III.57 Models of Set Theory

A model of set theory is, roughly speaking, a structure in which the usual AXIOMS OF SET THEORY [IV.22 §3.1] (that is, the axioms of ZF or ZFC) hold. To explain what this means, let us think first about groups. The axioms of group theory mention certain operations (such as multiplication and inversion), and a model of group theory is a set, equipped with such operations, such that the axioms hold. In other words, a model of group theory is nothing other than a group. So what does a “model of ZF” mean? The axioms of ZF mention one relation, namely “is an element of,” or “∈.” A model of ZF is a set M, on which there is a relation E, such that all the axioms of ZF hold in S if we replace “∈” by “E.”

However, there is one very important difference between these two sorts of model. When one first meets groups, one starts with some very simple examples, such as cyclic groups, or groups of symmetries of regular polygons, and one then builds up to more sophisticated examples such as the SYMMETRIC AND ALTERNATING GROUPS [III.68], and beyond. But this gentle process is not available for models of ZF. Indeed, since all of mathematics can be formulated in the language of ZF, it follows that every model of ZF has to contain a “copy” of the whole world of mathematics. This makes studying models of ZF rather difficult.

One aspect that is often found puzzling is the fact that a model of ZF is a set. This might seem to mean that there is a “universal” set (a set that has every set as a member), but from RUSSELL’S PARADOX [II.7 §2.1] it is easy to see that there can be no such set. The answer to this apparent problem is that the model M is indeed a set in the real mathematical universe, but that inside the model there is no universal set—in other words, there is no element x of M such that yEx for every element y of M. Thus, from the perspective of the model, the statement “there is no universal set” is true.

See MODEL THEORY [Iν.23] for more about models in general, and SET THEORY [IV.22] for more about models of set theory.