
Re: Using classes instead of sets
Posted:
Mar 28, 2013 11:48 PM


On Mar 28, 8:11 am, Frederick Williams <freddywilli...@btinternet.com> wrote: > pepste...@gmail.com wrote: > > > On Thursday, March 28, 2013 1:45:34 PM UTC, Frederick Williams wrote: > > ... > > > > Often one studies all groups, or all groups of a certain kind. Are > > > > those collections classes? > > ... > > > They are always classes and sometimes sets. A class is more general than a set so any collection which is a set is also a class. If we define isomorphic groups as being equal (as everyone does), then the collection of finite groups is a countably infinite set and we can talk about "the set of finite groups". > > > However, the collection of groups is "too big" to be a set. Hence that collection is a class which is not a set. "Too big" because it contains a subcollection which corresponds to the class of all ordinals. > > If groups could have classes for the collection of their elements, and > if we call such groups "Groups", then we couldn't call the collection of > Groups a set or a class, could we? I do not know if set theorists study > (what I shall call) superclasses, supersuperclasses, and so on; where a > superclass is a collection of classes in some theory, and a > supersuperclass is a collection of superclasses in that theory or some > other. > >  > When a true genius appears in the world, you may know him by > this sign, that the dunces are all in confederacy against him. > Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
http://groups.google.com/group/sci.math/msg/8c71a6292f793cb7?hl=en http://groups.google.com/group/sci.math/msg/a88f13c773dd1610?hl=en
Hmm, then what if the operation on groups is being composition, with the g * g = {} the identity, each group is its own inverse?
This is similar to the notion of the direct sum, where the direct sum of infinitely many copies of the natural integers is the empty set.
Grp, collection of all groups: naturally a group under composition with autoannihilation.
So, what's the natural operation on mathematical objects? Generally it is composition, as defined by their structure. Then, many algebraic structures are defined by closure of the operation as that a, b e A, a * b = A, then something like the natural integers and definition of successor seems as simple as that the operation is open or anticlosed: a + 1 not e a.
So, does the collection of groups Grp: _want_ to be a group?
Is addition simply open, in the finite, to be closed?
There's no winning strategy to the "group noun game": except don't play it.
Thank you: quite simple.
Regards,
Ross Finlayson

