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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