Guy asks:

>"It is my claim that indistinguishable elements constitute only a single
>element by definition under set theory."
>
>In other words, the intersection between a set of pears and a set of apples
>is empty, even if you recognize that both pears and apples are kinds of
>fruit, because each piece of fruit has been individually distinguished and
>there is no single piece that exists in both sets.
>
>"So, for example, the large set called fruit can be said to contain kinds of
>fruit, like apples and pears; but unless each apple and each pear is
>uniquely distinguished "apple" and "pear"
constitute only two elements of
>the set called "fruit".  You could also, as suggested by BOH, have each
>apple and each pear distinguished as individual elements in the set
"fruit",
>which would allow you to consider these as subsets with each containing many
>elements.  The intersection between these subsets would be empty as long as
>no particular element was both an apple and a pear."
>
>"I am sorry for repeating information I posted previously, but it still
>seems to me that this issue underlies the differences of opinion on how set
>intersection is done.  It also seems like a basic point that would have
>clear-cut validity or not.  I would appreciate it if someone could point to
>an authoritative resolution of this issue so that we could either get on
>with understanding JE's argument concerning kin selection or put it behind
>us."
>
>I would still like to have this resolved one way or the other to everyone's
>satisfaction by an authoritative reference.

You have it exactly right, but that should not be a surprise to anyone.
Defining what is and is not an element of a set is among the most basic aspects
of defining a set. To quote one math webpage:

======================================

What is an element of a set? An element of a set A is a method which, when
executed, yields a canonical element of A as result. 

When are two elements equal? Two elements a; b of a set A are equal if, when
executed, they yield equal canonical elements of A as results. 

It is interesting to note that one cannot construct a set if (s)he does not
know how to produce its elements

   --www.math.unipd.it/~silvio/papers/TypeTheory.ps 

======================================

Wirt Atmar
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