Ramble, Part 68: Stumble
Feb. 27th, 2009 12:41 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe year is 1902. The esteemed philosopher/mathematician Gottlob Frege is about to publish the second volume (of a planned three) of his magnum opus, Grundgesetze der Arithmetik ("Basic Laws of Arithmetic"), broadening and deepening the Logicist program. He has just received a letter from a younger colleague, an Englishman by the name of Bertrand Russell.
Cantor's construction of set theory - which Frege followed, albeit with some modifications - was a bit expansive in its definition of "set". A set, by this notion, was simply the collection of all things having such-and-such a property.
Russell's idea was a variant on the classic "Spanish barber" puzzle: the barber in a certain Spanish village makes it a rule to shave all and only the men in the village who do not shave themselves. Question, then: who shaves the barber? If he shaves himself, he breaks his rule; but if he doesn't, he also breaks it. Russell raised this to a higher level. Certain sets - e.g., the set of all integers - do not contain themselves as elements. Others - one might suggest the set of all sets - do. Let R be the set of all sets which do not contain themselves as elements. Question: is R an element of R? The problem is the same as that of the Spanish barber: if R is an element of R, then - by the definition of R - it shouldn't be; but if it isn't, then it should be.
Frege's formulation of the fundamentals of arithmetic allowed this construction, with its paradoxical consequence, to go forward; it was therefore inconsistent. Frege was reduced to adding an appendix to the volume, in which he modified his axioms to prevent this. Unfortunately, under the modified axioms, several of his proofs were no longer valid. (Furthermore, it was pointed out after his death that the modified axioms were still vulnerable to attacks like Russell's.) The third volume was never published.
The freedom with which Cantor's formulation allowed the creation of sets was the culprit. Indeed, the very notion of "the set of all sets" contradicts Cantor's demonstration that there is no largest cardinality, for how could there be a set whose cardinality was greater than that of the set of all sets? (The Italian mathematician Burali-Forti had noticed a similar problem involving the set of all Cantorian ordinals somewhat earlier, but his work did not seem to present as immediate a threat to Logicism as Russell's paradox.) Somehow, this freedom had to be restricted, if a firm and consistent logical foundation for mathematics was to be established. We'll discuss some approaches to that in the next Ramble.
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Cantor's construction of set theory - which Frege followed, albeit with some modifications - was a bit expansive in its definition of "set". A set, by this notion, was simply the collection of all things having such-and-such a property.
More precisely, given a predicate F(x), the collection of all x which made F(x) true was a set. This isn't much of a change, as the range of possible predicates is very wide.Russell's letter to Frege pointed out a serious problem with this.
Russell's idea was a variant on the classic "Spanish barber" puzzle: the barber in a certain Spanish village makes it a rule to shave all and only the men in the village who do not shave themselves. Question, then: who shaves the barber? If he shaves himself, he breaks his rule; but if he doesn't, he also breaks it. Russell raised this to a higher level. Certain sets - e.g., the set of all integers - do not contain themselves as elements. Others - one might suggest the set of all sets - do. Let R be the set of all sets which do not contain themselves as elements. Question: is R an element of R? The problem is the same as that of the Spanish barber: if R is an element of R, then - by the definition of R - it shouldn't be; but if it isn't, then it should be.
Frege's formulation of the fundamentals of arithmetic allowed this construction, with its paradoxical consequence, to go forward; it was therefore inconsistent. Frege was reduced to adding an appendix to the volume, in which he modified his axioms to prevent this. Unfortunately, under the modified axioms, several of his proofs were no longer valid. (Furthermore, it was pointed out after his death that the modified axioms were still vulnerable to attacks like Russell's.) The third volume was never published.
The freedom with which Cantor's formulation allowed the creation of sets was the culprit. Indeed, the very notion of "the set of all sets" contradicts Cantor's demonstration that there is no largest cardinality, for how could there be a set whose cardinality was greater than that of the set of all sets? (The Italian mathematician Burali-Forti had noticed a similar problem involving the set of all Cantorian ordinals somewhat earlier, but his work did not seem to present as immediate a threat to Logicism as Russell's paradox.) Somehow, this freedom had to be restricted, if a firm and consistent logical foundation for mathematics was to be established. We'll discuss some approaches to that in the next Ramble.
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Ramble Contents
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