Paradox

RUSSELL’S SET 327Russell realized that the liar paradox bears a more damningresemblance to a slight variant of this universal set. A set thatcontains everything must contain itself. Now consider a setthat includes all and only those sets that do not includethemselves as members. If this set contains itself as a member,then it does not contain itself as a member. But if it does notcontain itself as a member, then it does include itself as amember.Russell at first thought this derivation of a contradictionwas sophistical. He was not the first to have run across this typeof argument. In 1889 an assistant to Peano, Cesare Burali-Forti,was working on the theory of ordinal numbers. Ordinalsmeasure size like the ticket counters at butcher shops measurethe length of the queue. Ordinals do not convey informationabout the distance that lies between the members of thesequence. When three customers are ranked by first, second,third, the sequence is well ordered because there is a firstmember and a unique next position in line. To extend theconcept of well-ordering to infinite counters, we are careful notto require that there be a last member of the sequence. Forinstance, is well ordered. But there must be afirst member: < . . . , -2, -1, 0, 1, 2, . . . > is not well ordered. Thesequence of non-negative rational numbers singles out a first number but fails to singleout the second. The nonstandard ordering of the integers is not well ordered because it does notsingle out which whole succeeds all the natural numbers.However, a sequence that stacks all the evens before the odds,, is well ordered. The acceleratedsequence so popular in Zeno’s paradoxes iswell ordered. Burali-Forti notes that the sequences constitutingordinals can be ranked in size. First comes , second comes