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Topic: Using classes instead of sets
Replies: 26   Last Post: Apr 1, 2013 8:04 PM

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fom

Posts: 1,969
Registered: 12/4/12
Re: Using classes instead of sets
Posted: Mar 28, 2013 2:08 PM
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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.





















































































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