Re: set of a set etc.

Jasper wrote:
> Mark Nudelman wrote:
>> Jasper wrote:
>>> Mark Nudelman wrote:
>>>> Stephen J. Herschkorn wrote:
>>>>> Jasper wrote:
>>>>>
>>>>>> The description is what I would call formal, not conceptual. "My
>>>>>> cat" and the set of my cat {My cat} are different conceptually.
>>>>>> My cat likes milk. The "set of my cat" does not, yet the two
>>>>>> denotations are closely related. What is the conceptual
>>>>>> relationship between the two?
>>>>> Your cat is a member of the set of your cat. The set of your cat
>>>>> is not a member of your cat.
>>>>>
>>>>> Sets are collections. A collection is distinct from the objects
>>>>> therein (usually). Put a ring in a box. The box contains the
>>>>> ring; the box and the ring are not the same thing.
>>>>
>>>> Just to confuse matters, W.V.O. Quine in "Set Theory and Its Logic"
>>>> defines the law of extensionality and notes that a consequence of
>>>> it is that there is only one memberless object. That is, since
>>>> extensionality says that two things are identical if they have the
>>>> same members, and indivduals do not have members, all indivduals
>>>> are identical to the empty set and to each other. To avoid this,
>>>> he could treat an individual as a different sort of object than a
>>>> set, but instead he defines "x \in y" as meaning "x = y" when y is
>>>> an individual. A consequence of this is that individuals are
>>>> identical to their unit sets, that is, x = {x} but ONLY when x is
>>>> an individual. Of course, he retains x =/= {x} when x is a set.
>>>> He takes some pains to show why this is harmless, but it does seem
>>>> rather odd.
>>>>
>>>> --Mark
>>>
>>> Yes it does. Thanks for the input and the reference. What do you
>>> make of it?
>>
>> Well, Quine discusses other possible solutions. He mentions using
>> two different styles of variables, one for individuals and one for
>> sets, which is I think the most natural approach. This is probably
>> what many of your respondents have in mind when they point out that
>> a cat is not the same as the set of a cat, etc. (Otherwise, they'd
>> be forced to conclude that a cat is the same as the null set, since
>> neither has any members [potential jokes about tom cats at this
>> point notwithstanding].) Quine also mentions the possibility of
>> adding a predicate that asserts "individuality", or conversely,
>> "classitude", which would let us distinguish individuals from sets.
>> But he prefers his solution as more elegant, since it doesn't
>> require an extra predicate or separate variable styles. I quote
>> from Quine:
>>
>> We are interested in "x \in y" to begin with only for classes y;
>> such are the only cases of "x \in y" that are subject to
>> preconceptions worth respecting. If for the sake of smooth
>> systematization we see fit to assign meaning to further cases, let
>> us assign a meaning that maximizes the smoothness.... Let us rule "x
>> \in y" true or false according as x = y or x =/= y, when y is an
>> individual.... But what if y is an individual and z is the unit
>> class of y? On our new interpretation ... "x \in y" then becomes
>> true if and only if x is the individual y; so (Ax)(x \in y iff x \in
>> z) and therefore y = z. This result is prima facie unacceptable,
>> since y is an individual and z is a class. But actually it is a
>> harmless result; none of the utility of class theory is impaired by
>> counting an individual, its unit class, the unit class of that unit
>> class, and so on, as one and the same thing. True, we are well
>> advised now to adjust our terminology to the extent of ceasing to
>> explain "individual" as "nonclass"; let us take to saying that what
>> constitutes them individuals is not inclassitude, but identity with
>> their unit classes.... Everything comes to count as a class; still,
>> individuals remain marked off from other classes in being their own
>> sole members. ---End of quote
>>
>> The last point is a key one I think -- by this route, everything is
>> a class to Quine, which simplifies some things. It's important to
>> keep in mind that this is just the way Quine's axioms work. Many
>> (probably most) other axiomatic systems don't consider x = {x} to be
>> true, even if x is an individual. So there's no one true answer to
>> the question about what this means, it depends on the axiom system
>> you're using. But there's no argument when talking about
>> multi-element sets: for Quine as for everyone else, the set {x,y} is
>> different from the set {{x,y}}, since the first has two elements,
>> and the second has one element.
>>
>> --Mark
>
> Before I say anything could you please explain the construction "x \in
> y", especially the backslash.

Sorry, I was just using "\in" to mean the epsilon operator for set
membership.

> Also what distinction do you make
> between set and class? I've seen them used interchangably but there
> seems to be some distinction in Quine's useage.

Quine does distinguish between set and class. He usually prefers the word
"class" and reserves "set" for classes which are capable of being members of
other classes. (Not all classes are sets in his theory, in order to avoid
Russell's paradox.)

> Also, as you have emphasized, the issue doesn't seem to have any
> application to sets with more than one member. There is another
> situation with multiple member sets however in that it seems like they
> can somtimes be treated as if they were single member entities. I can
> say something like "I drove my car to work." or I could say "I drove
> the set of mechanical parts that I call my car to work".Here it seems
> that the set {the mechanical parts that I call my car} is being
> identified with the single object referred to with the expression "my
> car". (I don't think that the self referential aspect of the expresion
> is a concern here, I'm just using to to avoid a complete list of what
> the car parts actually are.) Do you have any thoughts or experience
> with the possible use of sets of components of things as being
> identified with the things that they comprise?

Sets are conceptual objects, not physical ones. The fact that a car is
physically composed of components has no bearing on what members it has,
unless you decide that it does by calling it a set of components rather than
a single entity. In other words, the set containing car is different from
the set containing the components of the car, even though a physical box
containing one car may be identical to a physical box containing the
components of the car. Sets have properties based on how they're defined --
investigation into their physical properties can't discover properties that
aren't derivable from their definitions.

> (Or of using a set of
> properties directly as a definition of a single entity?--As opposed to
> "For all x such that .. followed by the defining properties")

I'm not sure I follow this, but sets are usually defined by their
properties. E.g. the set of numbers divisible by 7 would normally specified
like that, the set of numbers which have the property that they're divisible
by 7.

--Mark


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