[FOM] 3 postings from Alexander Zenkin

Sat Feb 1 11:34:47 EST 2003

MESSAGE-1

From: alexzen at com2com.ru
To: fom at cs.nyu.edu
CC: Alasdair Urquhart [urquhart at cs.toronto.edu]
Subject: RE: [FOM] Cantor's argument

-----Original Message-----
From: Alexander Zenkin [mailto:alexzen at com2com.ru]
Sent: Saturday, February 01, 2003 12:08 AM
To: 'Alasdair Urquhart'
CC: fom at cs.nyu.edu
Subject: RE: [FOM] Cantor's argument

Indeed, a lot of 'something' had been done
in foundations of mathematics (more precisely - in meta=mathematics and
axiomatic set theory) in the last 100 years.
But it does not mean that today, 'in the early 21st century', anybody
knows how to solve the "Liar", the "Barber", the Russell's, Richard's,
Berry's, Grelling's etc. paradoxes.
I mean just to solve, but not to evade the paradoxes problem as it was
done by Russell, Hilbert, Brouwer, etc.

AZ

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf
Of Alasdair Urquhart
Sent: Friday, January 31, 2003 8:47 PM
To: fom at cs.nyu.edu
Subject: [FOM] Cantor's argument

[ . . .]
Arguments of this sort have been discussed since the
early 1900s, and form the basis of the Richard paradox,
the Berry paradox and the Grelling paradox.
It is unproductive to discuss such arguments
in the early 21st century as if nothing had been done
in foundations of mathematics in the last 100 years.

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MESSAGE-2

From: alexzen at com2com.ru
To: fom at cs.nyu.edu;
CC: Vladik Kreinovich [mailto:vladik at cs.utep.edu]
Subject: Re: [FOM] Cantor's argument

The 'Liar' is (A='I am a liar'):

[A -- > not-A]&[not-A -- > A]   (*)

The 'Barber' is (A = 'Barber must shave himself'):

[A -- > not-A]&[not-A -- > A]   (**)

So, the liar paradox is not completely different from the barber
(Russell's) paradox. Moreover, they are completely identical from the
formal point of view, are not?

AZ
-----Original Message-----
From: Vladik Kreinovich [mailto:vladik at cs.utep.edu]
Sent: Friday, January 31, 2003 11:55 PM
To: alexzen at com2com.ru
Cc: fom at cs.nyu.edu
Subject: RE: [FOM] Cantor's argument

 > The Epistle of Paul to Titus: "One of themselves < Cretians >, even a
 > prophet of their own, said, The Cretians are always liars, . . ."
 > Shortly: a Cretian states: "I am a Liar".
 > What 'instructions issued to the Liar' are inconsistent here?

The liar padarox is completely different from the barber (Russell's)
paradox.
What I said I said about Russell's padraodx, and you are right it is not

applicable to the liar paradox.

 > -----Original Message-----
 > From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf
 > Of Vladik Kreinovich
 > Sent: Friday, January 31, 2003 4:37 AM
 > To: fom at cs.nyu.edu; Dean.Buckner at btopenworld.com
 > Subject: Re: [FOM] Cantor's argument
 >
 > There actually is a well known rephrasing of Russell's paradox: a
barber
 > is assigned to a miliary unit with the task of shaving everyone who
does
 > not shave themselves. The question is: does this barber shave himslef
or
 > not? If he does, he violates his instruction because he then shaves a
 > person who shaves homself;
 > similarly, if he does not shave himself he also violates his
instructions.
 >
 > The conclusion here, of course, is not that the number of soldiers is
 > uncountable but that the instructions issued to the barber are, if
taken
 > literally, inconsistent.

MESSAGE-3

From: alexzen at com2com.ru
To: fom at cs.nyu.edu
CC: Vladik Kreinovich [mailto:vladik at cs.utep.edu]
Subject: Re: [FOM] Cantor's argument

The Epistle of Paul to Titus: "One of themselves < Cretians >, even a
prophet of their own, said, The Cretians are always liars, . . ."
Shortly: a Cretian states: "I am a Liar".
What 'instructions issued to the Liar' are inconsistent here?

AZ

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf
Of Vladik Kreinovich
Sent: Friday, January 31, 2003 4:37 AM
To: fom at cs.nyu.edu; Dean.Buckner at btopenworld.com
Subject: Re: [FOM] Cantor's argument

There actually is a well known rephrasing of Russell's paradox: a barber
is
assigned to a miliary unit with the task of shaving everyone who does
not shave themselves. The question is: does this barber shave himslef or
not? If he does, he violates his instruction because he then shaves a
person who shaves homself; similarly, if he does not shave himself he
also violates his instructions.
The conclusion here, of course, is not that the number of soldiers is
uncountable but that the instructions issued to the barber are, if taken

literally, inconsistent. 