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Re: [ontolog-forum] Visual Complexity

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Wed, 7 Feb 2007 23:11:27 -0600
Message-id: <C85CB486-5155-4B7B-90C1-9B56093A1727@xxxxxxxx>
> That is the distinction between set theory and mereology:
>> If sets are there in the world, then no - there's nothing
>> abstract going on.  I'm real.  Chuck is real.  And the set
>> of me and Chuck is real.
> In mereology, Bill and Chuck are each parts of the collection
> that consists of Bill and Chuck.  You can call that pair C,
> but C is not a new entity.  It is just Bill and Chuck.    (01)

John, that is not correct.  The mereological sum of Bill and Chuck --  
call it Bill+Chuck -- is typically defined in mereology as the  
smallest thing that has Bill and Chuck as parts.  (Equivalently, it  
is the unique thing X such that anything that overlaps X either  
overlaps Bill or overlaps Chuck.)  It is not "just Bill and Chuck".   
It is a third thing distinct from the two of them.    (02)

> But in set theory, if Bill and Chuck are members of the set S,
> then S is a third entity that is different from each of them.
> You have three things:  Bill, Chuck, and {Bill,Chuck}.    (03)

And in mereology you have Bill, Chuck, and Bill+Chuck.    (04)

> You
> can also construct a fourth thing, which is the set whose only
> member is the set whose members are Bill and Chuck: {{Bill,Chuck}}.    (05)

Now THERE is where mereology and set theory part company.  Mereology  
does not support an unbridled hierarchy of sums parallel to the  
hierarchy of sets in set theory.  As you note:    (06)

> You can form an infinite number of distinct sets whose starting
> elements are just Bill and Chuck:
>     Bill, Chuck, {Bill}, {Chuck}, {Bill,Chuck}, {{Bill},Chuck},
>     {Bill,{Chuck}}, {{Bill},{Chuck}}, {{Bill}}, {{Chuck}}....
> You don't even need to start with anything.  You can build up
> all of mathematics out of just the empty set:  {}, {{}}, {{},{}},
> {{{}}}, {{},{},{}}, {{{}},{{}},{{}}}...    (07)

By contrast, in mereology, the sum of Bill and Bill+Chuck is just Bill 
+Chuck; likewise, the sum of Bill's left arm and Bill is just Bill.  
In set theory, as you note, you get the distinct entities {Bill,  
{Bill, Chuck}} and {BillsLeftArm, Bill}.  But in mereology and set  
theory alike, the sum/set of Bill and Chuck is a third thing distinct  
from Bill and Chuck.    (08)

-chris    (09)

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