David Keyt Two 1) The Barber : In an all male army one member of a squad of soldiers is a barber who shaves all and only those members of the squad who do not shave themselves. Who shaves the barber? 2) What It Shows: i) There is a member of the squad who shaves all and only those members of the squad who do not shave themselves. Call him ‘John’. ii) For each member of the squad, x, John shaves x iff it is false that x shaves x. iii) But John is a member of the squad. iv) So John shaves John iff it is false that John shaves John, a contradiction. v) Therefore, (i) is false: there is no such barber. 3) The Barber's Paradox Resolved: a) Let’s make a minimal change in our description of the squad of soldiers: i) There is a someone who shaves all and only those members of the squad who do not shave themselves. Let’s call this person ‘Barber’. ii) So Barber shaves all and only those members of the squad who do not shave themselves. iii) That is to say, for all x, if x is a member of the squad, then Barber shaves x iff x does not shave x. iv) Hence, if Barber is a member of the squad, then Barber shaves Barber iff Barber does not shave Barber. v) But it is false that Barber shaves Barber iff Barber does not shave Barber. vi) Therefore, by modus tollens Barber is not a member of the squad. b) This looks like cheating. Is it? The situation is like this. The lieutenant who commands the squad wants every member of his squad to be clean shaven. His first idea is to appoint a member of the squad to shave everyone in the squad who does not shave himself. When he inadvertently appoints a logician to do this, the logician asks whether he is to shave himself. The lieutenant mumbles something inaudible, dismisses the logician, and hires a woman from the neighboring village as the squad’s barber.

1 4) Sets and properties: a) A set is a collection into a whole of definite, distinct objects (after (1845- 1918)). i) Distinctness: no two elements of the same set are identical; the same element cannot occur more than once. ii) Definiteness: for any given object a and any given set S it must be the case that either a is a member of S or a is not a member of S. Sets are sharply defined. b) Properties are signified by predicates (or sentence-forms or open sentences). Properties are intensional objects; sets are extensional objects. Sets that have difference members are distinct, but properties that have the same instances are not necessarily distinct. i) x is an equiangular plane triangle. ii) x is an equilateral plane triangle. iii) Every equilateral plane triangle is equiangular; and conversely. iv) Thus, the two predicates signify different properties, but define one and the same set. c) The principle of abstraction: Every property determines a set. 5) Russell’s paradox in the philosopher’s own words: “The [universal] class we are considering, which is to embrace everything, must embrace itself as one of its members. In other words, if there is such a thing as “everything,” then “everything” is something, and is a member of the class “everything.” But normally a class is not a member of itself. Mankind, for example, is not a man. Form now the assemblage of all classes which are not members of themselves. This is a class: is it a member of itself or not? If it is, it is one of those classes that are not members of themselves, i.e. it is not a member of itself. If it is not, it is not one of those classes that are not members of themselves, i.e. it is a member of itself. Thus of the two hypotheses--that it is, and that it is not, a member of itself--each implies its contradictory. This is a contradiction.” (, Introduction to Mathematical Philosophy [London, 1919], p. 136)

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6) Russell's paradox: i) Every property determines a set. ii) ‘x ∉ x’ is a predicate and hence signifies a property. iii) So there is a set y such that for all x, x ∈ y iff x ∉ x. iv) Call this set ‘ r’ (Russell’s set). v) Then for all x, x ∈ r iff x ∉ x. vi) Therefore, r ∈ r iff r ∉ r. vii) Consequently, (i) is false: not every property determines a set. 7) Russell’s Paradox Resolved: a) Russell’s paradox has exactly the same form as the barber paradox, and can be resolved in the same way. Mates adopts, without saying so, a version of the solution (or resolution) developed by von Neumann, Bernays, and Gödel (NBG ). According to this resolution there are three kinds of thing: (1) Individuals: such things as pencils and persons, which are not sets though they are members of sets. (2) Ordinary sets: such things as the set of pencils or the set of natural numbers or the set of human beings. Ordinary sets are members of further sets. Thus, the set of human beings is a member of the set of finite sets. (3) Ultimate sets: such things as the universal set, which have members but are not themselves members of further sets. b) For Mates (p. 36) an object is either an ordinary set or an individual. Thus, x is an object = df there is a set of which x is a member. c) Notice that ‘thing’ and ‘object’ are not synonyms. Not everything is an object. d) As we have seen, the unrestricted principle of abstraction is false. So, like the lieutenant of the squad, we settle for what we can get a restricted version of the principle: For every property there is a set whose members are just those objects —not, as in the original principle of abstraction, those things —that have the property.

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e) Paradox avoided: i) For every property there is a set whose members are just those objects that have the property. (Restricted principle of abstraction.) ii) Consequently, there is a set whose members are just those objects that have Russell’s property, that is, are not members of themselves. Let’s call this set ‘ r’ (Russell’s set). iii) Thus, for all x, if x is an object, then x ∈ r iff x ∉ x. iv) Hence, if r is an object, then r ∈ r iff r ∉ r. v) But it is false that r ∈ r iff r ∉ r. vi) Therefore, by modus tollens r is not an object. vii) So r is an ultimate set. 8) Most set theorists do no countenance individuals, restricting their universe(s) to collections. In this sort of set theory collections are called ‘classes’, and the collections we called ‘ordinary sets’ are labeled simply ‘sets’. Every set is a class, but not every class is a set. Proper classes are classes that are not sets; they are what we called ‘ultimate sets’. a) In class/set theory the principle of abstraction becomes: For every property there is a class whose members are just those sets that have the property. b) Russell’s paradox avoided: i) For every property there is a class whose members are just those sets that have the property. ii) Thus, there is a class K such that for every set x x ∈ K iff x ∉ x. iii) Call this class ‘ r’. iv) Hence for every set x x ∈ r iff x ∉ x. v) Suppose that r is a set. vi) Then r ∈ r iff r ∉ r, contradiction. vii) So r is not a set. viii) Thus r is a proper class.

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