Re: Does the phrase " Russell's paradox " should be replaced with another phrase?
Date: Fri, 5 Dec 2014 07:41:16 -0800 (PST)
Message-ID: <c082bc01-e24f-4791-890e-e9911ceeb41c_at_googlegroups.com>
Recently, I presented my solution of Russell's paradox in the Mathematical
Institute of the Serbian Academy of Sciences. Abstract of the presentation
can be found at the address:
http://www.mi.sanu.ac.rs/seminars/programs/seminar1.nov2014.htm
I would like to make a few comments related to my solution of the Russell's
paradox, after this event.
My solution contains the following two types of sets:
- Sets whose elements denote objects
- Classic mathematical sets, which have arbitrary elements. So, these are
the sets that belong to the current set theory
(i) Sets whose elements denote objects
This kind of sets is typical in databases and I consider them as a specific type of mathematical sets.
For these sets I have developed a theory of states, theory of identification, and the theory of decomposition of structures into atomic structures that are presented on this group. I took Entity / Relationship philosophy and mathematics, as it is described by Kurt Gödel in his work from the 1944, as the foundation for this type of sets.
This paper clearly confirms that the priority of ideas for Entity / Relationship model does not belong to Peter Chen. Entity / Relationship philosophy and mathematics belong to Kurt Gödel and probably to some other mathematicians and philosophers. Kurt Gödel published this work, before the theory of databases was created and before the occurrence of computers. We can say that Peter Chen applied these ideas in databases. Because of the great importance of this work and regarding the priority of the idea, I'll quote greater part of this Gödel's work: “By the theory of simple types I mean the doctrine which says that the objects of thought (or, in another interpretation, the symbolic expressions) are divided into types, namely: individuals, properties of individuals, relations between individuals, properties of such relations, etc. (with a similar hierarchy for extensions), and that sentences of the form: " a has the property φ ", " b bears the relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of types fitting together. Mixed types (such as classes containing individuals and classes as elements) and therefore also transfinite types (such as the class of all classes of finite types) are excluded. That the theory of simple types suffices for avoiding also the epistemological paradoxes is shown by a closer analysis of these.”
This Godel’s paper is from: Kurt Gödel: Collected Works: Volume II, Oxford University Press, 1990
Note that K. Godel called objects, properties and relationships as "the object of thought". In my paper from 2008, I introduced m-attributes, mentities, m-relationships and m-states, where m is the shortcut for a memory. Here the suffix “m” is in accordance with my definition of abstract objects.
In my opinion, these sets whose members denote objects actually form part of a new theory. More precisely this part of today's database theory is part of a new mathematical theory.
In my solution, serious and unresolved problems related to the identification of objects are precisely represented and their solution is given.
On this user group, for the first time, the wrong solutions, related to the identification of objects in RM / T, Anchor Modeling and OO databases are precisely explained and solved. In all these three approaches, it was used so-called surrogate key.
Leibniz’s Law enables work on the identification of entities. By using this law we assure that identification of objects is a mathematical discipline. Properties in Leibniz's Law, I have divided into two types: intrinsic and extrinsic properties.
(ii) Sets that are classic mathematical sets.
The members of these sets do not denote objects, that is, these members are arbitrary. Elements of sets are objects. Since the set can be an element of a set, then sets are also objects. Now in (3.3.3) we have to use m-entity instead of m-attribute. Note that I am using two semantic procedures.
The axioms of set theory
Note that if Russell's Paradox is solved, then the axioms of set theory must be changed.
Note that from Frege's definition of the concept and extension; we can
derive “comprehension” and “extensionality” for sets. For example, this can
be seen in the excellent and beautiful work about Frege, from John Burgess,
Princeton University.
On the other hand I also use my theory of identification, in the
construction of a set.
About some inaccuracy in Russell's paradox
In the example by which Russell presented a paradox in Frege's theory, in my opinion Russell used imprecise assertions. In Russell's paradox is not defined which concept he used.
It is often used example of a set of all sets. This set comes down to the example that is used in Russell's paradox. But we can set the question - what is the concept that determines the set of all sets and what is the appropriate extension. Whether in this example, Russell thought about the concept of all concepts? By the way it seems that such concept is nonsense. In order to better illustrate this lack of clarity, regarding to Russell's paradox, let's take an example that can be found in the English textbooks. Frege has defined construction of natural numbers. Now I will focus on the natural number 1. Frege defined the natural number 1 as:
{x: x is a set and x has one element}
Let us denote by [1] set of all one-element sets. Then {[1]} belongs [1].
Thus, we have that [1] belongs {[1]} belongs [1]
and this is not “nice”.
However, if we use precisely Frege's notation, we must determine which
concept we use in these two cases.
For example, if we use my definition for concepts, then we can see that:
In first case we can identify one simple element, which belongs to
one-element set…(1)
In second case we can identify set as element of one-element set…(2)
Now we can see that (1) has the object which is simple element and that (2)
has set as element. Is the concept for (1) same as concept for (2).
Obviously, these are two different concepts.
In Russell’s papers, it is not clear distinction between concepts and
extensions. Later Russell tried to solve this matter by using hierarchy of
types?
In my opinion there is no good definition about types. However, if somebody
thinks that Russell has good definition of type, I would have been grateful
to the person who can present such definition of the type on this user
group.
Vladimir Odrljin Received on Fri Dec 05 2014 - 16:41:16 CET