On Thursday, March 28, 2013 12:23:47 AM UTC, smn wrote: > On Wednesday, March 27, 2013 3:29:15 PM UTC-7, Paul wrote: > > > Most of the basic mathematical structures, for example topological spaces, fields, rings etc. assume an underlying set in their definitions. > > > > > > However, the surreal numbers don't form a set since they contain a copy of the ordinals. They form a class. Since I can't see a problem with the non-setness of surreal numbers, I wonder why definitions of other mathematical structures aren't more general and why the above categories are defined on sets rather than classes. > > > > > > Class -set theory is better then pure set theory for the general from for mathematical theories , its objects are classes ,say x,y,z etc on which there is a non-logical 2 place predicate "e" read as -is an element of for instance -xey :x is an element of y . "=" is a logical predicate ,x=y is interpreted as "x" and "y" denote the same class . x is a set means that for some y ,xey .There are many classes that are not sets, for example the class of all vector spaces over the Set of real numbers. However each vector space must be a set since it is an element of the class of all vector spaces . When one speaks of the class,say A of all all objects satisfying some condition it is only sets can be allowed in the class , since any such object ,say x which satisfies the given condition must be an element of A and thus must be a set. > > If you tried to form the class of all classes B satisfying say : x is not and element of x (Russell's example) then ,B can not be in B (if it were ,substituting B for x would give a contradiction. But then ,since B is not in B . it is one of the classes satisfying the condition so B is an element of B ; This is a contradiction to the system ; Hope this helps .See Wikipedia for references.Hope that helps.smn
Although this reply contains much interesting info, it doesn't answer my main concern. [Totally my fault for not expressing it directly enough, in the first place.] My point was meant to be that definitions of groups, rings, fields, topological spaces etc. generally begin with something like "Let X be a set.." Why? Why not say "Let X be a class..." It would be more general. Perhaps it might lead to Russell-style paradoxes. But, if that is the risk, why is it ok for surreal numbers to form a proper class which isn't a set?