Is Russell's paradox in fact fraud?

[vldm10]

1.
Nicolas Fillion in his publication:
„Les Enjeux de la Controverse Frege-Hilbert sur les Fondements de la
Geometrie“
on page 19, in the section "Zermelo-Russell Paradox", you can find the
following statements about this paradox:

„According to Russell, he discovered this paradox in 1901 (van Heijenoort,
1967a, p. 124). However, the first note about the existence of his discovery
is in the famous letter to Frege that he sent on 16 June 1902. „

The first published discussion of this paradox is to be found in 1903
(Russell, 1903).
This well-known paradox goes as follows: “

„Zermelo himself claimed in a note that he discovered the paradox before
Russell published it: “I had, however, discovered this antinomy myself,
independently of Russell, and had communicated it prior to 1903 to professor
Hilbert among others.”. “

„ (The decisive proof that Zermelo actually discovered the paradox
independently can be find in (Rang and Thomas, 1981) where it is shown that
he communicated it to Husserl, thanks to a note dated 16 April 1902 (2
months before Russell’s letter to Frege). “

2.
Jaan van Heijenoort (Jean Louis Maxime van Heijenoort), american
mathematician, was a pioneer historian of mathematical logic. Many of the
original papers are contained in his book: From Frege to Gödel. „Bertrand
Russell sow and approved the translation of of his 1902 letter to Frege. „
(look at the preface in this book).
In the letter to Frege, Russell communicates the paradox to Frege.

This book also contains Frege's response to the mentioned Rusell's letter.
Russell wrote this letter to Frege on June 16, 1902. Frege replied to
Russell on June 22, 1902.
In Frege's letter to Russell, there is the following part of the text:
„Incidentally, it seems to me that expression „a predicate is predicated of
itself“ is not exact. A predicate is as a rule a first-level function, and
this function requires an object as argument and cannot have itself as
argument (subject). „

From this text, it is clear that B. Russell did not understand the
fundamental elements of Frege's theory - that is, predicates, objects and
concepts. Russell also did not understand the relationship between objects,
concepts, and predicates.
We can notice that these same things did not understand Codd and his
followers Date and Darwen. Even more, I have not noticed that they have ever
mentioned concepts and objects, and relationships between concepts,
predicates, and objects.

3.
Barber paradox

(a)
Barber paradox was used by Bertrand Russell as an illustration of paradox.
The barber paradox is the folowing: "Barber shaves all those, who do not
shave themselves."
The question is, does the barber shave himself?

Since the barber is the only barber in town, then the upper sentence leads
to paradox.
But over time, somebody found a solution to this Russell's „illustration“ of
Barber paradox. This solution is Mary. We can notice that Mary is a woman
and she has no beard.

We can notice that the barber paradox today is presented differently from
the one presented by B. Russell - see the above definition of the barber
paradox labeled with (a).

Today's version of the barber paradox is properly presented, as follows:

(b)
The barber is a man in town who shaves all those, and only those, men in
town who do not shave themselves.
Who shaves the barber?

Still there are people who version of Barber Paradox labeled with (b)
represent as Russell's version. Of course, version (b) is a paradox, while
Russell's version (a) of the Barber paradox makes no sense.

4.
Gödel's „first incompleteness theorem“ first appeared 1931 in paper „Uber
formal unentscheidbare satze der Principia Mathematica“ (On formally
undecidable propositions of Principia Mathematica).

First Incompleteness Theorem: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved
nor disproved in F." (Raatikainen 2015).

Related to Godel's first incompleteness theorem we can notice the following:

(a)
In this theorem, Godel put Russell's theory, which is presented in Russell's
book Principia Mathematica, in the title of this Godel's work. In this way,
in fact, Godel has "thrown into a dust" this Russell's theory.

(b)
Godel's "first incompleteness theorem" did not mention Frege's theory. Frege
completely constructed the axiomatic system for propositional logic. This
axiomatic system is complete. Today, this axiomatic system is known as
"Frege-Lukasiewicz System". Frege has fully established Predicate Logic.
Kurt Godel also proved that the first order predicate logic is complete
theory.
---------------------------------------------------------------------------
Now, I can present my conclusion, which is based on presented facts. Ernst
Zermelo is the first scientist who has defined this paradox. Ernst Zermelo
is also the first who has solved this paradox. Therefore, this paradox
should has name Zermelo paradox instead of name Russell paradox.

It is shown that Russell did not understand some basic things that are
related to Zermelo paradox.

It has been shown that Godel proved that Principia Mathematica is incomplete
theoty. While Frege's theories - propositional logic and predicate logic are
complete theories.
----------------------------------------------------------------------------
Does Zermelo's Paradox have some relevance to database theory. The answer is
- yes, because concept, that is, the predicate, determines the plurality of
objects that satisfy that concept (predicate). We understand this plurality
as one object, which we call - set.
When we talk about Zermelo's solution to this paradox then we need to point
out two things:
(i) He solved this paradox at the level of set theory.
(ii) He solved this paradox by adding a new axiom to the axiomatic system
of set theory.

Zermelo uses this axiom as some kind of constraint for sets. Therefore, we
can set the following question: Are the axioms some kind of constrains. The
answer is - yes. Axioms are some kind of constrains for the corresponding
theory.

Vladimir Odrljin

[vldm10]

I presented my solution of "Russell's Paradox" at this user group. At that
time, I did not know that Zermelo was the first who discovered this paradox
and the first who solved it.
In this thread, I have underlined that Zermel's solution is given within the
set theory.
My solution is of a general nature. It is at Logic and Semantics level.
Why is it important to make a solution at a general level? The solution at
the general level enables the realization of these complex ideas in computer
technology, artificial intelligence, robotics and advanced software. Of
course, the general solution is most important for scientific theory.

Before presenting my main ideas in solving this paradox, I will present what
other scientists have rightly solved in this paradox.
After reading Russell's letter, Frege realized that there were some mistakes
in his work. In Frege's response to Russell's letter, he wrote the
following: „ ... Your discovery of the contradiction caused me the greatest
surprise and, I would almost say, consternation, since it has shaken the
basis on which I intended to build arithmetic...“.

However, no one has specified exactly where the mistake is. In the first
post of this thread I wrote that Russell did not understand the basic things
of paradox. By the end of his life, Frege remained convinced that he well
defined the concept. Zermelo introduced a kind of constrain but did not
explain the essence of the problem.
My solution is based on semantics and logic (not on sets) and in my opinion
these theories are on higher levels than the set theory. I wrote a lot about
my solution on this user group. Also, you can find my solution on my website
www.dbdesign10.com and www.dbdesign11.com

Note that concepts (predicates) determine plurality, that is a set. In my
solution, attention is paid to elements of sets, rather than to sets.
In the set theory, the basic concept is an object. Elements of a set are
objects. Since a set can be an element of another set, then a set is also an
object.
In my solution, I assume that elements of the set are names of objects
rather than objects.

In my solution, I use Leibniz's law in an altered form. I did four changes
to Leibniz's law:
(i) I introduced the state of the objects rather than the objects.
(ii) I introduced the atomic structures of the objects.
(iii) I introduced the extrinsic properties of the objects, which I added to
the intrinsic
properties of the object. So, the objects in my solution have
intrinsic and extrinsic properties.
(iv) I have introduced the history of events from objects.

Objects that are defined with these changes can also work as objects that
have Null values. In addition to Leibniz's law, I have also introduced the
identification of objects, attributes and relationships.

Regarding set theory and axioms associated with this paradox, I wrote about
this topic and also about identifying of elements of a set, in my thread "A
new way for the foundation of set theory".

Vladimir Odrljin

[vldm10]

At the end, it is necessary bring a conclusion because I think this area
remains with incomplete solutions in some its parts.
According to the above mentioned facts, Ernst Zermelo was the first to
discover this paradox and the first to solve this paradox.

In the above-mentioned work written by Nicole Fillion, the following facts
are particularly emphasized:
„The point of talking about this dispute here is not so much to establish
who discovered the paradox first, but only to show that the mathematicians
of the group of Gottingen were aware of such paradoxes in set theory, and
surely worked actively to solve the problem.“
As far as I know, E. Husserl and Hilbert were mathematicians who belonged to
Gottingen's group. At that time, most mathematicians have had opinion that
Hibert was the best mathematician. So, this powerful group knew for this
paradox and worked on it before Russell.
It should be noted that Frege just started to print three books in which he
wanted to show that mathematics can be completely constructed on the basis
of logic. At a time when Frege received a letter from Russell, the printing
of second volume was near completion. After that Frege remained with the
paradox and a bunch of printed books. So, Russell sent his letter right at
the time Frege had printed his three books.
After that, Russell wrote a book "Principia Mathematica" in which he also
tried to build mathematics that was built on the basis of logic.
But in 2005 there appeared the book with the name Fixing Frege written by
John Burgess, who, with some corrections, succeeded in realizing Frege's
great idea - mathematics can be built on the basis of logic. In my opinion,
this Frege project is the largest and most significant project in the
history of mathematics and probably in the history of science.

Here's how Frege reacted when he received Russell's letter and when he
finished printing his two books:
„Hardly anything more unfortunate can befall a scientific writer than to
have one of 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. ... “

After Russell's letter, Frege did not publish scientific papers for six
years.

Nowadays, more and more facts reveal that Frege had, in terms of volume, a
large correspondence with the most prominent mathematicians of his time,
using long, detailed and carefully written letters.
Unfortunately, during the bombing in 1945, most of Frege's correspondence
was lost.

At the end I will give my opinion about this paradox. Frege did not make any
mistake related to concepts. He defined the concepts well. The concept does
not construct elements of a set, but checks whether an object satisfies or
does not satisfies the concept. One concept generates a plurality of
objects.
In my opinion, Frege, Russell and Zermelo did not focus on the
identification of objects. It was a failure, because of which there was the
paradox and why they did not find the real reason for this paradox.

We notice the following:
- elements of a set are objects
- a set is an object
- a concept defines the plurality of objects that meet that concept.

We need to identify objects for several reasons. I will list only the two
most important ones:
(i) the concepts are the capital part of the semantics.
(ii) Memory is important for working with objects. I will mention only three
basic things in the work with every memory:
- The object must be stored in a memory. (Write)
- The object must be found in the memory. (Read)
- The object must be written in a language. Entering an object in some
memory allows the permanence of this object.

Databases work with real-world objects. But the database does not contain
real objects, but the names of real objects are stored. The same goes for
abstract objects and fictitious objects. Still, in real-world practice,
professionals usually say that objects are stored in a database, which is a
deep misunderstanding of databases and memories.

Today, real-life data has reached enormous size and become global. Because
of this, the sets that contain data about real objects become an important
part of the applied and theoretical mathematics. These data are from the
real world. But databases also contain data that are about abstract objects
...
Today, at the global level there is an identification for each car (VIN
number). Identification for each book (ISBN number). Identification of every
person (passport, social security number, memberships in any organization -
school, library, tax, medical numbers, telephone numbers ...), address
system, owners, bar-codes etc.
In short, a proper mathematical theory is required for the identification of
objects.

Vladimir Odrljin

[vldm10]

[vldm10]

I would like to emphasize that Zermelo-Fraenkel Aksioms (ZFC) are
constructed as follows:
They disable examples of paradoxes that Russell used as the main and only
arguments.
The axiom of separation and the axiom of well-foundedness created by
Zermelo, forbid the sets presented by B. Russell . B. Russell used the
following two examples, with which he tried to bring down the entire Frege's
theory:

1. Russell´s example:
R = {x : x is a set and x not belongs x}


2. Russell´s example
Set of all sets:

Did Zermelo have any reason to ban these two Russell examples? Yes, Zermelo
had solid reasons to ban Russell's examples. These reasons are as follows:

S belong S also S not belongs S are not well-grounded as the set is always
more complex than its elements. For this reason the set can not belong to
itself.
Note that the condition "x is a set" means "set of all sets" (at example 1).
If a set of all sets are marked with V then V belongs V. We have already
explained that this is not a well-founded statement.
------

I would now conclude this section with the following conclusions:
1. Russell's paradox is unfounded and is therefore erroneous.
2. Frege did not go wrong with his concepts, that is, with his predicates.
But in Frege's theory there is one missing part.

3. Zermelo found where the fault was in Russell's thinking. This is what I
have already explained in this message.
4. As I have claimed in this group, I have a solution to this problem
and I have published it in my papers, you can see it at www.dbdesign10.com
and at www.dbdesign11.com

My basic idea is that there are two procedures. It is a procedure that
determines the plurality and the procedure that determines the individual.
Frege solution for the procedure for plurality was well done. However
neither Frege, nor Zermelo nor Russell realized that the elements of the set
had to be identified. There are many other parts in my solution. I've
written all this on this user group since 2005, when I started to publish my
solutions. For example, I have a solved the construction of the atomic
structures for entities, relationships events, predicates, propositions and
concepts.

Set theory is fundamental for databases, especially for the Relational
Model. I would mention that the relation is defined as a set.

Vladimir Odrljin

[edpr...@gmail.com]

This might be useful to discuss with some mathematicians.
Have you tried writing this up and submitting to a Mathematics journal?


I agree that it is good to know the theory behind relational databases.
It is good to know abstract mathematics concepts and how they can apply
to the real world.

Enjoy,
ed

[vldm10]

I agree that here there are new moments of Set theory. These are primarily procedures for identifying elements of a set.
But it is also a problem related to databases, as elements of sets of data in a database have meaning in the real world.

Vladimir Odrljin