We prove a theorem about models with indiscernibles that are cofinal in a given linear order. We apply this theorem to obtain new independence results for Quine's set theory New Foundations, thus solving two open problems in this field.

Section 1 is devoted to the study of countable recursively saturated models with an automorphism moving every non-algebraic point. We show that every countable theory has such a model and exhibit necessary and sufficient conditions for the existence of automorphisms moving all non-algebraic points. Furthermore we show that there are many complete theories with the property that every countable recursively saturated model has such an automorphism. In Section 2 we apply our main theorem from Section 1 to models of Quine's (...) set theory New Foundations (NF) to answer an old consistency question. If NF is consistent, then it has a model in which the standard natural numbers are a definable subclass N of the model's set of internal natural numbers Nn. In addition, in this model the class of wellfounded sets is exactly $\bigcup_{n\in \mathbb{N}}\mathscr{P}^n(\varnothing)$. (shrink)