logic, i.e. something related finitary first order logic (as opposed to higher order logic and infinitary logic). $\endgroup$
    – Zhen Lin
    Jul 31, 2014 at 18:07

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I believe that all these usages of "elementary" refer back to the notion of an elementary class of structures. Let L be a language, and let $X$ be the class of L-structures. A subclass $\Sigma$ of $X$ is called elementary just in case there is a set $T$ of first-order axioms (in L) such that $\Sigma = \mathrm{Mod}(T)$.

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