 These two properties (1) and (2) are just the properties of first-order logic that allowed Abraham Robinson to build his nonstandard analysis. The background is that Leibniz, when he invented differential and integral calculus, used infinitesimals, i.e. numbers that are greater than 0 and smaller than all of 1/2, 1/3, 1/4 etc. Unfortunately there are no such real numbers. During the nineteenth century all definitions and proofs in the Leibniz style were rewritten to talk of limits instead of infinitesimals. Now let \(\mathbb{R}\) be the structure consisting of the field of real numbers together with any structural features we care to give names to: certainly plus and times, maybe the ordering, the set of integers, the functions sin and log, etc. Let \(L\) be the first-order language whose signature is that of \(\mathbb{R}\). Because of the expressive strength of \(L\), we can write down any number of theorems of calculus as sentences of \(L\). Because of the expressive weakness of \(L\), there is no way that we can express in \(L\) that \(\mathbb{R}\) has no infinitesimals. In fact Robinson used the compactness theorem to build a structure \(\mathbb{R}'\) that is a model of exactly the same sentences of \(L\) as \(\mathbb{R}\), but which has infinitesimals. As Robinson showed, we can copy Leibniz’s arguments using the infinitesimals in \(\mathbb{R}'\), and so prove that various theorems of calculus are true in \(\mathbb{R}'\). But these theorems are expressible in \(L\), so they must also be true in \(\mathbb{R}\).