NetNews Usenet Archive 1992 #26/NN_1992_26.iso/spool/sci/logic/1959

Internet Message Format | 1992-11-08 | 1.8 KB

Path: sparky!uunet!noc.near.net!nic.umass.edu!m2c!jjmhome!smds!rh
From: [e-mail] (Richard Harter)
Newsgroups: sci.logic
Subject: Re: Impredicativity - was: Russell's Paradox
Message-ID: <[e-mail]>
Date: 8 Nov 92 07:41:20 GMT
References: <[e-mail]> <[e-mail]>
Reply-To: [e-mail] (Richard Harter)
Organization: Software Maintenance & Development Systems, Inc.
Lines: 24

In article <[e-mail]> [e-mail] (Randall Holmes) writes:

>The open sentence "x is not an element of x" is in fact true or false
>for each x, in any reasonable set theory.  But it does not define a
>set.  The reason it does not define a set is that _if it defined a
>set_ there would be a problem with substituting some particular x (the
>extension of this property) into the sentence.  There is in fact no
>instance in either ZFC or NFU in which "~x E x" fails to be true or
>false for some x; in ZFC, it is _always true_, and in NFU it is
>sometimes true and sometimes false.

This is the classical approach to dealing with the paradoxes, i.e. to
claim that there are open sentences whose extensions are not sets.
However one can also take the view that "x is a member of y" is not
a proper relationship.  In other words it is not the case that for
all x,y either T(x a member of y) or F(x a member of y).  I have come
to have the opinion that the paradoxes establish that membership is
not a two valued function, and more generally, truth is not a predicate.

--
Richard Harter: SMDS Inc.  Net address: [e-mail] Phone: [phone]
US Mail: SMDS Inc., PO Box 555, Concord MA 01742.    Fax: [phone]
In the fields of Hell where the grass grows high
Are the graves of dreams allowed to die.