Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!mnetor!uunet!seismo!mcvax!unido!ecrcvax!andy From: andy@ecrcvax.UUCP (Andrew Dwelly) Newsgroups: sci.math,sci.math.symbolic,sci.philosophy.tech Subject: Re: Russell's set of sets which... paradox Message-ID: <423@ecrcvax.UUCP> Date: Mon, 27-Jul-87 06:34:18 EDT Article-I.D.: ecrcvax.423 Posted: Mon Jul 27 06:34:18 1987 Date-Received: Tue, 28-Jul-87 02:09:34 EDT References: <1214@utx1.UUCP> <6678@reed.UUCP> Reply-To: andy@ecrcvax.UUCP (Andrew Dwelly) Organization: ECRC, Munich 81, West Germany Lines: 35 Keywords: set theory, paradox, logic Xref: mnetor sci.math:1662 sci.math.symbolic:107 sci.philosophy.tech:307 Regarding this paradox, G Spencer Brown, makes an interesting comment in his book "The laws of form" (Dutton, New York) "Recalling Russell's connection with the Theory of Types, it was with some trepidation that I approached him in 1967 with the proof it was unnecessary. To my relief he was delighted. The Theory was, he said, the most arbitary thing he and Whitehead had ever had to do, not really a theory but a stopgap, and he was glad to have lived long enough to see the matter resolved. Put as simply as I can make it the resolution is as follows....." Spencer Brown introduces the idea of an imaginary boolean, the counterpart of an imaginary number. He justifies this by showing that all the self referential paradoxes "solved" by the theory of types are no worse than "This statement is false" (This step is not included in the text, is it valid ?). He then draws the readers attention to the counterpart in ordinary equation theory. X^2 = 0 transposing X^2 = -1 dividing both sides by X X = -1/X which is self referential. It is common knowledge that the introduction of imaginary numbers tidies up this kind of paradox. Hence imaginary booleans. The whole area of the "laws of form" seems to have not been touched since. Does anyone know of new applications/developments/refutations ??? Andy