Center for Science in Society

Working Group on Information

There is a fair amount of confusion about what counts as a paradox. For some a paradox is just any conclusion that seems absurd and that has an argument to support it. Others talk of a person being in a paradox (better facing a paradox) when from assumptions that they believe, by methods of reasoning they find acceptable, they establish conclusions that they find unacceptable, perhaps because those conclusions are seen as absurd. Note that if you don't accept the assumptions, or the methods of reasoning or the absurdity of the conclusion, then there is no paradox for you to face. Paradox, if you will, is in the eye of the beholder. An antinomy is a paradox that is absurd because the conclusion is a self-contradiction. There are those who talk about Russell's antinomy. In all of this there is a problem about what this paradox/antinomy actually is. Normally, a paradox/antinomy involves some reasoning. For example, Russell's antinomy is the reasoning from the set theoretic principle that every condition determines a set to the contradiction obtained from the consequence of this principle that there is a set of all sets that are not members of themselves. Sadly, Russell's antinomy is often identified not with the reasoning but with the proposition that there is a set of all sets that are not members of themselves. Historically, the point of Russell's antinomy was that the set theoretic proposition is false.( This was more like a disaster than a paradox for Frege who had proposed the proposition in print.) Once you see that the proposition is false(i.e. the point of the reasoning), there is no longer an antinomy; and interestingly, from this perspective, Russell's antinomy was not an antinomy for Russell. It is worth noting that one standard presentation of the reasoning for Russell's antinomy is just indirect reasoning for the proposition that there is no set of all sets that are not members of themselves. Thus, we suppose, for the purposes of argument, that there is such a set, and derive a contradiction. When done in this way, the paradox/antinomy seems to vanish.

Contrary to the "history" one is taught as a graduate student, the reasoning for Russell's antinomy was known at least in Germany for some years before Russell noticed it. The problem was that Frege was intellectually isolated in the German mathematical community and Russell , like most of the English, was innocent of any knowledge of mathematics in the rest of the world. The concern in the rest of the world (among set theorists) with this reasoning was motivated by the recognition that it was very close to the diagonal reasoning(this is not obvious, but can be explained in a few minutes) that had been developed earlier by Cantor and the worry that there might be something wrong with diagonal reasoning.

None of this is, in the current context, very important except that it is worth being careful to distinguish between the history of the subject and our heritage. Our heritage is the story we tell when we want to explain the royal road to our own work.

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