Is Russell's paradox in fact fraud?
Date: Tue, 19 Sep 2017 15:56:31 -0700 (PDT)
Message-ID: <167c7040-8d78-45e4-8e9e-0c5241096057_at_googlegroups.com>
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 Received on Wed Sep 20 2017 - 00:56:31 CEST