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Tom
Registered Member

Date Joined Jul 2001
Total Posts : 10353

Posted 10/16/2004 1:09 PM (GMT -6)

Hey guys (and thank you anewbeginning for jumping in!)-

I got an interesting note from a mathematician who dismisses Cantor’s theorem altogether. I don’t think he’d mind my quoting him:

“The problem is that mathematicians allow sets to be objects. In fact in most such discussions, sets must be used to create the objects upon which mathematics is based. The trouble is when sets are allowed to group objects and to also be objects; then there can never be a set which contains all objects. Most mathematicians have trouble allowing objects to “exist” first and then to say that the set is the fabricated mechanism to group objects but a set in itself is not an object. That is what I am saying; sets are not objects and they do not “create” objects but rather they group together objects. So sets do not group sets and do not contain sets as members. Thus the very definition of a power set is faulty because it is a set of sets and not a set of objects.” David Lowe

Makes sense.

Tom

Alan Rhoda
Registered Member

Date Joined May 2002
Total Posts : 680

Posted 10/18/2004 2:35 PM (GMT -6)

quote:
Originally posted by TGB

I got an interesting note from a mathematician who dismisses Cantor’s theorem altogether. I don’t think he’d mind my quoting him:

“The problem is that mathematicians allow sets to be objects. In fact in most such discussions, sets must be used to create the objects upon which mathematics is based. The trouble is when sets are allowed to group objects and to also be objects; then there can never be a set which contains all objects. Most mathematicians have trouble allowing objects to “exist” first and then to say that the set is the fabricated mechanism to group objects but a set in itself is not an object. That is what I am saying; sets are not objects and they do not “create” objects but rather they group together objects. So sets do not group sets and do not contain sets as members. Thus the very definition of a power set is faulty because it is a set of sets and not a set of objects.” David Lowe

I don't think this works, but it points in the direction of a solution that may work. First, it seems to me that sets are objects of a certain sort, namely, abstract objects.

Second, a power set is a coherent notion--it is simply a set containing all the subsets of another set.

Now, what Lowe points to when he says "The trouble is when sets are allowed to group objects and to also be objects; then there can never be a set which contains all objects" is that we need what Russell called a theory of types, or better, we need to distinguish between different levels of abstraction. Simply put, a power "set" is not a set in the same sense as the set is it the power set of. Let's say our first set S is a set of concrete objects, say, oranges. Let's say that this is a level-0 set. The power set of S, P(S), is a level-1 set of level-0 sets. And the power set of that P(P(S)) is a level-2 set of level-1 sets. As long as we rigorously bear in mind the level of abstraction and don't confuse, say, level-1 sets with level-2 sets, we avoid the problem that Lowe identifies.

So sets are objects (abstract ones) and they do create objects (objects at a higher level of abstraction). Paradoxes result only when one takes the term "set" to be univocal when applied both to a set S and its power set P(S).

So I don't think Lowe's point solves Grim's problem. To solve it I think one has to take one of two approaches: (1) take the tack suggested (but not developed) by Plantinga and argue that propositions (and therefore truths) are not the right kinds of things to qualify for set membership, or (2) argue that sets, qua abstract, contain no new information, and so propositions about sets of sets of sets of sets ... of oranges gives us no more information than do propositions about oranges.

Alan

Lee Merrill
booted

Date Joined Jun 2001
Total Posts : 723

Posted 10/23/2004 5:04 PM (GMT -6)

This seems to me to be a contradiction in the premises, for this to work, you have hold there are true statements we can think of, that are not in the set of all truths. And if you start with a contradiction, you can wind up anywhere...

Blessings,
Lee

==================================================
"Even now we seem to have dim glimpses into regions
from which we receive no word to bring away" (George MacDonald)

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