Alan Dow

One of the many gifts from Alan to Set-theoretic Topology is the use of elementarity. For a while this was even known as “Dow’s method of elementary submodels”. But Alan would be, was, and still is the first to protest that the Löwenheim-Skolem theorem predates him by a few decades. We have for the longest time been familiar with recursive constructions where often beforehand a sequence of situations/sets is set up and during the construction witnesses to bad things will be eliminated or witnesses to good things will be embraced. In the end we consider such a situation and realize that it was basically dealt with during the construction. A very good example is the Pol-Shapirovskĭı proof of Arhangel′skĭı’s theorem on the cardinality of compact first-countable spaces. What set theorists realised was that one can reduce the length of such proofs considerably by an application of the Löwenheim-Skolem theorem to a model of ‘enough set theory’: its proof is the ultimate closing-off argument where one deals with all possible situations in one go (even ones that will never occur in your problem at hand). But, and this is where this method gets its power, you will certainly have dealt with every eventuality related to your problem. Basically what is left is to perform what would have been the final step of your old recursive argument. This requires some familiarity with first-order logic and model theory, so that you know how far you can go with your arguments. But the time spent learning that will pay itself back handsomely in time saved later.