Axiom of regularity
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In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads:
The axiom of regularity together with the
The axiom is the contribution of
Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the
In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of sets that are elements of themselves.
Elementary implications of regularity
No set is an element of itself
Let A be a set, and apply the axiom of regularity to {A}, which is a set by the axiom of pairing. We see that there must be an element of {A} which is disjoint from {A}. Since the only element of {A} is A, it must be that A is disjoint from {A}. So, since , we cannot have A ∈ A (by the definition of disjoint).
No infinite descending sequence of sets exists
Suppose, to the contrary, that there is a function, f, on the natural numbers with f(n+1) an element of f(n) for each n. Define S = {f(n): n a natural number}, the range of f, which can be seen to be a set from the axiom schema of replacement. Applying the axiom of regularity to S, let B be an element of S which is disjoint from S. By the definition of S, B must be f(k) for some natural number k. However, we are given that f(k) contains f(k+1) which is also an element of S. So f(k+1) is in the intersection of f(k) and S. This contradicts the fact that they are disjoint sets. Since our supposition led to a contradiction, there must not be any such function, f.
The nonexistence of a set containing itself can be seen as a special case where the sequence is infinite and constant.
Notice that this argument only applies to functions f that can be represented as sets as opposed to undefinable classes. The
Simpler set-theoretic definition of the ordered pair
The axiom of regularity enables defining the ordered pair (a,b) as {a,{a,b}}; see
Every set has an ordinal rank
This was actually the original form of the axiom in von Neumann's axiomatization.
Suppose x is any set. Let t be the
For every two sets, only one can be an element of the other
Let X and Y be sets. Then apply the axiom of regularity to the set {X,Y} (which exists by the axiom of pairing). We see there must be an element of {X,Y} which is also disjoint from it. It must be either X or Y. By the definition of disjoint then, we must have either Y is not an element of X or vice versa.
The axiom of dependent choice and no infinite descending sequence of sets implies regularity
Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-empty intersection with S. We define a binary relation R on S by , which is entire by assumption. Thus, by the axiom of dependent choice, there is some sequence (an) in S satisfying anRan+1 for all n in N. As this is an infinite descending chain, we arrive at a contradiction and so, no such S exists.
Regularity and the rest of ZF(C) axioms
Regularity was shown to be relatively consistent with the rest of ZF by Skolem (1923) and von Neumann (1929), meaning that if ZF without regularity is consistent, then ZF (with regularity) is also consistent. For his proof in modern notation see Vaught (2001, §10.1) for instance.
The axiom of regularity was also shown to be independent from the other axioms of ZF(C), assuming they are consistent. The result was announced by Paul Bernays in 1941, although he did not publish a proof until 1954. The proof involves (and led to the study of) Rieger-Bernays permutation models (or method), which were used for other proofs of independence for non-well-founded systems (Rathjen 2004, p. 193 and Forster 2003, pp. 210–212).
Regularity and Russell's paradox
In the presence of the axiom schema of separation, Russell's paradox becomes a proof that there is no set of all sets. The axiom of regularity together with the axiom of pairing also prohibit such a universal set. However, Russell's paradox yields a proof that there is no "set of all sets" using the axiom schema of separation alone, without any additional axioms. In particular, ZF without the axiom of regularity already prohibits such a universal set.
If a theory is extended by adding an axiom or axioms, then any (possibly undesirable) consequences of the original theory remain consequences of the extended theory. In particular, if ZF without regularity is extended by adding regularity to get ZF, then any contradiction (such as Russell's paradox) which followed from the original theory would still follow in the extended theory.
The existence of
Regularity, the cumulative hierarchy, and types
In ZF it can be proven that the class , called the von Neumann universe, is equal to the class of all sets. This statement is even equivalent to the axiom of regularity (if we work in ZF with this axiom omitted). From any model which does not satisfy axiom of regularity, a model which satisfies it can be constructed by taking only sets in .
Herbert Enderton (1977, p. 206) wrote that "The idea of rank is a descendant of Russell's concept of type". Comparing ZF with type theory, Alasdair Urquhart wrote that "Zermelo's system has the notational advantage of not containing any explicitly typed variables, although in fact it can be seen as having an implicit type structure built into it, at least if the axiom of regularity is included. The details of this implicit typing are spelled out in [Zermelo 1930], and again in a well-known article of George Boolos [Boolos 1971]."[2]
Dana Scott (1974) went further and claimed that:
The truth is that there is only one satisfactory way of avoiding the paradoxes: namely, the use of some form of the theory of types. That was at the basis of both Russell's and Zermelo's intuitions. Indeed the best way to regard Zermelo's theory is as a simplification and extension of Russell's. (We mean Russell's simple theory of types, of course.) The simplification was to make the types cumulative. Thus mixing of types is easier and annoying repetitions are avoided. Once the later types are allowed to accumulate the earlier ones, we can then easily imagine extending the types into the transfinite—just how far we want to go must necessarily be left open. Now Russell made his types explicit in his notation and Zermelo left them implicit. [emphasis in original]
In the same paper, Scott shows that an axiomatic system based on the inherent properties of the cumulative hierarchy turns out to be equivalent to ZF, including regularity.[3]
History
The concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimanoff (1917) cf. Lévy (2002, p. 68) and Hallett (1996, §4.4, esp. p. 186, 188). Mirimanoff called a set x "regular" (French: "ordinaire") if every descending chain x ∋ x1 ∋ x2 ∋ ... is finite. Mirimanoff however did not consider his notion of regularity (and well-foundedness) as an axiom to be observed by all sets;[4] in later papers Mirimanoff also explored what are now called non-well-founded sets ("extraordinaire" in Mirimanoff's terminology).[5]
Skolem (1923) and von Neumann (1925) pointed out that non-well-founded sets are superfluous (on p. 404 in van Heijenoort's translation) and in the same publication von Neumann gives an axiom (p. 412 in translation) which excludes some, but not all, non-well-founded sets.[6] In a subsequent publication, von Neumann (1929, p. 231) gave an equivalent but more complex version of the Axiom of Class Foundation, cf. Suppes (1972, p. 53) and Lévy (2002, p. 72):
- .
The contemporary and final form of the axiom is due to Zermelo (1930).
Regularity in the presence of urelements
See also
References
- ^ Rieger 2011, pp. 175, 178.
- ^ Urquhart 2003, p. 305.
- ^ Lévy 2002, p. 73.
- ^ Halbeisen 2012, pp. 62–63.
- ^ Sangiorgi 2011, pp. 17–19, 26.
- ^ Rieger 2011, p. 179.
Sources
- S2CID 250344277
- S2CID 250351655
- JSTOR 2025204reprinted in Boolos, George (1998), Logic, Logic and Logic, Harvard University Press, pp. 13–29
- Enderton, Herbert B. (1977), Elements of Set Theory, Academic Press
- Forster, T. (2003), Logic, induction and sets, Cambridge University Press
- Halbeisen, Lorenz J. (2012), Combinatorial Set Theory: With a Gentle Introduction to Forcing, Springer
- Hallett, Michael (1996) [first published 1984], Cantorian set theory and limitation of size, Oxford University Press, ISBN 978-0-19-853283-5
- ISBN 978-3-540-44085-7
- ISBN 978-0-444-86839-8
- ISBN 978-0-486-42079-0
- Mirimanoff, Dmitry (1917), "Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles", L'Enseignement Mathématique, 19: 37–52
- Rathjen, M. (2004), "Predicativity, Circularity, and Anti-Foundation" (PDF), in Link, Godehard (ed.), One Hundred Years of Russell ́s Paradox: Mathematics, Logic, Philosophy, Walter de Gruyter, ISBN 978-3-11-019968-0, archived(PDF) from the original on 2022-10-09
- Rieger, Adam (2011), "Paradox, ZF, and the Axiom of Foundation" (PDF), in DeVidi, David; Hallett, Michael; Clark, Peter (eds.), Logic, Mathematics, Philosophy, Vintage Enthusiasms. Essays in Honour of John L. Bell., The Western Ontario Series in Philosophy of Science, vol. 75, pp. 171–187, ISBN 978-94-007-0213-4
- Riegger, L. (1957), "A contribution to Gödel's axiomatic set theory" (PDF), Czechoslovak Mathematical Journal, 7 (3): 323–357,
- Sangiorgi, Davide (2011), "Origins of bisimulation and coinduction", in Sangiorgi, Davide; Rutten, Jan (eds.), Advanced Topics in Bisimulation and Coinduction, Cambridge University Press
- Scott, Dana Stewart (1974), "Axiomatizing set theory", Axiomatic set theory. Proceedings of Symposia in Pure Mathematics Volume 13, Part II, pp. 207–214
- Skolem, Thoralf (1923), Axiomatized set theory Reprinted in From Frege to Gödel, van Heijenoort, 1967, in English translation by Stefan Bauer-Mengelberg, pp. 291–301.
- ISBN 978-0-486-61630-8
- Urquhart, Alasdair (2003), "The Theory of Types", in Griffin, Nicholas (ed.), The Cambridge Companion to Bertrand Russell, Cambridge University Press
- Vaught, Robert L. (2001), Set Theory: An Introduction (2nd ed.), Springer, ISBN 978-0-8176-4256-3
- von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240; translation in van Heijenoort, Jean (1967), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, pp. 393–413
- S2CID 120784562
- S2CID 199545822
- (PDF) from the original on 2022-10-09; translation in Ewald, W.B., ed. (1996), From Kant to Hilbert: A Source Book in the Foundations of Mathematics Vol. 2, Clarendon Press, pp. 1219–33
External links
- Axiom of foundation at PlanetMath.
- Inhabited set and the axiom of foundation on nLab