Date: Mar 28, 2013 2:08 PM Author: fom Subject: Re: Using classes instead of sets On 3/28/2013 9:43 AM, pepstein5@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.

>

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,

{x|F(x)}

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

Ax(x=x)

and the logical theory of identity is a presupposition of set

theory. The standard account of that theory can be found at

the link,

http://plato.stanford.edu/entries/identity-relative/#1

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.

I hope this has helped.