On 3/28/2013 9:43 AM, email@example.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. >
In spite of the many opinions out there, it would be correct to say that the question is hard to answer because the history of the subject led to fixes based on formalism and undefined language primitives interpreted according to such wonderful phrases as "definition-in-use".
Having said that, consider the class specification of naive set theory,
A class is the collection of objects satisfying a grammatical form with one free variable.
From my perspective, a non-standard view, you run into the problem right here with the notion of "object" and the nature of identity criteria. Having said that, let me get standard again.
The apparent paradoxes arising from naive class specification led to Zermelo's axiomatization. Zermelo spoke of "definite" uses of logic in his paper which had bee criticized for being unclear. I have not read all of the material that I should have on this, but the result is that "definite" is now established by the compositionality of recursively-defined formulas whose only predicate symbols are 'e' and '='.
So, now one interprets F(x) in the naive class specification according to the language used to express the given set theory.
Cantor and others had been aware that one could not form a set of ordinals containing all ordinals. This did not concern him as it did others since it had not defeated his definition of a set for all infinite sets. But, it did lead him to speak of consistent and inconsistent multiplicities. He became careful to speak of "finished" sets, and, he came to understand the problem in terms of "limitation of size".
The big debates started, of course, with Russell's paradox. As George Greene will be quick to point out, that paradox is not a limitation of size issue. But, it had been addressed within set theory along with certain other paradoxes by this means.
That is why there are no proper classes in Zermelo set theory or its descendants. And, that is why you use the scare quotes when you say that classes are "too big" to be sets.
What then would be a class? It is simply a grammatical form.
In Jech, you will find a quick nod toward "all work with classes can be done by manipulating formulas" before introducing a shorthand for the use of bolded capital letters for discussing classes.
Later developments, about which I know less, introduced a two-sorted logic for classes and introduced the distinction by which sets are a species of class that can be the element of a class.
It is at this point where the question of "object" and identity criteria become more pronounced. Historically, although it is rarely stated axiomatically, the universal class is described by
and the logical theory of identity is a presupposition of set theory. The standard account of that theory can be found at the link,
What admits singular reference to proper classes as a sort is the axiom of extension and its converse,
AxAy(x=y <- Az(zex <-> zey))
AxAy(x=y -> Az(zex <-> zey))
The converse is thought to be obtained through the logicist interpretation of Leibniz' law explained in the link.
What should be observed here is that the axiom of extension does not apply to "objects" in the Fregean definition of object identity. That would be expressed as
AxAy(x=y <-> Az(xez <-> yez))
The general sense here is the subtle distinction between the statements,
"A set is determined by its elements"
"A set is a collection taken as an object"
When singular reference to proper classes is introduced with a two-sorted logic, those statements might be recharacterized according to
"A class is determined by its elements"
"A set is a class taken as an object"
Returning to the restrictions imposed by Zermelo, the notion of classes in terms of formulas gives the axiom schema of separation. Any class specification applied to an existing set forms a new set which is a subset of the existing set. That is axiomatic in the Zermelo system.
Such a set formation strategy is finds support in the two-sorted characterization without further elaboration because of the power set axiom. Class specifications determine subclasses. The subclasses of sets are collected into a set by the axiom. Thus, the subclasses of sets are elements of a set and, therefore, are sets themselves.
When classes are merely grammatical forms, they are realized only with respect to models. Given a model, the class specification has a meaningful interpretation relative to the domain of the model.
One may compare this to the Aristotelian framework. A genus such as "animal" is prior to its species, "aquatic", "winged", and "footed". This is because there would still be "animal" even if there were no fish in the sea. But, Aristotelian "substance" is associated with the individuals to which the logic is applied. The hierarchy is vacuous without the individuals.
So, Russellian type theory is based on building a logical structure through types beginning with individuals. Then there are classes of individuals followed by classes of classes of individuals and so on. In set theory, this becomes better secured with the introduction of the axiom of regularity/foundation. It is this axiom that introduces well-founded sets and provides for the definition of cumulative hierarchies upon which the modern model theory of set theory primarily depends.
Now, category theory is different from set theory. There is some convoluted treatment of how to deal with sets and the size limitations. Category theory does not arise from the topological/arithmetical area of mathematics. It is based on algebraic relations that relate objects of the same type through morphisms.
I am not anywhere near qualified to address the relationships between category theory and its relationship to the limitation of size issues in set theory. Perhaps someone else can fill that void to further answer your questions.