This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than 35 years. The first three were "Set-theoretical foundations of category theory" (1969), "Categorical foundations and foundations of category theory" (1977), and much more recently, "Typical ambiguity: Trying to have your cake and eat it too" (2004). The present paper summarizes the results from a long (in two senses) unpublished manuscript,"Some formal systems for the unlimited theory of structures and categories' (1974), referred to below simply as "Unlimited". That MS can be found in full on my home page at http://math.stanford.edu/ feferman/papers/Unlimited.pdf; the lengthy proof of its main consistency result is omitted here but the methods involved are outlined briefly in the Appendix below.