Some of my e-messages to:
[ARISTOTLE] - list;
[Bertrand RUSSEL] - list;
[FOM]-list = [Fondations Of Mathematics]-list;
[HM]-list = [History of Mathematics]-list;
and others.

###################################################

19 February 2001.

TO:
"Robert Tragesser" <rtragesser@hotmail.com>
[FOM] "FoM" <fom@math.psu.edu>,
[HM] Historia Matematica <historia-matematica@chasque.apc.org>,
[EDULIST] edulist@thrace.ee.duth.gr , hellasmath@platon.ee.duth.gr
[Aristotle Society] Aristotelian Society <aristotl@sas.ac.uk>,
[Russell Society] russell-l-admin@informer2.cis.McMaster.CA;
russell-l@mailman.McMaster.CA
Kenneth Blackwell <blackwk@mcmail.cis.mcmaster.ca>
science-wars@egroups.com
"Moshe' Machover" <moshe.machover@kcl.ac.uk>,
"Martin Davis" <martin@eipye.com>,
Everdell@aol.com
"Prof. Michael R. Matthews" <m.matthews@unsw.edu.au>
"Vladimir Kanovei" <kanovei@wmwap1.math.uni-wuppertal.de>
"Wilfrid Hodges" <w.hodges@qmw.ac.uk>
"The Bulletin of Symbolic Logic" <asl@math.uiuc.edu>; <ablass@umich.edu>

TO: [Foundation Of Mathematics]-list.

I would like to touch two theses of the initial Robert Tragesser's message
"Subject: FOM: Cantor's Theorem & Paradoxes & Continuum Hypothesis",
on the Date: Sat, 10 Feb 2001.

THESIS I."Now the proof of Cantor's Theorem really seems to be very much more intractably dependent on tricks of logic and language that actually block the way to intuitive mathematics, to mathematical insight, in the way that the RAA <Reductio ad absurdum> proof of the irrationality of the square root of 2 did not. Hence my urge to say that the logical tricks (not to say illegitimate moves) exploited in the proof of Cantor's Theorem, variations of which drive the Russell Paradox, perhaps count as paradigmatic examples of the uses of logic and language that worried Brouwer.
THESIS II. "I'm curious about the provenance and evolution of this style of <Cantor's> argument".

As to the THESIS I.
As is known, there are the following two, - logically absolutely different, - versions of Reductio ad Absurdum (RAA).

ARISTOTLE'S RAA-1.
We intend to prove a statement, say A. Further:
1) Assume the contradictory of A, NOT-A.
2) Then, from that "NOT-A" as from a common formal premise, we deduce a series of formal consequenses, say

NOT-A --> B -->B1 -->B2 -->... --> Bk (1)

Till what k? - Nobody knows beforehand! Remark, the pure intuitive "minimization" of k is a high mathematical art in the Polia sense: "Before we can realize a proof, we must guess (discover) a way by which to realize the proof".
3) But if we deduced such a formal consequence Bk for which (usually)it is already known that NOT-Bk is an authentic TRUTH, then we will get a desired, "purposeful" CONTRADICTION-1 between Bk and NOT-Bk.
4) From the contradiction-1, by the law of contradiction, we conclude that Bk is an authentic FALSITY.
5) From the last, according to Aristotle's modus tollens rule, we conclude that the assumption NOT-A is FALSE.
6) From the last and from the contradiction-2 between NOT-A and A, we conclude that A is an authentic truth.
7) Q.E.D.

META-MATHEMATICAL RAA-2.
We intend to prove a statement, say A. Further:
1) Assume the contradictory NOT-A.
2) Then, from that "NOT-A" as from a formal premise, we deduce as formal consequenses a series of the following quite special form:

NOT-A --> B --> B1 --> B2 --> ... --> Bk (1*)

where IT TURNS OUT that Bk = NOT-B. That is, in brief, we have the scheme:

NOT-A --> B --> NOT-B (2)

3) Usually (see Kleene, Bourbaki, etc.), from the CONTRADICTION-1 between NOT-B and B, they conclude that the assumption NOT-A is FALSE.
4) From the last and from the contradiction-2 between NOT-A and A, they conclude that A is an "authentic" truth.
5) Q.E.D.

Observe the following.
1) In Aristotle's RAA-1 they prove the FALSITY of a formal consequence Bk, and then conclude from this the falsity of the assumption NOT-A by means of modus tollens rule; in the RAA-2 they prove the TRUTH of a formal consequence Bk = NOT-B, and conclude to the FALSITY of assumption NOT-A directly from the CONTRADICTION between not-B and B. So, there is a very essential difference between RAA-1 and RAA-2. BTW, the rule of negation introduction (the meta-mathematical "Reduction ad absurdum") formalizes just the RAA-2, but not the Aristotle RAA-1. Therefore to state that predicate calculus of modern symbolic logic is an adequate formalization of classical Aristotle's logic is some incorrectly.
2) From Pythagoras' time, all mathematics which produces such results which can, in the end, be tested by a number or an experiment is based on the RAA-1. Otherwise, as is well-known, the Epimenides "Liar", Russell "Barber", etc., and numerous well-known meta-mathematical theorems (of Cantor, Goedel, Church, Tarski, etc.) are based on the RAA-2.

Further, Robert Tragesser states that "the proof of Cantor's Theorem <based on the RAA-2> really seems to be [...] dependent on tricks of logic and language that actually block the way to intuitive mathematics, to mathematical insight","drive the Russell Paradox", and "worried <not only> Brouwer".

I agree, and I believe that there is a deep (almost genetic!) ground for such a worry.
The point is that a human-being from his first steps in the world is taught to comprehend the following absolute logical truth. -

ARISTOTLE'S POSTULATE-1. If even one premise (an assumption, as you please) of a formal deduction is not authentic (i.e., not proved), then any formal consequence of such a premise also will not be authentic (in the framework of that deduction).

From this point of view, the scheme (2) is a formal deduction of an inauthentic consequence NOT-B from the inauthentic premise B, BTW one is a negation of another. Just therefore such a "deduction" of a consequence, which is a negation of its own inauthentic formal premise (NOT-B from B), and such a contradiction between these two inauthentic formal consequences (NOT-B and B) seem to be a quasi-logical trick, which worries our intuition, and contradicts our generic experience. BTW, for example, the famous "Liar" is described by the scheme (here L="I am a Liar"):
[ L --> NOT-L ]&[ NOT-L --> L].
As it's easy to see, the fragment [B --> NOT-B] of the meta-mathematical RAA-2 (2) is an obvious half of the "Liar", and our intuition feels this dangerous resemblance very fine. Of course, the "Liar", "Barber", etc. are not the quarks of modern theoretical physics, but so far independent "halfs" of these paradoxes too are not observed in the nature ...
I am sure that now, after such the manifest prompting, any student will be able to construct the second half [NOT-B --> B] of a new set-theoretical "Liar". And not in "saving suburban areas of meta-mathematics" (according to Kleene, Fraenkel and Bar-Hillel, etc.), but in the core of modern set theory - in the framework of Cantor's argument itself.

However, I don't completely agree with Robert Tragesser that our "logic and language" are responsible for this trick. I think that the reason is a very special structure of such the paradoxical statement A and of special "rules" of inference (e.g., the Cantor diagonal method, the Goedel numeralwise expressibility, etc. ), based on the construction "self-applicability + negation". The last problem is considered in my paper (A.A.Zenkin, New Approach to the Paradoxes Problem Analysis. - Voprosy Filosofii (Philosophy Problems), 2000, No. 10, 79-90.), where first the necessary and SUFFICIENT conditions of the "Liar" paradoxicality and of the paradoxicality as a whole are formulated. In particular, it is shown that the self-applicability (or predicability by Russell) is a necessary, but not sufficient, condition of the paradoxicality. Moreover, the "predicability + negation" are also necessary, but not sufficient, conditions of the paradoxicality. However, it's enough to state that now it doesn't need to exclude self-applicable notions and the excluded middle law from mathematics, and, say, to prove the consistency of the FINITE numbers arithmetic by means of a TRANS-finite induction up to the famous Cantor transfinite integer e_0, where

e_0 = w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^w^...,

and w is the minimal TRANSFINITE integer (Cantor's "OMEGA"), which has no relation to Peano's arithmetic of FINITE natural numbers.

The foundations of that trick are already were discussing in history of mathematics during for a long time: by Aristotle ("Infinitum actu non datur"), Gauss ("I object ... against the usage of the infinite magnitude as some completed thing that is not permissible ever in mathematics"), Poincare ("There is not the actual intinifte; cantorians fogot about that and fell into contradictions"), the pupil and colleague of Russell and Frege, Wittgenstein ("...Cantor's <diagonal> argument has no deductive content at all. "), etc.

However, there are some well-known mathematical FACTs that make vulnerable the position of all opponents of Cantor's proof scheme (2). One of the facts is explicitly mentioned by Robert Tragesser: "the RAA proof of the irrationality of the square root of 2 ...". Indeed, it occurs as follows.--

PYTHAGORAS' THEOREM-1. The sqrt(2) is not rational.
PROOF-1 (by RAA-2). Assume that sqrt(2) is rational.
Then [[sqrt(2) = m/n]&[(m,n)=1]] --> [2 = m^2/n^2] --> [2n^2 = m^2] --> [m=2m1] --> [n=2n1] --> ... --> [(m,n) =/= 1].
So, the obtained CONTRADICTION between a formal consequence [(m,n) =/=1] of the assumption and the assumption itself [(m,n)=1], according to RAA-2, "proves" the statement of Theorem 1. Q.E.D.

Fortunately, the following, less well known fact occurs in the familiar Peano arithmetic.

THEOREM 2. For any m>n>=1: 2n^2 =/=m^2.
PROOF-2 (by Aristotle's RAA-1). Assume that 2n^2 = m^2.
Then:
[2n^2 = m^2] --> [m=2m1]--> [n2 = 2m1^2] --> [n= 2n1] --> [2n1^2 = m1^2], where m>m1, n>n1,
and further
[2n1^2 = m1^2] --> [2n2^2 = m2^2] --> ... --> [2ni^2 = mi^2],
where the regression n>n1>n2> ... >ni takes place, and there exists such i that ni<1. The last statement [ni<1] is FALSE, since it contradicts to the DEFINITION of the natural number as integer >= 1.
By modus tollens rule of RAA-1, from the proved FALSITY of the formal consequence [ni<1] we conclude to the FALSITY of the assumption [2n^2 = m^2]. Hence, by the law of contradiction, we have [2n^2 =/= m^2]. Q.E.D.

COROLLARY 1. [2n^2 =/= m^2] --> [2 =/= (m^2/n^2)] --> [sqrt(2) =/= m/n] for any m>n >=1. That is sqrt(2) is not a rational number or, if you prefer, is an irrational number.

As is easy to see, the Corollary 1 proves the Pythagoras Theorem-1.

Thus, one of the principal, as it were purely arithmetical, argument in favour of the RAA-2 being legitimate becomes invalid.

Now some words as to Robert Tragesser's Thesis II.
I believe that his intellectual curiosity "about the provenance and evolution of this <RAA-2> style of <Cantor's> argument" is extremely important for the Foundations of Mathematics, since if it will turn out that all other RAA-2 proofs in real mathematics can be proved by RAA-1, so that the legitimacy of RAA-2 usage in mathematics will become quite doubtful. I believe that the last problem merits a special investigation, and hope [HM]-listmembers will help to find out other proofs like the RAA-2 proof-1 of PYTHAGORAS' THEOREM-1 above.
At once remark here that other famous Pythagoras Theorem about the (potential) infinity (according to Aristotle, NON-finity) of the set of natural numbers, and the Euclidean Theorem about the infinity of the set of prime numbers are usually also proved by means of the RAA-2. But as is well known the both of the Theorems have direct proofs of their NON-finity (of the kind "if n then n+1") in the framework of Peano's arithmetics, and consequently also don't need the use of RAA-2 proofs.

Now about the evolution of Cantor's diagonal procedure.

In his two posts to FOM-list (12 Feb 01 02:35:58 and 12 Feb 01 20:20:45), V.Kanovei wrote:

BEGIN ^^^^^^^^^^^^^^^^^^^^^
"Once again, the proof <of Cantor's theorem> has two important issues,
1) technically - the diagonal method (which in fact was used, perhaps, earlier by Du Bois Reymond in "scaling" sequences, near 1872)
2) foundationally - the postulate that all elements of P(N) (or P(X),[...]) are "already given" and no new ones can appear in the course of the proof.
The rest is a couple of lines of ordinary transformations.
Something similar to 2) appears in many paradoxes, say those of Liar and Barber in the form of a mess between "has been" and "has been and will always be".
END^^^^^^^^^^^^^^^^^^^^^^^^

I don't agree with all these V.Kanovei's points.
1) the diagonal method has been used before Cantor not by Du Bois Reymond, but by Pythagoras (see below).
2) The V.Kanovei foundationally "important issue" in such a formulation, simply leads to the ancient problem of "speculative philosophy": is the notion itself of actual infinity consistent or it is not? The problem has been discussed from Aristotle's time without a success. So, as the history shows, the problem doesn't have a solution in such speculative formulation.
However, as is known, Cantor claimed all infinite sets actual, and first explicitly used that actuality in his proof as the statement "a given enumeration contains ALL real numbers", and then utilized it as a formal premise for the "constructive and algorithmical" building (deduction, creation, definition, etc.) of an (anti-diagonal) real number which differs from ALL elements of the given enumeration, and, consequently, it isn't contained in that enumeration. BTW, this "ALL" is a necessary condition (point 1) of Cantor's RAA-2 proof, since Cantor's diagonal method is simply inapplicable to potentially infinite sets of real numbers.
3) "The rest" <of the famous Cantor diagonal procedure> is not "a couple of lines of ordinary transformations", but the only line : -).
Indeed, using, for simplicity, the binary number system, we can write the main "constructive and algorithmical core" of Cantor's diagonal procedure (for real numbers) as follows:

IF a diagonal digit is 0 THEN the (corresponding Cantor's) anti-diagonal digit is 1, and vice versa.

Further, V.Kanovei wrote:
"Hardly one can find *tricks* of anything (especially in plural) in few lines of the <Cantor's> diagonal argument.

I believe that V.Kanovei here is not right at all for the following reasons.
In the paper "Mistake of Georg Cantor" (Voprosy Filosofii (Philosophy Problems), 2000, No. 2, 165-168), I proved (emphasize here, - proved, but I was not talking or claiming "some about something"), in particular, that
1) the famous Cantor diagonal procedure (CDP) is literally applied to any FINITE sequences of real number, that is the CDP doesn't distingwish FINITE sets from INFINITE sets just by their cardinalities, and, moreover, CDP uses only just the actuality of "the given enumeration containing ALL real numbers", and doesn't use the INFINITY of that enumeration,
2) in such the case, the CDP, without any change of its "constructive and algorithmical" nature and its final results, can be formulated as follows:

for any i>=1 we have a real number (future Cantor's anti-diagonal one), say, x*, which differs from the first i elements of the given enumeration by its (of Cantor's number x*) first i digits.

Now, consider
PYTHAGORAS' THEOREM-2. The set N={1,2,3,...} is infinite (more precisely, is NON-finite, according to Aristotle).
PROOF-3 (by the RAA-2). Assume that N is finite.
Then all elements of the set N can be written as the sequence ("enumeration"):

1,2,3, ..., n. (4)

Define a new mathematical object by the following (DIAGONAL) rule:

n*=1, and for any i>1 [n* = i+1]. (5)

Applying the rule (5) to the "enumeration" (4), we shall construct the diagonal number n*=n+1, which, according to Peano, is a natural number, i.e., it belongs to N, but, again according to Peano, for every i n* is greater than (and differs from) all first i elements of (4). As a result we have n*>n, and consequently n* differs from all elements of (4), i.e., the "enumeration" (4) doesn't contain all elements of the set N. The obtained contradiction, according to the RAA-2, "proves" that our assumption is false. Q.E.D.

As it's easy to see, the Cantor's argument is a deductive model (in Tarski's sense) of the traditional RAA-2 proof of the Pythagoras Theorem-2.
So, it was not Cantor, nor even Du Bois Reymond, but just Pythagoras who was a real inventor of the famous Diagonal procedure. Why didn't the Pythagorian Diagonal procedure ever lead to paradoxes? - Because before Cantor nobody counted the set N as an ACTUALLY infinite set containing ALL its elements.

Now we get one of a lot of the "pathological incidents" in Cantor's "Study of Transfinitum" which were predicted by Brouwer (see, e.g., the Fraenkel and Bar-Hillel "Foundations of Set Theory", 1966, p. 315. The Russian translation.).
Indeed, if we trust in Aristotle's "Infinitum actu non datur", there is no problem: simply consecutively applying the Pythagorian Diagonal Method to enumeration (4), we shall be creating new "diagonal elements" n+2, n+3, and so on, and adding them to the enumeration without any contradiction with Aristotle's NON-finity of N.
But if we, together with Cantor, state that N is an ACTUAL infinite set, i.e., N and the "enumeration" (4) contains ALL natural numbers, then we immediately get the contradiction generated by application of RAA-2 to such an ACTUALLY infinite set:

the new mathematical object n*, constructed according to the diagonal rule (5), using the only constructive, finite, Peano's operation "+1" (BTW, defined for finite numbers only!), is a finite natural number which differs from ALL finite natural numbers.

There are two ways to resolve the contradiction.
1) the new mathematical object n*, produced by applying the operation "+1" to all FINITE natural numbers, is not a finite number. It's very natural to call such the NEW object n* a (minimal) NON-finite or, say, TRANS-finite integer or a non-standard natural number, and to denote it by a NEW symbol, say, "OMEGA".
Thus, if N is actual, then we get (for the first time!) a rigorous proof of the existence of Cantor's "omega" by means of the Pythagoras Diagonal Method.
Further, according to the well-known Peano axiom: "IF "something" is an integer, THEN "something"+1 is an integer too", we can go on the well known, "transfinite" building:

"OMEGA", "OMEGA"+1, "OMEGA"+2, ..., and so on up to the famous e_0 (see above), and even much further.

2) In just such a logical situation with real numbers, Cantor prefers another solution: he states that his anti-diagonal object x* is, by definition, a real number differing, by construction, from all real numbers of the enumeraion of ALL real numbers, and concludes that no enumeration of real numbers contains all real numbers, interpreting the last as an uncountability of the real number set.
As was said above, applying this "logic" to natural numbers, we prove that no sequence (enumeration) of natural numbers contains all natural numbers. The last can also be interpreted as a direct consequence of Aristotle's axiom: the set of "all" natural numbers is POTENTIALLY infinite, i.e., it can't contain all its elements, by definition.

Thus, the evolution of the Diagonal Method is as follows:

FROM: Pythagoras, Potential Infinity + RAA-2, with no paradox,
TO: Cantor, Actual Infinity + RAA-2, with a lot of paradoxes, and
TO: the Great Third Crisis in Foundation of Mathematics
which, according to D.Hilbert (it was said in 1925), H.Weil (it was repeated in 1946), A.Fraenkel and Y.Bar-Hillel (it was repeated in 1958), and, how the present FOM-discussion shows, "is going on today".

Best regards,

-- AZ

###################################################

From: "Alexander Zenkin" <alexzen@com2com.ru>
To: "Michael Detlefsen" <detlefsen.1@nd.edu>
Subject: Re: conference announcement
Date: Sat, 24 Feb 2001 18:05:54 +0300

Dear the Conference Director, Michael Detlefsen,

I would like to take part in your unique conference "LOGICISM AND THE PARADOXES: A REAPPRAISAL".
I have what to say (enclosed) and hope this will be interesting for the conference participants. Could I hope to get a stipend for me and Anton A.Zenkin (he is my permanent co-author and translator, BTW I am not too young to be without his helping, alas) to defray the expenses connected with our participation in the conference.

Best regards,

Prof. Alexander Zenkin
Leading Research Scientist,
Computing Center of the RAS,
Profsoyuznaya street, 156-3-151,
117465, Moscow,
Russia

e-mail: alexzen@com2com.ru

THE ENCLOSED ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

ABOUT THE NATURE OF THE "LIAR", "BARBER", etc.

Alexander A.Zenkin (alexzen@com2com.ru)

Computing Center of the Russian Academy of Sciences

1) As is known, the famous Epimenides "Liar" sounds so: "I am a Liar". - "Am I a Liar?"
IF I am a Liar THEN my statetment is false, and, consequently, I am not a Liar.
But IF I am not a Liar, THEN my statement is true, and, consequently, I am a Liar.
Write the said shortly (here A="I am a Liar"):

[A -->NOT- A] &[NOT- A -->A] (1)

2) Consider the Russell "Barber": a village barber must shave all those dwellers of his village who don't shave themselves.
Denote: V is a set of all dwellers of a given village, and P(x1, x2) = "x1 shaves x2".
Then the barber y Î V is defined by the formula:

For any x Î V

[[P(x,x) -->NOT-P(y,x)]&[ NOT-P(x,x) -->P(y,x)]] (2)

It is a very normal definition of a logical object called a barber. But if
we ask whether the baber himself must shave self, i.e., if we assume a case
x=y in (2), we have the paradox

[[P(y,y) --> NOT-P(y,y)]&[ NOT-P(y,y) -->P(y,y)]] (3)

of the same type like the "Liar" (1).

BTW, the "Barber" has the following evident set-theoretical interpretation.
Suppose N1={x | P(x,x)} and N2={x | NOT-P(x,x)}, where N=N1+N2 is a set of all sets, and predicate P(a,b) = "a is an element of b". Then the "diagonal" set y of all sets that are not elements of themselves is defined as:

"x Î N [P(x,x) --> ØP(x,y) ] & [ØP(x,x) --> P(x,y)],

and we have the same "Barber" (3) in the common set-theoretical form:

[[y Î N1] --> [y Ï N1]] & [ [y Ï N1] --> [y Î N1]] (3a)

3) From this pure formal point of view, the known Ramsey classification which refers "Liar" and "Barber" to different classes is quite doubtful [1]. Therefore we use the automated classification system VISAD (VISual Analysis of Data) [2] to make a new computer classification of main paradoxes of logic and mathematics. The interactive system VISAD did make the following
prompt. In a natural language, we frequently use the equivalences of the type: "small" is "NOT-large", "foolish" is "NOT-clever", "false" is "NOT-true", etc., therefore we have changed "I am a Liar" into "I am a NOT-Truer", "the given statement is false" into "the given statement is NOT true", etc. Then, the system VISAD generated automatically two classes of paradoxes, say K1 and K2, and formulated a notion about every class as a set of the most important characteristics of objects of the corresponding class. Thus, the "Liar" and all Russel's paradoxes including the "Barber" were placed into the class K1. The automatically created notion on the class K1 contained two main characteristics: "SELF-applicabilitity" and "negation",
or, shortly, the logical construction "SELF+ NOT".
4) Using a prompt being contained in some very deep G.Soros ideas as to a feedback influence of a logic of our decisions upon the complex systems behaviour [3], we interpreted that logical construction "SELF+ NOT" as a negative feedback (FB). As is well known, the simplest and direct PHYSICAL realization of such the negative FB is the analogue computer summator with two inputs, one of which is a negative FB (denote such the summator M_P). In the usual situation, such the M_P transforms any input voltage, say X, into the output voltage, say Y, according to the well-known law of addition of input voltages: Y= - (X+Y), from where Y= -1/2X. So, the M_P is a PHYSICAL model of the LOGICAL construction "SELF+ NOT", and the existence itself of such the really working physical device proves the

THEOREM 1. The properties of the self-applicability (SELF) and the negation (NOT) of statements are necessary, but not sufficient conditions of their paradoxicality.

5) As is known the electric current velocity V<300 000 km/sec. Assume that V=oo. In such the case a) our model M_P becomes NON-physical, therefore b) the "current" runs from the summator input to its output without any resistance, delaies and a loss of a magnitude, and only inverts a sign of its voltage (the last is saved as the main property of M_P). So, our M_P begins to work as a switch of the output voltage sign with an infinite frequency, and, if the input "voltage" is equal, say, to -1 Volt, we have a new "physical" paradox:

[-1 -->] 1 -->0 -->1 -->0 -->1 -->0 -->... (4)

This paradox proves the

THEOREM 2. The necessary and SUFFICIENT conditions of the transformation of a really working physical system (the summator) into a NON-physical paradoxical system (4) are
1) the SELF-applicability (FeedBack) of the input signal X,
2) the negative character (NOT) of this FeedBack, and
3) the oversteping of a physical parameter (here - V=oo) the limits of PHYSICS, i.e., such the parameter becomes NON-physical.

6) Using A.Tarski's theory of deductive systems [4], we proved that M_P is an isomorphic model, say M_L, of the theory of logical proof. Indeed, if the input signal of M_L is an (inauthentic) statement X="S is P", the summing block is a logical proof block, then the output signal of M_L is Y="S is (or isn't) P" is the proved statement with an authentic value of the copula of the statement Y. BTW, the proof is understood in the usual logic sense as a sequence

f_1, f2_, ... , f_n, (5)

where for any i f_i is an axiom, a proved theorem, or follows from preceding formulas, and f_n = Y; the natural isomorphic analogy of the physical summing up of votages in M_P is the logical assignment operation of the proved connective (is/isn't) of Y to X in M_L. Remark that the model M_L works as an invertor only in the cases when we didn't guess a right value of the copula in the input signal X (what is according to the arbitrarity of a voltage sign of the input X in M_P). It's also very natural, that under, this isomorphism between M_P and M_L, the velocity of a logical proof is defined as V=1/n, where n is a length of a proof in (5).

7) Now, if the input logical signal is X="X is false", i.e., the "Liar" (according to Kleene [5]) in its strong form, then we have V=1/n=oo, i.e., n=0, our normal model M_L is turned into a qusi-logical switch generating the output signal Y of the form (here Y="X is false"= the "Liar"= the "Barber" = etc.):

Y --> NOT-Y --> Y --> NOT-Y --> Y --> NOT-Y --> Y -->. . . (6)

The last allowed to prove the following

THEOREM 3. The necessary and SUFFICIENT conditions of the transformation of a really working logical system (the proof block) into a NON-logical paradoxical system (6) are
1) the SELF-applicability (FeedBack) of the input statemenet X,
2) the negative character (NOT) of this FeedBack, and
3) the oversteping of a logical parameter (here - n=0) the limits of LOGICS, i.e., such the parameter becomes NON-logical.

Why and how does the length of a "proof" in (5) turn out n=0 for the "Liar"? There is the following simple and natural explantion of that fact.
1) Any statement X="S is P" has two information channel (through the subject S and the predicate P) to elicit an information from an outer world (a data-base) which is necessary for constructing the proof (5).
2) Any self-applicable statement X="X is P" has the only channel (through the predicate P only) to construct the proof (5). But it is not enough to be paradoxical. For example, the self-applicable statement X="X contains 1000 symbols" is false one without any paradoxicality.
3) The "Liar" X="X is false" has no connection with an outer world at all: the subject channel is closed due to the self-applicability, but all the content of the predicate-constant P="to be false" becomes exhausted by the constant statement-indication: "(any) current value of the copula of X is wrong", that is too has no connection with the outer world. It can be said that the "Liar" is a Kant noumen, a quasi-logical "thing in self". Just therefore the "proof" (5) for the "Liar" has n=0.
It's remarkable, that from this point of view, the statement X="X is true" is a consistent statement, but also is one having no connections with the outer world, i.e., it is too a Kant logical "thing in self" with an empty content which is unprovable not thanks to Goedel, but solely because there is no possibility to construct any proof of the kind (5), and therefore n=0 in (5).

So, the true nature of the "Liar", "Barber", etc. is not an arbitrary pared-down (by quite obvious pure psychological reasons) fragment (1) or (3), but it is the (potentially) infinite "argumentation" of the kind (6). And there are not logical or mathematical reasons to stop the process (6). Some non-trivial connections between the "Liar" and G.Cantor's proof of the real number uncountability are considered in the papers [6-8].

REFERENCES

1. A.A.Fraenkel, Y. Bar-Hillel, Foundation of Set Theory. - North- Holland Publishing Company, Amsterdam, 1958.
2. A.I.Zenkin, A.A.Zenkin, On One Method for Optimal Classifications Construction. - Banach Center Publications, Vol. 7, 197-204 (1982), Poland.
3. G.Soros, Soviet System: To Open Society. - Moscow, Political Literature Publishing, 1991.
4. A.Tarski, Introduction To Logic and Methodology of Deductive Sciences. - Moscow, 1948.
5. Stephen Cole Kleene, IntroductionTo Metamathematics. - D.Van. Nostrand Company, Inc., New York - Toronto, 1952
6. A.A.Zenkin, The Time-Sharing Principle and Analysis of One Class of Quasi-Finite Reliable Reasonings (with G.Cantor's Theorem on the Uncountability as an Example) - Doklady Mathematics, vol 56, No. 2, pp. 763-765 (1997). Translated from Doklady Akademii Nauk, Vol 356, No. 6, pp. 733 - 735.(1997).
7. A.A.Zenkin, George Cantor's Mistake. - Voprosy Filosofii (Philosophy Problems), 2000, No. 2, 165-168.
8. A.A.Zenkin, New Approach to the Paradoxes Problem Analysis. - Voprosy Filosofii (Philosophy Problems), 2000, No. 10, 79-90.

THE END^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

----- Original Message -----
From: Michael Detlefsen <detlefsen.1@nd.edu>
To: <fom@math.psu.edu>
Sent: Friday, February 23, 2001 4:19 PM
Subject: FOM: conference announcement

> Dear FOMers,
>
> A few days ago, I sent out a conference announcement that contained a few
> errors of detail. Below please find the corrected announcement.
>
> Best regards,
> Mic Detlefsen
>
>
> ++++++++++++++++++++++++++++++++++++++++++
>
> LOGICISM AND THE PARADOXES: A REAPPRAISAL
> University of Notre Dame
> March 29-31, 2001
>

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
"Professional snobbery leads to intellectual blindness and to moral degradation." - J. Wiser