Ramble, Part 69: Escape?
Apr. 5th, 2009 05:32 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThere are two sardonic answers to the puzzle of the Spanish Barber. One is to posit that the barber does not live in the village; one pictures a chain of villages, each drawing its barber from the next one up the hill. The other is that the barber is a woman, and hence does not require shaving. The two principal responses to Russell's Paradox mirror these, after a fashion.
The first method is that of Bertrand Russell himself. A set S is defined by a predicate function F(x); it is the set of all x which make F(x) true. Every such predicate has a domain - the collection of all things for which it is sensible to ask whether F(x) is true. (One doesn't seriously ask whether, say, the square root of two is purple.) The theory of types supposes that there is a hierarchy - there are sets, and predicate functions, of type 0, type 1, and so on - and presumes that, if the domain of a predicate function includes sets of type n, the function (and the set it defines) must be at least of type n+1. Russell's Paradox dissolves; you cannot speak of the set {S: S is not an element of S}, but only of the set R = {S: S is not an element of S and S is of type n}. Is R an element of R? No, because R is of type n+1. It only satisfies half of the criterion for membership in R, and there is no paradox. We are then faced with the problem of determining the type of any given set or function, but this is surely less of a hurdle than the paradox was.
The second method is to allow that collections ("classes") can be defined using any arbitrary predicate (or some very broad class of predicates), but to assert that only certain classes - "sets" - can be elements of classes. All others are "proper classes". This does away with the paradox by allowing the possibility - indeed, the necessity - of declaring that R is a proper class. In essence, a proper class is one that is "too big" to be a set. There is no set of all sets; there is no class of all classes; but there is a (proper) class of all sets. Again, there is a bit of a problem associated with determining which classes are sets and which are proper classes. (There is even the possibility of disagreement over this issue; some classes are necessarily proper classes, some must be sets if we are to do anything substantive, but there may be classes whose status is a matter of taste.)
Both of these approaches are useable; both have a slightly kludgy air about them. Unfortunately, both fail at the original task of saving Logicism. We'll return to this later, but we should turn now to the other main schools of thought.
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The first method is that of Bertrand Russell himself. A set S is defined by a predicate function F(x); it is the set of all x which make F(x) true. Every such predicate has a domain - the collection of all things for which it is sensible to ask whether F(x) is true. (One doesn't seriously ask whether, say, the square root of two is purple.) The theory of types supposes that there is a hierarchy - there are sets, and predicate functions, of type 0, type 1, and so on - and presumes that, if the domain of a predicate function includes sets of type n, the function (and the set it defines) must be at least of type n+1. Russell's Paradox dissolves; you cannot speak of the set {S: S is not an element of S}, but only of the set R = {S: S is not an element of S and S is of type n}. Is R an element of R? No, because R is of type n+1. It only satisfies half of the criterion for membership in R, and there is no paradox. We are then faced with the problem of determining the type of any given set or function, but this is surely less of a hurdle than the paradox was.
The second method is to allow that collections ("classes") can be defined using any arbitrary predicate (or some very broad class of predicates), but to assert that only certain classes - "sets" - can be elements of classes. All others are "proper classes". This does away with the paradox by allowing the possibility - indeed, the necessity - of declaring that R is a proper class. In essence, a proper class is one that is "too big" to be a set. There is no set of all sets; there is no class of all classes; but there is a (proper) class of all sets. Again, there is a bit of a problem associated with determining which classes are sets and which are proper classes. (There is even the possibility of disagreement over this issue; some classes are necessarily proper classes, some must be sets if we are to do anything substantive, but there may be classes whose status is a matter of taste.)
Both of these approaches are useable; both have a slightly kludgy air about them. Unfortunately, both fail at the original task of saving Logicism. We'll return to this later, but we should turn now to the other main schools of thought.
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Date: 2009-04-06 02:54 am (UTC)