Saturday, June 16, 2007

Russell's Paradox

The basic concept in mathematics is the concept of a plurality, a number of things. Prima facie each plurality, being one plurality, is also a unity, one thing. E.g. when we think of the extension of the concept of a plurality, i.e. the totality of all the pluralities, we are clearly regarding each plurality as one thing, one member of that totality. That totality must have, as one of its members, itself (since it is a plurality), but clearly many pluralities are not (in that way) self-membered. So now consider the totality of all the non-self-membered pluralities—is that totality a member of itself? Russell’s paradox is that if it is then (by its definition) it should not be, and if it is not then (similarly) it ought to be.
......By analogy with the Grelling-Nelson paradox, perhaps we should have considered the totality of all the other non-self-membered pluralities—but then what of the plurality whose members are that totality and that totality’s members? This paradox is not so easily resolved. Still, the Grelling-Nelson paradox may nonetheless indicate the best resolution. Think of the ordinary pluralities, of cats and dogs, and tables and chairs—whilst any such plurality will be some definite number of definite things, the totality of all the pluralities would be unlikely to be so well-behaved, because even a merely possible object is a thing of some kind, and would therefore be a member of various pluralities.
......E.g. even though some chairs are definitely red, we cannot consider, not as a definite plurality, the totality of all the definitely red possible chairs, because of the fairly obvious borderline cases, such as we are not thinking of when we think of something that is definitely a red chair. Clearly, in order to justify our initial belief, in the totality of all the pluralities, we would need to rule out even the possibility of continuous transitions between individual things (such as a single white cloud, in an otherwise blue sky) and stuff that is too fuzzy to be one or more things, to rule that out for all possible kinds of thing. So, it may well be that Russell’s paradox just shows that such a belief cannot be justified. (2nd Aug: More of the same posted.)

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