Table of contents |
2 Burali Forti paradox 3 Endnotes 4 References 5 See also 6 External links |
Cantor is quoted as saying:
Cantor's view
Cantor also mentioned the idea in his famous letter to Richard Dedekind 28 July 1899*:
This seems paradoxical, and is closely related Cesar Burali Forti's "paradox" that there can be no greatest ordinal number. There is a quick fix in Zermelo's system by his Axiom of Separation, which stipulates that sets cannot be independently defined by any arbitrary logically definable notion, but must be separated as subsets of sets already "given". This, he says, eliminates contradictory ideas like "the set of all sets" or "the set of all ordinal numbers".
But it is a philosophical problem. It is a problem for the view that a set of individuals must exist, so long as the individuals exist. Moreover, Zermelo's fix commits us to rather mysterious objects called "proper classes". The expression "x is a set" is the name of such a class, what sort of object is it? So is the object named by "x is a thing". Is it a thing or not?
As A.W. Moore notes, there can be no end to the process of set formation, and thus no such thing as the totality of all sets, or the set hierarchy. Any such totality would itself have to be a set, thus lying somewhere within the hierarchy and thus failing to contain every set.
Endnotes
References
See also
External links