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Tuesday, December 30, 2008

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Rather than ask what has to be added to a mere multiplicity to arrive at a set---an approach to the idea of a set from below, as it were---we could make an approach from above. Presumably we have no difficulty thinking of a house as a unity over many bricks (plus other stuff which we can ignore for the sake of an example). The bricks have complex relations between them---one is above another or abuts another, and so on, but let's abstract these details away. What we are left with is a house, a unity, thought of as a set of bricks, also therefore a unity. A set is the abstraction arrived at when the relations between the parts of a one-over-many are forgotten, and set theory is an account of how such abstractions behave. (I'm using 'part' here in the engineering sense of 'component' rather than its mereological sense)

Bill,

You wrote:

"... This power set can be displayed as follows:

{{ }, {a, b}, {a}, {b}}.

... although {a} and {b} are members of the power set, a and b are not."

Let’s have the set S = {a}.
The power set *S (of the set S) = {{ }, {a}}.
And the power set **S (of the power set *S) = {{ }, {{ }, {a}}, {{ }}, {{a}}}.
Am I right?

Now compare your claim I cited with G. Oppy (books.google.com/books?id=FPU9tzW-2HAC, pp. 23-24):
"A set x is transitive if every member of x is a subset of x, that is, every member of a member of x is a member of x." (Oppy, probably following Drake's 1974 Set Theory, employs this definition in a recursive definition of ordinal sets.) But you, Bill, seem to assume, contrary to Oppy, that for every set x, every member --which is not empty set -- of x is NOT a subset of x, and that every member -- which is not empty set -- of a member of x is NOT a member of x. For instance, empty set is both a member of and a subset of *S, but {a} is not among its subsets , though {a} is its member. *S is a member of **S, empty set is a member of both, but {a} is a member of *S and not of **S. Am I right? Is Oppy wrong? Is the way he proceeds standard?

Vlastimil,

Yes, you are right about the power set **S. But then I have trouble following you. In paragraph #1 above I repeat the well known point that math. sets are entities over and above their members. Then, in para. #2 I give an argument why this must be the case. The argument was:

1. Sets can have sets as members without having their members as members; ergo
2. A set and its members are distinct.

To put that more clearly:

1*. Some sets have sets as members without having their members as members; ergo
2*. Some sets are such that they are distinct from their members.

You write, "your claim." What claim do you take me to be making? And how are transitive sets relevant to it?

David,

I hope you are not suggesting that a brick house is a set. That can't be right for two reasons. First, a house can gain or lose parts while retaining its identity; a set by contrast cannot gain or lose members. If it does it ceases to exist. Second, a modal point. A house that actually has such-and-such parts might have had different parts (e.g. one of the red bricks might have had a yellow replacement). But no set might have had a different membership than the one it actually has.

But I may not have understood you.

Happy New Year.

Bill,

I do not challenge your argument (1*)-(2*).

By "your claim" I mean the last line of my quotation from your post ("although {a} and {b} are members of the power set, a and b are not"). It seems to me that this line implicitly states the principle I stated (for every set x, every member --which is not empty set -- of x is NOT a subset of x, and that every member -- which is not empty set -- of a member of x is NOT a member of x). Maybe I'm wrong.

And Oppy's transitive sets seem to be in conflict with the principle. Again, maybe I'm wrong; maybe I don’t understand the concept of transitive set.

Let's consider the following.

Your set = {a, b}.
Your power set = {{ }, {a, b}, {a}, {b}}.

My set S = {a}.
The power set *S = {{ }, {a}}.
The power set **S = {{ }, {{ }, {a}}, {{ }}, {{a}}}.

In this list, no set which have members which have members (i.e., all the power sets) is transitive. For instance, your power set has your set as a member, but your power set does not have any member of your set as a member. **S has *S as a member, but **S doesn't have all the members of *S as members. *S has S as a member, but *S does not have all the members of S as members. A layman like me naturally asks whether there are any transitive sets at all, or, rather, how do they look like.

Bill and David,

Could you please supply some example(s) of clearly transitive set(s) (by enumeration of their members)?

Hello Bill, and Happy New Year in return.

Can I refer you to my further comment on your empty set post? I am suggesting that a brick house is a set, but the 'is' here is not the 'is' of identity. Rather, it's the 'is' of abstraction, or perhaps of metaphor. What I mean by this is that a house, in some aspects, 'behaves like' a set of bricks. The problem is that the idea of 'set' is such a thin extension to the idea of multiplicity that it's hard to pin down quite what it adds. Suppose we want to know the weight of a house. Common sense tells us that we just have to add up the weights of the individual bricks. But this is a special case of a general rule: if a material thing consists of a multiplicity of disjoint sub-things then its weight is the sum of the weights of the individual sub-things. This can be formalised in the language of sets of objects and functions from objects to real numbers representing weights. We have a metaphor for a house taken from the world of mathematical objects. Like any metaphor, it claims to express only a limited aspect of the thing in question. So, Yes, the house, for the purpose of calculating its weight at a particular instant, is the set of bricks (with their own particular weights) that it's actually made of at that moment. How else would the engineering disciplines, and applied maths in general, succeed?

Addendum

Bill,

You wrote in your reply to me:

"I repeat the well known point that math. sets are entities over and above their members. Then, in para. #2 I give an argument why this must be the case."

Is the well known point supposed to be that ALL math. sets are entities over and above their members? If so, note that your conclusin (2*) is only that SOME math. sets are such that they are distinct from their members.

Secondly, what are the logical relations btw the words "over and above" and "distinct from" as you are using them? Do they mean the same? If not, what does "over and above" mean?

Hi Bill,

May I take another stab at explaining my original comment here? The problem is to understand how a set can be a unity over and above its members, even though it appears to be wholly constituted by its members. Let's return to the house made exclusively of bricks. We can think of the house as a unity of the bricks themselves plus the way they are arranged to make the house. I think you have said something very close to this in past posts. The 'way they are arranged to make the house' comes down to relations between the bricks, or alternatively, between the individual bricks and the unity that is the house. It will include such facts as that a certain brick belongs to the house, and that another brick lies in the topmost course of the chimney, and so on. Let's now imagine paring away these relations by a process of gradual forgetting or ignoring. At first we will retain the fact that a brick belongs to the house but forget where it sits in relation to the house. If we carry this abstraction process to the very end we arrive at a mere plurality of bricks with no relations between them or the house. But (and this is partly inspired by Alex's comment on the empty set posting) we can stop just short of this when what remains is just the plurality of bricks and the is-part-of relation with the house. At this point what we have left looks just like a mathematical set---a one-over-many consisting of a plurality plus a membership relation. This appears to be the 'thinnest' extension possible that turns a mere plurality into a unity. To summarise, we have a sorites-like argument: If we can understand the house as a unity over and above its elements and also understand that there is a continuum from the house to a set, then we ought to be able to accept a set as a unity over and above its members.

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