Dominus Paradoxum

Now I know this would not count as a formal inconsistency, but it nevertheless does seem inconsistent.

Question: Are sets simply identical to their members, or are they something over and above them which 'unites' them into one entity? At first it would seem that they must be something more, otherwise there could not be an "empty" set. Yet if they are something more than their members, then how can there be any guarantee that sets which have the same extention are the same set?

Witt

Dominus Paradoxum:
Now I know this would not count as a formal inconsistency, but it nevertheless does seem inconsistent.

Question: Are sets simply identical to their members, or are they something over and above them which 'unites' them into one entity? At first it would seem that they must be something more, otherwise there could not be an "empty" set. Yet if they are something more than their members, then how can there be any guarantee that sets which have the same extention are the same set?
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I agree that we have a problem with set identity.

The axiom of extensionality assures us that: classes that have the same members are identical, independent of the predicate that determines that class.

1. (All x)(x e {x:Fx} <-> x e {x:Gx}) -> {x:Fx}={x:Gx}.

2. (All x)(x e {x:Fx} <-> x e {x:Fx}) -> {x:Fx}={x:Fx}.

3. (All x)(x e {} <-> x e {}) -> {}={}.
This is OK because we assume that {} exists by axiom, i.e. it is self identical.

But, it is false for the Russell class..the class of those classes which are not members of themselves.

4. (All x)(x e {x:~(x e x)} <-> x e {x:~(x e x)}) -> {x:~(x e x)}={x:~(x e x)}, is a contradiction.

x e {x:~(x e x)} is contradictory for all x!

5. (All x)(x e {x:~(x e x)} <-> x e {}) -> {x:~(x e x)}={}, is contradictory.

The premise in both 4 and 5 is tautologous and the conclusion false.

That is, the axiom of extensionality expressed by 1 and 2 is invalid.

What do you think of this argument.

Witt

John Page

I agree with Witt, my own rationale is that a set is not identical to its members. Each member has its own identity and each set has its own identity.

In set theory, Identity is determined a set of qualities for which we test for equivalence (the identity relation). It seems obvious to me by these definitions (challenge, please ) that a set's qualities can never be identical with it's member's qualities, even when a set comprises other sets.

By example. We can say that a tea set comprises its members (cups, saucers, plates, teapot whatever) - but this merely confers a name on a bunch of stuff that we can instantiate as the concept of a tea set. The (concept of a) tea set, however, does not have the same identity as its members.

Let's take the proposition "I am the sum of my parts." We are saying that the terms "I" and "sum of my parts" both refer to you. Similar to the tea set, however, the concept of "I" does not have the same identity as the actual sum of my parts.

Clear?

Cheers, John

Dominus Paradoxum

I agree that the concept is not. But for the purpose of the arguement I was assuming that a set was supposed to be something actual.

John Page

But sets are concepts - there may be a bunch of objects that have similar qualities but their forming a set is a mental opertaion formed in your mind/brain through comparison of each of the members with the archetype for the set.

Mind experiment. You have never seen a tree before. You will not recognize the entity "tree". You turn a corner and are confronted with a forest. You will not see a forest (set of trees). After study, I expect, you will see the trees for the woods.

So there are actual things and sets of actual things but it seems that one's mind must contain the concept of them to know them.

Cheers, John

Peter Soderqvist

It seems to me that a forest is a set, and its members are trees! Human body is a set, and its members are the body parts, this body is an identity which consists of body parts, this body is even identical to its members, and vice versa, but the body parts are not identical to each other, because an arm is not a leg!

John Page

As set is to member, so forest is to tree.
As set is to member, so body is to body part.

A "body" is a concept. "parts" is a concept. The relation between them ("is the sum of its") is a concept.
We can say that "this body" refers to the same matter as the term "these body parts", and this is the meaning of "This body is the sum of its parts". As before in this thread, language may employ multiple names that refer to the same 'thing'.

The confusion, I think, arises when we try to employ the same tactic to assert that the concepts themselves are identical. Now, all the objects we are discussing here are objects of the mind. The world outside the mind "is what it is", it is the mind that thinks one part might be the "same" as another, but this can only be the same in quality, not actuality. (For if they were the same in actuality, the mind would not be able to tell them apart).

Cheers, John