On 2013-03-28, email@example.com <firstname.lastname@example.org> wrote: > 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?
Having particular classes does not necessarily lead to contradictions, but making it too easy to form classes can. Since we know that any sufficiently general model, and having the integers makes it general, cannot show its own consistency, the Morse-Kelley axioms which allow more formation of proper classes than does NBG, and in which NBG can be proved, clearly must have these problems. Even the weaker version, only allowing sets, but allowing large cardinals, has the same problems.
> paul Epstein
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