In his classic paper “Adjointness in foundations” (1969), F. William Lawvere writes of ‘the familiar Galois connection between sets of axioms and classes of models, for a fixed [signature]’. The idea might be familiar to category theorists, but it isn’t easy to find a clear account of what it involves. So, inspired by a talk by Nathan Bowler last term, I’ve put together a piece on Galois connections to explain. All comments, corrections, suggestions for improvements/additions very welcome. It’s part of a planned longer piece about order and ordinals.[Later] Thanks to Luca Incurvati for catching a daft thinko and a few typos too! [Later again] I’ve replaced the previous version with a notationally slightly prettier version.