Models of Set Theory with Atoms

Abstract

In this chapter, we shall construct various models of Set Theory in which the Axiom of Choice fails. In particular, we shall construct a model in which C(ℵ₀,2) fails, and another one in which a cardinal $\mathfrak {m}$ exists such that $\mathfrak {m}^{2} < [\mathfrak {m}]^{2}$. These somewhat strange models are constructed in a similar way to models of ZF (see the cumulative hierarchy introduced in Chapter 3). However, instead of starting with the empty set (in order to build the cumulative hierarchy) we start with a set of atoms and define a certain group $G$of permutations of these atoms. Roughly speaking, a set x is in the model if x is “stable” under certain subgroups $H \subseteq G$ (i.e., for all permutations $\pi \in H$, πx=x). In this way we can make sure that some particular sets (e.g., choice functions for a given family in the model) do not belong to the model. Unfortunately, since we have to introduce atoms to construct these models, we do not get models of ZF; however, using the Jech–Sochor Embedding Theorem 17.2, we can embed arbitrarily large fragments of these models into models of ZF, which is sufficient for our purposes.