Abstract
The view of set theory as a foundation for mathematics emerged early in the thinking of the originators of the theory and is now a pillar of contemporary orthodoxy. As such, it is enshrined in the opening pages of most recent textbooks; to take a few illustrative examples: All branches of mathematics are developed, consciously or unconsciously, in set theory. (Levy, 1979, p. 3) Set theory is the foundation of mathematics. All mathematical concepts are defined in terms of the primitive notions of set and membership … From [the] axioms, all known mathematics may be derived. (Kunen, 1980, p. xi).
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Notes
- 1.
There are those who propose category theory as an alternative to set theoretic foundations, but at least for now, this has not changed the fact that set theory is so viewed.
- 2.
That is, \(n + 0 = n\) and \(n + Sm = S(n + m)\) for addition, \(n \times 0 = 0\) and \(n \times Sm = (n \times m) + n\) for multiplication.
- 3.
For details, see Enderton (1977, chapter 4). This formulation is preferable to Frege’s because the system in which it is framed, ZFC, is not prone to Russell’s (or any other known) paradox. In the cumulative hierarchy corresponding to ZFC, Frege’s candidates for the numbers do not exist. This is because, for example, there are new three-element sets formed at every stage of the hierarchy, so there is no stage of the hierarchy at which the set of all three-element sets could be formed.
- 4.
The integers are the positive and negative whole numbers. The underlying idea is to let \(< n, m>\), where n and m are natural numbers, represent the integer \(n - m\). So e.g. the integer \(< n, >\) is less than the integer <n’,m’> iff n + m’ is less than n’ + m as natural numbers. See Enderton (1977, p. 90–101), for the niceties.
- 5.
Here the idea is to let \(< a, b>\), where a and b are integers, represent the fraction a/b. For details, see Enderton (1977, p. 101–11).
- 6.
See Dedekind (1872). A Dedekind cut is a pair (A,B) of sets of rationals such that A and B are non-empty and disjoint, every element of A is less than every element of B, and every rational is in either A or B. So, for example, if A is the set of all negative rationals and all positive rationals whose square is less than 2, and B is the rest of the rationals, then (A,B) is the cut corresponding to \(\surd 2\). For more, see Enderton (1977, p. 111–120).
- 7.
This would be a metaphysical reading, as it purports to reveal the true nature of numbers, etc.
- 8.
See Zermelo (1908). On this account, other adjustments must also be made; e.g. the successor of n is {n}, not \(n \cup \{n\}\).
- 9.
Von Neumann’s version is in fact preferred because it carries over directly to the transfinite ordinals.
- 10.
We noted earlier that Cantor and Dedekind gave distinct but equally workable accounts of the real numbers.
- 11.
- 12.
- 13.
I come back to this issue in part II of my book Maddy (1997).
- 14.
- 15.
Here a point on the line corresponds to the real number that measures the distance to that point from an arbitrarily chosen origin using an arbitrarily chosen unit of length. For plane geometry, a point corresponds to an ordered pair of numbers determined in the familiar style of Cartesian coordinates.
- 16.
e.g. by erecting an isosceles right triangle on the unit length of a line, then using a compass to lay out the length of the hypotenuse onto the line, we can generate a point that corresponds to no rational number. (If it did, i.e. if a/b is in lowest terms and \((a/b)^{2} = 2\), then \(a^{2} = 2b^{2}\), so a is even. But then \(a = 2c\), for some c, so \(b^{2} = 2c^{2}\), and b is also even. This contradicts the assumption that a/b is in lowest terms.)
- 17.
See Boyer (1949) for more.
- 18.
Gödel’s first incompleteness theorem says that for any sufficiently strong theory, there is a sentence it cannot prove or disprove. The second incompleteness theorem, cited here, follows from the first. See Enderton (1972) for a textbook treatment.
- 19.
Detlefsen (in his (1986)) defends Hilbert’s programme for establishing the consistency of classical mathematics by arguing that a criticism based on Gödel’s second theorem need not be viewed as conclusive. So far as I know, no one has succeeded in exploiting the loopholes Detlefsen identifies.
- 20.
- 21.
- 22.
Even MacLane allows this much: ‘The rich multiplicity of mathematical objects and the proofs of theorems about them can be set out formally with absolute precision on a remarkably parsimonious base’ (1986, p. 358). He is referring to ZFC.
- 23.
It needn’t even include the claim that set theory is the only theory that could serve as this sort of foundation.
- 24.
A striking example: forms of the Axiom of Choice turn up in the fundamental assumptions of algebra, topology, analysis, and logic, as well as set theory. See Moore (1982) for a description of how these interconnections were discovered, and their impact on the various fields.
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Maddy, P. (2011). Set Theory as a Foundation. 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_3
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