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Topic: What are sets? again
Replies: 21   Last Post: Dec 9, 2012 10:12 AM

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fom

Posts: 1,969
Registered: 12/4/12
Re: What are sets? again
Posted: Dec 4, 2012 5:50 AM
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On 12/2/2012 9:38 PM, William Elliot wrote:
> On Sun, 2 Dec 2012, Zuhair wrote:
>> On Dec 2, 10:44?am, William Elliot <ma...@panix.com> wrote:
>>> On Sat, 1 Dec 2012, Zuhair wrote:
>
>>>>>> The following is an account about what sets are,
>>>
>>>>>> Language: FOL + P, Rp
>>>>>> P stands for "is part of"

>>>
>>>>> Does P represent "subset of" or "member of"?
>>>> Neither.
>>>
>>>> P represents "is part of"
>>>> review mereology to understand that relation informally.

>>> What is simple jargon, a brief intuitive description of "is a part of".
>
>> Just read Varzi's article on Mereology:
>> http://plato.stanford.edu/entries/mereology/

>
> It's long winded as philosophy usually is.
> Basically, "is a part of" is a (partial) order.
> "Subset" is the better interpretation that "is member of".
>
> So I'll take it as "subset" unless you give a useful
> interpretation within 300 words or less.
>

>> The relation "is part of" is well understood philosophically speaking,
>> it has natural examples.

>
> For example?
>

>> I think Varzi's account on it is nice and interesting really. You can
>> also read David Lewis account on it. The discipline of Mereology is
>> well established.

>
> What's the point of mereology?
>


Arguably, mereology as an investigation into foundational
mathematics is from Lesniewski

Lesniewski wrote several papers criticizing Russell's
Principia as basically being incoherent

He did his own investigations along the lines of logical
structure of sentences having existential import that
are different from both Frege and Russell

Subsequently, he characterized a notion of class that was
different from Russell's

At first, he tried to characterize his ideas in traditional
logical formats but ultimately began to pursue it
using formal syntax

This actually gets dense and I have not really
examined it. But, for example he begins a system
he calls protothetic with

A1. ((p <-> r) <-> (q <-> p)) <-> (r <-> q)

A2. (p <-> (q <-> r)) <-> ((p <-> q) <-> r)

and then lists 79 theorems about logical equivalence.

He then switches notation to quantify over propositional
variables

A1. ApAqAr(((p <-> r) <-> (q <-> p)) <-> (r <-> q))

A2. ApAqAr((p <-> (q <-> r)) <-> ((p <-> q) <-> r))

So that he can extend the system using variables ranging
over truth-functions

A3. AGAp(AF(G(p,p) <->
((Ar(F(r,r) <-> G(p,p))
<->
(Ar(F(r,r) <-> G(p <-> Aq(q),p))))
<-> Aq(G(q,p)))

He then lists 422 theorems in order to obtain the
three logical axioms of Lukasiewicz grounding
the usual theory of deduction based on implication
and negation. His primary stated goal at the
outset for extending the original system was to
obtain

ApAqAF((p <-> q) <-> (F(p) <-> F(q)))

and the equivalent was obtained at step 381

Next, he turns to his system of ontology
grounded upon the logical foundation of his
protothetic. His only axiom is

A0. AZAz((Z class of z) <-> (
(-(AY(-(Y class of Z)))
/\
AYAX(((Y class of Z) /\ (X class of Z)) -> (Y class of X)))
/\
AY((Y class of Z) -> (Y class of z))
))


In his exposition he comments on Russell's
paradox:

"... can be strengthened in ontology by
means of the easily proved sentence which
says that:

AZAz((Z class of z) -> (Z class of Z))


(I call this the 'ontological identity sentence';
it should be noticed that the yet stronger thesis

AZ(Z class of Z)

is not provable in ontology -- indeed, its
negation is provable.) In connection with this
sentence, I want to emphasize expressly that in
ontology there is always a very good possibility
of proving theses having a single component of
the type (Z class of Z) or (what is indifferently
the same in ontology) (z class of z). This does
not, however, lead to a contradiction via the
well-known schema of Principia Mathematica
because the definition directives of ontology
have been appropriately formulated so that
no thesis of the type

AZ((Z class of x) <-> -(Z class of Z))

can be obtained."


This assertion might best be viewed much like
the situation with general relativity. Philosophers
who have been looking at Lesniewski's systems
have not run into any contradictions such as
Russell's (at least, in so far as Peter Simons
has reported accurately)

The character of his predicate in ontology
allows him to formulate terminological
explanations about ontology within the language
of ontology.

The axiom of ontology, and the equivalent axioms
he discusses in his exposition derive from
his analysis of Russell's paradox.

The Lesniewskian notion of class is based upon
a part relation and this is the formal mereology
associated with his investigations:


A1:
If P is a part of object Q, then Q is not a
part of object P

A2:
If P is a part of object Q, and Q is a part of object R,
then P is a part of object R

D1:
P is an ingredient of an object Q when and only when,
P is the same object as Q or is a part of object Q

D2:
P is the class of objects p, when and only when the
following conditions are fulfilled:

a)
P is an object

b)
every p is an ingredient of object P

c)
for any Q, if Q is an ingredient of object P, then
some ingredient of object Q is an ingredient of
some p



These conditions formalize a statement Lesniewski
made in his analysis of Russell's paradox:

"I use the expressions "the set of all objects m"
and "the class of all objects m" to denote every
object P which fulfills the two following
conditions:

1) every m is an ingredient of the object P

2) if I is an ingredient of object P, then
some ingredient of object I is an ingredient
of some m"



Since ingredient is effectively the reflexive
subset relation in set theory by Zermelo's
1908 language, you can see why Zuhair chose the
language he did to describe atoms. In an
atomistic theory, I and m must at least share
some atom of P

I expressed this idea in a formal sense long
ago only to be flamed (singed by you and firebombed
by someone else)

What actually makes mereology work is something that
is associated with the constructible universe. It is
called almost universality. Of course, that is not
how Lesniewski referred to it:

"Lukasiewicz writes in his book as follows: 'we say
of objects belonging to a particular class, that
they are subordinated to that class'

"It most often happens that a class is not subordinated
to itself, as being a collection of elements, it
generally possesses different features from each of
its elements separately. A collection of men is not
a man, a collection of triangles is not a triangle,
etc. In some cases, it happens in fact to be otherwise.
Let us consider e.g., the conception of a 'full class',
i.e., a class to which belong, in general, some
individuals. For not all classes are full, some
being empty; e.g., the classes: "mountain of pure
gold', 'perpetual motion machine', 'square circle',
are empty, because there are no individuals which
belong to those classes. One can then distinguish
among them those classes to which belong some
individuals, and form the conception of a 'full
class'. Under this conception fall, as individuals,
whose classes, e.g., the class of men, the class
of triangles, the class of first even number (which
contains only one element, the number 2), etc.
A collection of all those classes constitutes
a new class, namely 'the class of full classes'.
So that the class of full classes is also a full
class and therefore is subordinated to itself."

In almost universal models of set theory, every subclass of
the universe is an element of the universe. Thus,
"is an element or is equal to" is the same as
"is a proper part or is equal to". So, the satisfaction
predicate can be reflexive containment... except for
one thing. The identity predicate in the theory can
not be based on extensionality. It must be based on
first-order object identity as described by Frege and
this cannot arise just because one invokes the
ontological position of a "theory of identity". The
reason that it must be based on object identity is that
reference to the universe can only be made if, as
Lesniewski has observed

P is an object

In an almost universal model, every proper part
of the universe is an element of a class of which
the universe is not an element. Thus every proper
part of the universe is distinguished from the
universe on the basis of object identity. Since
to be a 'full class' the universe can have no
other parts, it is unique and may be denoted
by a singular term.

I have a very strong suspicion that no one will
ever derive a Russellian paradox in Lesniewskian
mereology. This is especially true if one considers
George Greene's explanation that the paradox arises
from grammatical form. As we have seen, Lesniewski
specifically devised his mereology to circumvent the
grammatical forms he thought would be problematic.















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