A note on Kunen
Both of Kunen's textbooks use nonstandard notation. The most important deviations are:
- R(α) denotes the levels of the cumulative hierarchy of sets, normally denoted by Vα
- ON denotes the class of Ordinals
- HF denotes the set of "hereditarily finite sets", usually denoted by Vω
- ZC denotes ZFC − Replacement. Z denotes ZF − Replacement. ZFC- is ZFC − Foundation. ZF- is ZF − Foundation.
Moreover, Kunen sometimes puts emphasis on first ZFC without Foundation, and then showing that the cumulative hierarchy of sets V satisfies ZFC with Foundation. We will not be concerned with this and you can always assume that Foundation is a basic ZFC axiom and skip any discussion of Kunen that relates to Foundation. Whenever Kunen assumes ZFC- or ZF-, you can exchange it with ZFC or ZF, respectively.