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From: g-rh@cca.UUCP (Richard Harter)
Newsgroups: net.philosophy,net.math
Subject: Re: Ineffable numbers
Message-ID: <5086@cca.UUCP>
Date: Wed, 27-Nov-85 05:56:28 EST
Article-I.D.: cca.5086
Posted: Wed Nov 27 05:56:28 1985
Date-Received: Sat, 30-Nov-85 00:27:26 EST
References: <2467@sjuvax.UUCP> <11700014@inmet.UUCP> <667@spar.UUCP> <>
Reply-To: g-rh@cca.UUCP (Richard Harter)
Organization: Computer Corp. of America, Cambridge
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Xref: watmath net.philosophy:3263 net.math:2569

In article <> lazarus@brahms.UUCP (Andrew J Lazarus) writes:

>These so-called 'ineffable' numbers date back to one of the
>turn-of-the-century paradoxes.  I think it's the Burali-Forti
>paradox, but that course was many years ago and I don't remember.
>

The Burali-Forti paradox: Consider the set of all ordinals, W.
This set is ordered by the well-ordered order relationship of
ordinals and has, in turn, and ordinal greater than that of any
ordinal in W.  If ordinals are defined in the usual manner in set
theory, this can be stated more succinctly by saying that the set
of all ordinals, W, is an ordinal larger than any member of W.

The Cantor paradox:  Cantor's theorem says that the set of all 
subsets of a given set has a cardinality greater than that of
the given set.  Consider the universal set, V, of all sets.
It is equal to its set of all subsets and hence its cardinality
is the same as that of its set of all subsets.

The Russell paradox:  Consider the set of all sets which do not
contain themselves as members.  From its definition it contains
itself if and only if it does not contain itself.  Rusell's set
arises naturally in applying the Cantor diagonalization process
used in proving Cantor's theorm to the universal set V.

	These paradoxes (antimonies is the term usually used)
	are the major ones of naive set theory.  They are avoided
	in axiomatic set theory by restricting the admissable
	predicates which determine sets.  There are also a
	class of so-called semantic paradoxes of which the most
	important is Richard's paradox.  This is, in effect,
	the paradox being discussed here.

Richard's paradox:  Consider the set of all numbers between o and 1
that can be uniquely characterized by English words.  Let R be the
enumerated set of such numbers.  Define r as the real number between
0 and 1 whose n'th digit is the cyclic sequent of the n-th digit
of the n-th number in the enumeration under consideration.  Then
r has been defined by the above English sentence and yet is not a
member of R by construction.

	Semantic paradoxes are resolved by recognizing that not
	all definitions are effectively computable.  In the case
	of Richard's paradox the problem is that there are English
	sentences which purport to specify numbers but for which
	there is not way to determine what that number is.  The
	particular importance of Richard's paradox is that Godel's
	incompleteness theorem is based on a formalization of
	Richard's paradox.

				Richard Harter, SMDS
				decvax!cca!g-rh