Abstract
A theory (ETHOS) is developed in which the concepts of operation, function, and single-valued relation are distinguished. In ETHOS operations (and functions) and sets are treated on a par; the former are not ‘reduced’ to sets of ordered pairs, as in set theory, but neither do sets vanish altogether, as in category theory. The theory, which is formulated constructively and based on intuitionistic logic, provides a natural framework for investigating the Axiom of Choice.
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Notes
- 1.
In the eleventh edition (1913) of the Encyclopedia Britannica we find the mathematical concept of function defined as a variable number, the value of which depends upon the values of one or more other variable numbers. This is essentially Dirichlet’s definition.
- 2.
Nevertheless, it is of interest to observe that the old term fungible, itself derived from “function”, is defined to mean “capable of mutual substitution” or “interchangeable”; that is, “having the same function”. This would seem to indicate that the idea of function involved is extensional in the sense specified above.
- 3.
- 4.
More generally, a function can be conceived of as an operation defined on a set respecting a given equivalence relation on that set. Taking the equivalence relation is to represent the idea of “possessing the same value”, an operation respecting such is then extensional in the sense that its outputs depend only on the “values” of its inputs.
- 5.
Here the term “function symbol” (which, strictly speaking, should be “operation symbol”) is being used with its usual syntactic sense in formal systems.
- 6.
- 7.
Here and in the sequel we employ standard set-theoretical terms such as “equivalence relation” and symbols and terms such as ⊆ for inclusion.
- 8.
Here a set X is said to be nonempty if \(\exists x.\;x \in X\).
- 9.
This is the principle of detachability introduced in (Bell, 2009).
References
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Bell, J.L. 2009. The axiom of choice. London: College Publications.
Feferman, S. 1975. A language and axioms for explicit mathematics. In Algebra and logic, Springer lecture notes in mathematics 450, ed. John Crossley, 87–139. Berlin: Springer.
Feferman, S. 1979. Constructive theories of functions and classes. In Logic colloquium ’78, eds. M. Boffa, D. van Dalen, and K. McAloon, 159–224. Amsterdam: North-Holland.
Martin-Löf, P. 2006. 100 years of Zermelo’s axiom of choice: what was the problem with it? The Computer Journal 49(3): 345–350.
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Bell, J.L. (2012). The Axiom of Choice in an Elementary Theory of Operations and Sets. In: Frappier, M., Brown, D., DiSalle, R. (eds) Analysis and Interpretation in the Exact Sciences. The Western Ontario Series in Philosophy of Science, vol 78. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2582-9_9
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