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 The present paper summarizes the results from a long (in two senses) unpublished manuscript,“Some formal systems for the unlimited theory of structures and categories” (1974), referred to below simply as “Unlimited”.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.
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Notes
- 1.
See, for example, my collection of essays, In the Light of Logic (Feferman, 1998).
- 2.
Bénabou (1985) proposes more specific requirements which need to be considered for a full scale foundation of naïve category theory.
- 3.
Though inaccessible cardinals are not met in ordinary mathematical practice, working set-theorists accept their existence without hesitation as constituting a natural extension of the ZFC axioms, and indeed as only the first in a series of progressively stronger extensions. Gödel (1947) was an early proponent of this idea.
- 4.
Just one universe of this kind is assumed in my 1969 paper; that is all one needs for the applications. In the 2004 paper, I assumed a sequence of such universes \(U_n \in U_{n+1}\) for each \(n \in \omega\), in order to relate the idea more directly to Russell’s idea of typical ambiguity.
- 5.
Lower case letters will also be used for classes in some contexts below.
- 6.
Actually, NFU is quite weak, proof-theoretically, compared to PA (Solovay, unpublished). As shown by Enayat (2004), one can obtain an extension of NFU equivalent in strength to PA by adding “every set is finite” and “every Cantorian set is strongly Cantorian” as axioms (cf. the final section below for the notions of Cantorian and strongly Cantorian sets in the framework of NFU).
- 7.
Independently, Holmes (1991) showed that NFUp is interpretable in NFU + Inf, giving a more direct proof of Theorem 1 assuming Jensen’s work.
- 8.
As in Mac Lane (1971) we use lower-case letters \(f, g, h, \ldots\) for morphisms in an abstract category, but this does not signal a new kind of variable in NFUp. Similarly, in the next section, where we use \(a, b, \ldots\) for objects in a category and η for natural transformations.
- 9.
In the syntax of S* lower case letters are now used only in this way.
- 10.
This has been suggested to me by Randall Holmes.
- 11.
An axiom stating that all sets are Cantorian was first studied by Henson (1973). A related “axiom of counting” was introduced by Rosser (1953) in order to develop a smooth theory of finite cardinals in NF. It states that the set of finite cardinals is strongly Cantorian; that set is Cantorian in NF and in NFU + Infinity. (I am indebted to Ali Enayat for this background information.)
References
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Acknowledgments
I wish to thank Ali Enayat, Thomas Forster, Randall Holmes, Robert Solovay and Sergei Tupailo for their helpful comments on a draft of this article. I am especially grateful to Shivaram Lingamneni for his work on preparing a LaTeX version of this paper.
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Appendix
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 NFUp 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 \((x, y) = \{\{x\}, \{x, y\}\} \in c_i\).
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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.
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Feferman, S. (2011). Enriched Stratified Systems for the Foundations of Category Theory. In: Sommaruga, G. (eds) Foundational Theories of Classical and Constructive Mathematics. The Western Ontario Series in Philosophy of Science, vol 76. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0431-2_6
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