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Re: [ontolog-forum] Ontological Assumptions of FOL

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Sun, 18 Mar 2007 12:20:29 -0500
Message-id: <6574B789-6CF8-4335-B3CF-E68496E34E06@xxxxxxxx>
> The critical distinction between set theory and mereology
> is that the axioms of set theory generate something new:
>     For any x, the set {x} is distinct from x.    (01)

Yes, in well-founded set theories like Zermelo-Fraenkel set theory,  
but singletons are perhaps the wrong thing to focus on to highlight  
the distinction in question, as there are in fact so-called non-well- 
founded set theories in which x = {x}.  (See, e.g., the fine book  
Vicious Circles by Moss and Barwise that I think I've mentioned before.)    (02)

The critical difference between set theory and mereology (I know that  
you yourself know this, John) is that, in the former, you get, so to  
speak, an "upward" ontological explosion.  For example, in mereology,  
starting with a, b, and c, we can infer the existence of new objects a 
+b, b+c, a+c, and a+b+c.  Importantly, though, any sum formed from  
these objects is identical to one of the sums we've already got.   
Thus, e.g., (a+b)+(b+c) = a+b+c.  Sum formation, so to say, stops  
after one iteration.  In set theory, by contrast, we get analogues  
{a,b}, {b,c}, {a,c}, {a,b,c} of the four sums above, but, critically,  
the set {{a,b},{b,c}} formed from {a,b} and {b,c} is *not* identical  
to {a,b,c}.  It therefore can itself serve as a member of such new  
sets as {{a,b},{{a,b},{b,c}}} and off we go into an infinite,  
iterative hierarchy of sets starting with only our modest stock of a,  
b, and c.    (03)

Chris Menzel    (04)

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