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In article <25916@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:
>In article  <1992Nov04.233228.16942@Cookie.secapl.com> Frank Adams writes:
>]In article <24780@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:
>]>This (or so I claim) is the single, simple, obvious cause of Russel's
>]>paradox.  In particular, R = {x : x not elem x} is a definition of
>]>type (2) that contains an implicit reference to R because x quantifies
>]>over a set that contains R.  (Or you could show the recursion in the
>]>definition of elem as I did before)...
>]
>]It appears to me that you are complaining not about elementhood being a
>]proposition, but about the unrestricted range of the quantification in the
>]axiom of comprehension.  Note that your complaint about the implicit
>]reference to R is just as applicable to R = {x : x = x}, which gives the
>]universal set without using elementhood.
>
>No, I am not complaining about elementhood being a proposition, I am
>complaining about a syntactic circularity begin treated as though it
>had semantic significance.  Neither am I concerned about the
>unrestricted range of the quantification.  As the set {x : x = x}
>clearly shows, there is no inconsistency inherent in either
>self-membership or impredicativity.  I claim that {x : x = x} is a
>perfectly reasonable set and that the mere fact that ZF proves the
>opposite is enough to show that ZF is not an adequate axiomatization
>of set theory.

Dare I concur?

>-- 
>					David Gudeman
>gudeman@cs.arizona.edu



-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu

In article <25916@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:
>No, I am not complaining about elementhood being a proposition, I am
>complaining about a syntactic circularity begin treated as though it
>had semantic significance.  Neither am I concerned about the
>unrestricted range of the quantification.  As the set {x : x = x}
>clearly shows, there is no inconsistency inherent in either
>self-membership or impredicativity.  I claim that {x : x = x} is a
>perfectly reasonable set and that the mere fact that ZF proves the
>opposite is enough to show that ZF is not an adequate axiomatization
>of set theory.

All right; but what would you put in it's place?  I'm not willing to call
something "set theory" unless it has *some* recognizable form of the axiom
of comprehension.  Can you formalize your ideas?

In article <1992Nov09.172532.43648@Cookie.secapl.com> frank@Cookie.secapl.com (Frank Adams) writes:
>In article <25916@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:
>>No, I am not complaining about elementhood being a proposition, I am
>>complaining about a syntactic circularity begin treated as though it
>>had semantic significance.  Neither am I concerned about the
>>unrestricted range of the quantification.  As the set {x : x = x}
>>clearly shows, there is no inconsistency inherent in either
>>self-membership or impredicativity.  I claim that {x : x = x} is a
>>perfectly reasonable set and that the mere fact that ZF proves the
>>opposite is enough to show that ZF is not an adequate axiomatization
>>of set theory.
>
>All right; but what would you put in it's place?  I'm not willing to call
>something "set theory" unless it has *some* recognizable form of the axiom
>of comprehension.  Can you formalize your ideas?

i.  Russell's paradox does have _semantic_ consequences.  Whatever
membership is, there is no set of all sets that are not members of
themselves.  Certainly the mechanics of applying the paradox are
syntactical -- so what?

ii.  NFU:

	Axiom 1.  Objects with elements that have the same elements
	are equal.  (this allows multiple objects with no elements).

	Axiom 2.  Stratified comprehension (any instance of
comprehension which would make sense in the simple theory of types
under some assignment of types to its variables holds -- further
details available on request).

	Axiom 3.  The universe is infinite.

	Axiom 4.  The universe can be well-ordered (choice).

	This is a set theory with the same consistency strength as
Russell's theory of types; it is untyped (a set definition makes sense
if it would make sense in the theory of types, but objects in this
theory are not typed).  {x|x=x}, the universe, exists (the definition
would make sense in TT); {x|~x E x}, the Russell class does not exist
(the definition would not make sense in TT -- if the first x had type
n, the second would have type n+1).

	This theory is known to be consistent (R. B. Jensen, On the
consistency of a slight (?) variation of Quine's New Foundations,
Synthese, vol. 19), and adequate for most mathematical purposes.  It
-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu

In article <1992Nov10.001234.18488@guinness.idbsu.edu> holmes@garnet.idbsu.edu (Randall Holmes) writes:
>In article <1992Nov09.172532.43648@Cookie.secapl.com> frank@Cookie.secapl.com (Frank Adams) writes:
>>All right; but what would you put in it's place?  I'm not willing to call
>>something "set theory" unless it has *some* recognizable form of the axiom
>>of comprehension.  Can you formalize your ideas?
>
>ii.  NFU:

Yes, Randall, I know that's what *you* would put in its place.  If you can
show that NF[U]'s comprehension rule is free from the kind of objection David
is making, I will grant that you have given a good reason for taking NFU
seriously.  It isn't enough to show that a few examples work, however.

In <1992Nov10.001234.18488@guinness.idbsu.edu> holmes@garnet.idbsu.edu (Randall Holmes) writes:

>i.  Russell's paradox does have _semantic_ consequences.  Whatever
>membership is, there is no set of all sets that are not members of
>themselves.  Certainly the mechanics of applying the paradox are
>syntactical -- so what?

I don't see that the Russell paradox of itself requires the
non-existence of this object (RP).
The most I could deduce from it is that the predicate

	RP is a member of RP

has no truth value. I.e. the function 'is a member of' is not total.

--
When soldiers form lines or hollow squares, you call it reason.
When wild geese in flight take the form of a letter V, you say instinct.
When the homogeneous atoms of a mineral arrange themselves into shapes
mathematically perfect you have nothing to say. You have not even invented a name to conceal your heroic unreason."