Re: Does the phrase " Russell's paradox " should be replaced with another phrase?
Date: Tue, 11 Feb 2014 09:10:22 -0800 (PST)
Message-ID: <a895e4a4-39cb-4d2a-b2b5-3f247104263e_at_googlegroups.com>
My solution, introduces two semantic procedures in working with concepts, say ProcedureA and ProcedureB. ProcedureA uses "the relation of satisfaction", which checks whether a particular attribute satisfies the given concept. ProcedureB performs the identification of the attribute for which we want to determine whether it satisfies the given concept. Obviously B. Russell did not use ProcedureB, Russell only used ProcedureA. Generally, the existing theory only uses ProcedureA, that is, it checks only a concept, and does not check the elements that are the input into the concept. The work of these two procedures is formally presented in my paper "Database design and data model founded on concept and knowledge constructs“, from 2008, at http://www.dbdesign11.com , by using the following formula:
S (the m-attribute, the concept of the property) = T iff the m-attribute matches the entity’s attribute. … (3.3.3)
Later in mentioned paper, the formula (3.3.3) in addition to attributes is used also for the entities, relationships and m-states. Thus, the concepts of attributes, entities, relationships and states have been introduced recursively. Note that in this hierarchy attributes are atomic; they are indivisible objects when they are in the concept and when they are elements of the corresponding extension. In my papers, I defined that factual sentences are also atomic sentences. This was based on the decomposition into the binary data structures. These sentences are atomic in terms of their meaning and their truth-value. Sometimes, factual sentences are atomic because they are facts about the attributes.
- Russell’s Paradox
With regard to Russell's Paradox, I would like to point out that Frege made
clear distinction between a concept and its extension. In my post from December
14, 2013, in my critique of Peter Chen's theory of conceptual model, I showed
how to construct the corresponding set from a concept. Note that in Chen's
paper, it is not known what it is a concept. A similar situation exists in the
current database theory, concepts are not defined. I also wrote that the
predicate denotes concept and the predicate gets the meaning, using the concept
(according to Frege). Note also that Frege distinguishes "logical axioms" from
the axioms of Set Theory.
In my post from September 24, 2013 I presented a schema of predicates of
unsaturated expressions. This schema shows that the concepts can be very
complex. The complexity of predicates enables that the theory of databases (and
the modern mathematics) formalizes the very complex objects and their whole
world, and not only numbers.
Frege's concepts are usually represented through properties. Usually concepts
are represented in the following way: that for given property P there exists a
set consisting of those and only those objects that have the property P. The
usual notation has the following form: {x|P(x)}.
For the construction of his paradox, Russell applied: “x not belong x“ property. This is a property that set x is not a member of itself. So we form a set S = {x | x not belong x}. This formula can be written in the following way: For every set x we have: (x belongs S) ≡ (x not belong x)
Since this is true for every x, we can take S for x, then we have contradiction
(S belongs S) ≡ (S not belongs S)
==================================
Where did Russell make a mistake in this proof? The mistake is that Russell puts in the concept, objects that can not be identify. He could not identify the above-mentioned set S. Set S does not actually exist, and therefore it can not satisfy the concept.
My formula, labeled by (3.3.3), argues that the inputs in concepts are only those elements that we can identify. (Note that subject also must exist for a concept). Thus Frege did not make a mistake with the concepts. In his theory lacks the part that relates to the identification of the elements, which are input into the procedure for the concept. So Frege was on the right track, but did not realize the importance of identifying elements that are input into a concept's. In contrast to Frege, Russell doubted in concepts.
2. Identification
As I have already noted, the identification in my model is mostly determined by
formula (3.3.3). In my solution I use m-attributes, m-entities, m-relationships
and m-states. The prefix "m" is placed to suggest that these objects are stored
in memory. m-objects that are stored in the computer's memory are modeled so
that they comply with the man's image of the corresponding real object. Thus,
the meaning has a great importance in the identification of real objects. In
addition to m-attributes, in my paper, I've also defined a universal attributes
and particular attributes. Particular attribute is defined as an attribute of
the particular entity. For example, it may be the red color of a particular car.
Universal attribute is man's innate ability to perceive certain attributes. For
example, one can determine whether two cars have the same color. The main
purpose of the universal attributes is to identify the corresponding particular
attribute of a real entity. Of course, these attributes can be abstract, for
example, one can identify a number.
In my conceptual model, in addition to concepts, identification plays a vital
role.
In my data model, with respect to the identification, all the objects are divided into the following groups:
(i) Attributes. Attributes are atomic objects, and each attribute is an
identifier. Formula (3.3.3) defines the attribute as an identifier and specifies
the procedure for identifying of attributes.
(ii) Complex objects. Entities, relationships and states are complex objects in
my data model. They are defined recursively, according to the complexity of
these objects. Each complex object has an identifier that is constructed
according to identifiers of its parts. For example, an entity consists of
attributes, each of these attribute is an identifier. A set of these attributes
(identifiers) specifies the entity identifier.
A Relationship is constructed from the entities. These entities determine the
identifier of the relationship.
At the end, the identifier of a state is constructed of elements that determine
the state.
Frege's idea about the concept and extension, allows a significant expansion of
mathematics. In addition to the numbers, we can now introduce the various
properties and their corresponding extensions with objects that belong to these
extensions. For example, we can work with the concept of color or a concept of
car.
In the equivalence (3.3.3), there are four possible cases in the analysis of
truth-values of the equivalence. In the real life, this equivalence will only
work when both of its sentences are true. So if m-attribute satisfies the
concept, and m-attribute is identified, then the whole semantics works OK. So
here we have a connecting between what is imagined and what is real.
Let's imagine that we have to construct a semantic machine that has these two
semantic procedures. Then the concept, roughly speaking, constructs a set and
the procedure for the identification must know (identify) the elements of the
set. So, in addition to the set, we need to know the elements that are input
into the concept. If we exclude some elements (attributes), then the
functionality of this semantic machine can change dramatically, no matter what
the concept remained the same. Suppose now, that the semantic machine can have
thousands of concepts, and that there is no identification of attributes, which
are input into these concepts – it will be very bad.
Let's take one example that might be naive, but that is an example from real
life. It is known that various beings see the world in different ways. Some
insects can see maybe a thousand different colors. Man can see maybe a hundred.
Leo sees mainly two colors, black and White. (Maybe that's reasons why they have
problems with zebras). So, all these beings have the same concept of color, but
different elements that satisfy the concept of color. These small differences in
attributes cause huge differences in the real life.
Conclusion: The concept and procedures that are associated with the concept, we need to understand as one totality (a whole), and not as one procedure. These procedures refer to the following link: Subject - The real World.
3. The construction of the numbers
Peano's axioms did not give the construction of natural numbers, but they
defined axioms, the natural numbers must satisfy. G. Frege made the first
construction of natural numbers. He did it in the following way:
0 = {Ø}. Representation of 0 is the set of all sets containing 0 elements. Only such set is empty set.
The set n is representation of the natural number n and n consists of all sets with n elements.
--
Today, the natural number is constructed by using successor function. Given a
set x, the successor of x is the set x+ = x U {x}.
Note that (x belongs x U{x}) < = > (x belongs x) OR (x = x). Thus x+
contains x as a subset and contains x as element.
Note that for this element x, must be satisfied the following:
(x not belong x).
==================================
How did Frege came to this construction of natural numbers? I will show it on
the example of the number 3. On some building we can notice certain group of 3
windows. So, 3 denotes these windows. Note, that according to Frege theory the
name denotes the corresponding real world object and give the sense of it. Note
that the name 3 can denote any three windows on this building. The name 3 can
denote any three windows on the world. The name three can denote any 3 doors,
cars etc. B. Russell wrote that "Frege identified with number 3-s a plurality of
pluralities, and number in general, of which 3 is an instance, is a plurality of
pluralities of pluralities. The elementary grammatical mistake of confounding
this with the simple plurality of a given triad made the whole philosophy of
number, before Frege, a tissue of nonsense in strictest sense of term
"nonsense". "
Thus, this explanation suggests why Frege represents a natural number N as "all
sets with n elements." With examples about colors and numbers, I wanted to show
that the concept and identification of attributes are strongly associated with
the real world. They are semantic procedures.
I think my solution of Russell's Paradox is essential. It is not on the formal
level. In contrast, my solution is at the level of important semantic
procedures, which are the important link mind-the real world.
In the current Set theory (say Zermelo-Fraenkel) there are formulas that are not
of a "set type". For formula, namely for property p (x) is said to be "set type"
if there is a set whom belong all sets with the property p (x).
In Set theory can be proved the following two theorems:
Formulas (x not belongs x) and (x = x) are not “set type”.
4. Some historical facts about Russell's Paradox
When he submit his capital work to printing, in the late evening of the day,
Frege received a letter from Russell in which Russell briefly informs Frege that
he had found a paradox in Frege’s work. The paradox refers to the main axiom in
Frege's theory. Here is what Frege wrote about this event: “Hardly anything more
unfortunate can befall a scientific writer than to have one of the foundations
of his edifice shaken after the work is finished. This was the position I was
placed in by a letter of Mr. Bertrand Russell, just when the printing of this
volume was nearing its completion ... I should gladly have dispensed with this
foundation if I had known of any substitute for it. And even now I do not see
how arithmetic can be scientifically established; how numbers can be apprehended
as logical objects, and brought under review; unless we are permitted – at least
conditionally – to pass from a concept to its extension. ....“
From this Frege's text two things are clear: First, Frege kept his ideas on
concepts and extensions. Second, Frege predicted certain restrictions on the
level of concept / extension, as Zermelo later did with axioms of set theory.
After the happenings of Russell’s Paradox, Frege abandoned completion of what
was to have been his greatest achievement and his work of large scale. He
produced no work for next seven years. He spent his last years as broken and
bitter man, without scientific interest as he had before.
In the spring of 1908th, director of Universities of Jena, Mr. Dr. von Eggeling,
in charge of awarding medals and awards in such circumstances, announced the
following: “Mr Frege does not belong any of medals, since his scholarly
activities are of the lower level and the University has no specific benefits of
them.” In 1908 Frege was 60 years old.
In one period Frege has held lectures at Jena University. Rudolf Carnap and a
retired colonel, who was involved in contemporary logical theory as a hobby,
were the only students at these lectures. Carnap described Gottlob Fregea as a
professor who was a great mathematician and philosopher. Note that R. Carnap is
recognized as a great logician. Many scientists put Carnap on the second place
at the list of great logician, behind Frege.
If anyone has any criticism regarding this post, then feel free to post it. For
me this is a very important topic.
Vladimir Odrljin
Received on Tue Feb 11 2014 - 18:10:22 CET