This paper argues against the view that the standard explanation of phase transitions in statistical mechanics may be considered a causal explanation, a distortion that can nevertheless successfully represent causal relations.

We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such.