Abstract
This paper discusses some of the ways in which Martin-Löf type theory differs from set theory. The discussion concentrates on conceptual, rather than technical, differences. It revolves around four topics: sets versus types; syntax; functions; and identity. The difference between sets and types is spelt out as the difference between unified pluralities and kinds, or sorts. A detailed comparison is then offered of the syntax of the two languages. Emphasis is put on the distinction between proposition and judgement, drawn by type theory, but not by set theory. Unlike set theory, type theory treats the notion of function as primitive. It is shown that certain inconveniences pertaining to function application that afflicts the set-theoretical account of functions are thus avoided. Finally, the distinction, drawn in type theory, between judgemental and propositional identity is discussed. It is argued that the criterion of identity for a domain cannot be formulated in terms of propositional identity. It follows that the axiom of extensionality cannot be taken as a statement of the criterion of identity for sets.
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Notes
- 1.
Cantor (1882, pp. 114–115): “Eine Mannichfaltigkeit (ein Inbegriff, eine Menge) von Elementen, die irgend welcher Begriffssphäre angehören, nenne ich wohldefinirt, wenn auf Grund ihrer Definition und in Folge des logischen Princips vom ausgeschlossenen Dritten es als intern bestimmt angesehen werden muss, sowohl ob irgend ein derselben Begriffssphäre angehöriges Object zu der gedachten Mannichfaltigkeit als Element gehört oder nicht, wie auch ob zwei zur Menge gehörige Objecte, trotz formaler Unterschiede in der Art des Gegebenseins einander gleich sind oder nicht.”
- 2.
Cantor (1883, p. 587): “Unter einer Mannichfaltigkeit oder Menge verstehe ich nämlich allgemein jedes Viele, welches sich als Eines denken lässt…”
- 3.
- 4.
- 5.
- 6.
The idea of a many-sorted language is implicit in Hilbert (1899). An early, perhaps the first, use of the term “many-sorted logic” (or, its German cousin), as well as a definition of the thing itself, can be found in Schmidt (1938). Schmidt (p. 485) emphasizes that he uses “sort” instead of “type” or “kind” so as to avoid the association of levels characteristic of type theory.
- 7.
Cf. Liddell & Scott’s Greek–English Lexicon, s.v. “τúπως”.
- 8.
Cf. Oxford English Dictionary, s.v. “type”.
- 9.
The use of the term “set” for types of individuals, perhaps inspired by Bishop (1967, p. 13), goes back to Martin-Löf (1984). “Domain” was used for the same purpose in Diller and Troelstra (1984). One reason why this, rather natural, terminology was not taken up by Martin-Löf himself seems to have been the use of the term “domain” for a very different notion in Dana Scott’s so-called theory of domains (e.g. Scott 1982).
- 10.
For a general discussion of dependent types, see Aspinall and Hofmann (2005).
- 11.
In the reconstruction of Frege’s type hierarchy given by Dummett (1973, pp. 44–45) there is a separate type of propositions.
- 12.
- 13.
In homotopy type theory there is also a more specific conception of propositions. A so-called mere proposition is a type of individuals that, intuitively, has at most one element; see The Univalent Foundations Program (2013, ch. 3).
- 14.
- 15.
For the distinction between function and operator, see Church (1956, §§ 3, 6).
- 16.
Dependent function types introduce a further variable-binding operator; thus with dependent function types present, there are two variable-binding operators.
- 17.
- 18.
The term “judgement” in this sense was introduced by Martin-Löf (1982). The term had of course been used in logic before, but it had fallen out of fashion during, say, the first decades of the twentieth century; see e.g. Carnap (1934, § 1). For the early history of the use of the term in logic, see Martin-Löf (2011); for aspects of the later history, see Sundholm (2009).
- 19.
- 20.
In the literature, also the term “proof-object” is used, following Diller and Troelstra (1984).
- 21.
This contrasts with so-called justification logic (cf. Artemov and Fitting 2016), where a: A, understood as “a is a justification of A”, is a proposition. Thus, for instance b: (a: A) and (a: A) ⊃A, are well-formed formulae there.
- 22.
See Curry (1963, ch. 2.C) for such an account of the metamathematical “⊢A”.
- 23.
- 24.
- 25.
- 26.
Cantor’s use of the term “mode of givenness” (Art des Gegebenseins) might have inspired Frege’s use of the same term in his elucidation of the notion of sense; see Sundholm (2001, pp. 65–67).
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Acknowledgements
I am grateful to Deborah Kant for inviting me to contribute to this volume. The critical comments of two anonymous readers on an earlier draft helped me in the preparation of the final version of the paper. While writing the paper I have been supported by grant nr. 17-18344Y from the Czech Science Foundation, GAČR.
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Klev, A. (2019). A Comparison of Type Theory with Set Theory. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_12
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