I've been reading a little bit into mathematical paradoxes lately, and I'm not sure what to make of Bertrand's paradox (regarding the extraordinary set R). I understand the proof, but does this paradox extend to other areas of thought (on that note, this question might belong in the philosophy section)? One example of an extraordinary set given in the text I was reading was the "idea" of a set that includes all ideas. Is this truly an extraordinary set?

EDIT: I meant to title this "Bertrand's Paradox" but I hit enter instead of the apostrophe. Sorry for the typo, feel free to change it.

Last edited:

HallsofIvy
Homework Helper

It essentially shows that "naive set theory", in which a set exists as long as it is possible to give a rule by which anything can be determined to be in or not in the set, has problems. If you are willing to identify the "idea" of a set with the set itself, then the "set of all ideas" would include itself and so is an "extraordinary set". One attempt to get around that is "class theory" in which one "set" is not allowed to contain another set. Instead you get a "hierarchy" of classes with ordinary sets of non-set objects are at the lowest rung and each level can contain classes at a lower level.

Certainly, any discipline can be expressed in terms of sets and so set, in that sense, form a basis for all disciplines. I think, however, it would be a stretch to assert that Russel's paradox ("Bertrand" is Bertrand Russel's first name and ideas, theorems, paradoxes, etc. are not normally labeled by first names!) plays any important role in most disciplines.

However, the question was Bertrand's Paradox. (Promulgated by Joseph Bertrand.)
For all I know, there is a Joseph's Paradox as well, but that also was not the question.

That was my fault, I meant Russell's Paradox. No idea why I used his first name, I guess I wasn't thinking, but I was definitely talking about the problem involving set theory. Thanks for the answer, HallsofIvy, I haven't read much (anything) about class theory, but it seems interesting, I'll have to read up a little more on it. Seems like it may get me back into programming...

HallsofIvy