What's a class (as opposed to a set?) Does it get around Russell's 'paradox'?

Question

Please provide an example of a class, if you can.

And I realize that in the past, class and set were used synonymously in various contexts. But no longer.

Answer 1

The technical aspects depend a bit on which set/class theory you are working with, i.e. the particular axioms. For Zormelo-Fraenkl set theory, classes don't technically exist in the theory; they are simply abbreviations for relations among sets. So, A\in {x|p(x)} simply means p(A) is true. Containment between classes is then just an implication between relations, etc. Since classes don't technically exist, you can't actually talk about one class being a member of another; only sets being members of classes.

In Godel-Bernays class theory, the undefined term is 'class'. Sets are defined as those classes that are elements of other classes. In this axiom system, if you have a property p(x), you can define the class {x:p(x)} to be the collection of all *sets* with the property p(x).

Both formulations avoid the Russell paradox since A={x:x\not\in x} is a collection of sets but is a proper class itself (i.e. a class which is not a set). Hence A\not\in A but this does not imply that A\in A since A is not a set.

Answer 2

Mathematician is right. Tom does not know what he is talking about. the problem being self belonging is neither a secret nor it is omitted from set theory books due to laziness or any kind of conspiracy. However his explanation of how it can be overcome is at best scatchy and mathematician's explanation is far superior.

Let me explain why self belonging is not such a perverse thing for a set. After all mathematicians used to talk about the "set of everything" or the "universal set" until they realized that everything involved sets too and that meant the set of everything had to belong to itself too. Then of course the Russell paradox was only one step ahead and we had to come up with an axiomatic set theory which required us to make clear what a set is. Although somewhat unnatural, the solution was not to allow the designation of "set" for arbitrary collection of objects. Sets are the only collections which belong to some other collection and the rest are called classes. This eases the pain but of course there is a price to pay. For example you no longer can measure how big a collection (class) is. Also of course not everything about set theory is settled. Far from it, we know that we can not come up with a finite collection of axioms to describe what it means to be a set. And some possible axioms are just too mind boggling for us. Will we accept Continuum hypothesis or the generalized continuum hypothesis as a property of sets? In terms of intution we have close to none for this question... So don't get the misconception that we invented the classes and now foundations of mathematics is in safe hands.

On an other note, I am really tired of people preaching conspiracy in mathematics. Although mathematicians are quite capable of making mistakes like anyone else, there is no discipline in the world that can compare to mathematics when it comes to the ability of sorting out what is right or wrong in time. People who jump up and down claiming to have found big mistakes in simple things have very very little understanding of mathematics. Please be wary to these people. They will hinder your intellectual development a lot if you are not careful. I think this is also quite clear if you compare mathematician's clear and concise explanation to Tom's slogans calling you to "logical explanations".

Answer 3

doug_donaghue's answer is partly correct.

1. Russell's paradox is not a paradox.
2. The definition of set has always been illogical. It is only partly correct.
3. Russell uses the word manifold to refer to a set - this is incorrect and shows that contrary to popular thought, he was not a very good logician.
4. Yes, a class does not succumb to the problem of sets being "members of themselves".

Unfortunately there are no good mathematicians around today. If there were, they would tell you that to allow a set to be a member of itself is very problematic. Why? Answer is simple: The distinction between set and element in this instance is blurred. For if a set is a member of itself, then it contains itself as an element. Of course this is rubbish, no? The set contains elements which may be other sets. The paradox arises when you have that the empty set contains itself. Nonsense! The empty set contains no elements. Thus if you think of these terms in this logical way, you will not be confused. Again, most of your lecturers are arrogant and conceited fools who will teach you that the paradox is true.

Just read the irrelevant rubbish 'mathematician' has written and you will see a response that is typical to one your lecturer might provide!

Firat_c: I have a BS in mathematics. As for your rambling, it means nothing to me. I challenge every aspiring mathematician to prove everything. This means thinking for themselves. I guess this is something you know very little about given your rants.

Answer 4

There's not really a huge difference. Class usually refers to a structure within a set or a particular type of mapping between sets (which may preserve various structutres), whereas a set usually refers to a collection of objects.

Russels paradox only crops up when you are dealing with self-referential quantities.


Doug

Answer 5

sushi class.