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Re: [ontolog-forum] Universal Basic Semantic Structures

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>, "Matthew West" <dr.matthew.west@xxxxxxxxx>
Cc: "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Avril Styrman" <Avril.Styrman@xxxxxxxxxxx>
Date: Sun, 30 Sep 2012 23:26:07 +0300
Message-id: <20120930232607.159453iaynxx0gyn.astyrman@xxxxxxxxxxxxxxxxxxx>
Matthew, John, Chris,    (01)

John already told it, but to clarify once more, when you have only  
rank 1 sets, and you decide that you do not use the memberOf operator,  
and you forget the atoms that are not members of sets, then you can  
apply sets and discrete/atomistic mereology in an identical way. But  
once I suggest this, people immediately say "what about a in {a}?"  
This shows that unnecessary elements should not be included, only  
because they are a potential source of confusion. In general, the less  
unnecessary confusion, the better. That's why it's probably better to  
follow John's advise and use 'set' only to refer to Cantor-originated  
stuff. But all Cantor-originated stuff does not have to suffer from  
the problems that have been pointed out by a long parade of the  
greatest philosophers in this planet.    (02)

I'm only interested in structural features of set theory, i.e., about  
applying set theoretic structures to the concrete measurable reality.  
I'm not talking about set theory as a foundation of mathematics at  
all. Having this view, only a very diminished version of set theory is  
required. Here is one version, which I call finitist set theory (FST):    (03)

http://www.cs.helsinki.fi/u/astyrman/fst.pdf    (04)

It is currently under evaluation. From the viewpoint of a  
mathematician who is concentrated on ZF(C), this is just an  
unimportant and uninteresting theory that can be encoded by using ZF.  
 From a structural point of view, it is the minimal theory that can be  
used in building granular structures, and it has several advantages  
compared to ZFU and KPU (set theories which accommodate ur-elements).    (05)

1. It is finite, i.e., there are no transfinite sets. You explicitly  
assign the number of atoms (ur-elements) and the maximum rank. Thus,  
axiom of foundation is not needed: it is needed only to exclude some  
implications of the transfinite hierarchy, but because there are only  
finitely many sets, there is no danger of non-wellfounded structures  
in the first place.    (06)

2. The disclusion of empty set is inherited from mereology. That is a  
great simplification: empty set is not needed, and thus there is no  
use to have it. It only messes up conceptual modeling.    (07)

3. All set theories I'm aware of (except FST) have copied the axiom of  
union from ZF. It follow from ZF's union that in order to have all  
rank n sets, also rank n+1 sets have to exist. In FST, you define the  
maximum rank, and the axioms give all sets from rank 1 to n. This was  
done by modifying the axiom of union. Also the axiom of pairing was  
diminished into the axiom of singleton sets. Pairing is just  
unnecessarily strong.    (08)

FST should be seen as a nonproblematic foundation of granularity. In  
general, you can encode whatever with ZFC. Then again, if you only  
need the encoded version, the question rises that for what purpose do  
you need ZFC. In the big picture, you have natural language and you  
can make any theory with it. So, when talking about modeling the  
measurable reality, why do we need to have ZFC as a mid-layer between  
NL and e.g. FST? Scientists should not be concentrated too much on  
ZFC. If someone needs granularity, she can have it, without having to  
learn ZFC. Take a biologist, chemist, a computer scientist, or anyone  
who is interested in modeling the measurable reality. It's not  
necessary for them to spend time on learning and undestanding ZFC:  
they can enjoy the fruitful features of set theory without ZFC.    (09)

-Avril    (010)

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