Enriched Stratified Systems for the Foundations of Category Theory

Abstract

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 The present paper summarizes the results from a long (in two senses) unpublished manuscript,“Some formal systems for the unlimited theory of structures and categories” (1974), referred to below simply as “Unlimited”.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.

Appendix

The methods used to prove Theorem 2(i), the consistency of S^*, in “Unlimited”, are by an extension of those applied by Jensen (1969). They consist of three parts:

• 1. Specker (1962) reduced the consistency of NF to the existence of models $M_T = (\langle U_i \rangle, \langle \in_i \rangle)_{i \in Z}$ of type theory with types i ranging over the set of all integers, $Z = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$, where $\in_i \subseteq U_i \times U_{i+1}$, for which M _T satisfies the axioms of typed comprehension and extensionality, and in addition has a type-shifting automorphism $\sigma: U_i \to U_{i+1}$ for all $i \in Z$. The model of NF constructed from M _T is defined to be $M^* = (U_0, \in^*)$ where for $a, b \in U_0$, $a \in^* b \leftrightarrow a \in_0 \sigma(b)$. Jensen observed that if M _T satisfies extensionality only for non-empty classes, then M ^* is a model of NFU.

• 2. Ehrenfeucht and Mostowski (1956) applied the infinite Ramsey theorem to obtain models of first-order theories with indiscernibles $\{c_i\}_{i \in I}$ in given orderings $(I, <)$. When these models are generated by Skolem functions from the indiscernibles we get elementary substructures having automorphisms induced by those of $(I, <)$. Jensen applied the Ehrenfeucht-Mostowski theorem to obtain models M of Zermelo set theory plus the Skolem function axioms having indiscernibles c _i in order type $(Z, <)$ and shifting automorphism induced by $\sigma(c_i) = c_{i+1}$. A Z-typed model as required for the Specker construction of M ^* is formed by taking $U_i = \{x \mid x \in_M c_i\}$. Jensen showed that one can also arrange to have M a model of the axioms of Infinity and Choice, which leads to M ^* having the same properties. Thus NFU is consistent with Infinity and Choice. In order to satisfy NFU_p it is only necessary to ensure of the model M that if $x, y \in c_i$ then $\{x\}$ and $\{x, y\} \in c_i$, hence $(x, y) = \{\{x\}, \{x, y\}\} \in c_i$.

• 3. In part II of his paper, Jensen showed how, given any ordinal α, one can construct M ^* satisfying these conditions which is an end-extension of α; this uses the Erdös-Rado (1956) generalization of the Ramsey theorem to certain infinite partitions. These methods were extended in “Unlimited” to construct M ^* which are end-extensions of any given transitive set A. The main theorem needed for this and proved in the Appendix of “Unlimited” is in terms of models of $L_{\infty, \omega}$ with indiscernibles satisfying certain prescribed properties. The formulation of that theorem is too technical to present here. The particular transitive set used in the application to Theorem 2(i) above is the cumulative hierarchy up to a strongly inaccessible cardinal κ. The proof also assumes the existence of a strongly inaccessible cardinal δ greater than κ.