🚀 go-pugleaf

Lighten up Chris.  Just because you don't understand someone, doesn't
mean that it is necessarily the other person who is confused.  Maybe,
just _maybe_ the problem is with your preconceptions rather than my
background.

Here are a few salient points: (1) I am not talking about axiomatized
set theories, I'm talking about intuitive, metaphysical set theory.
Your points about specific axiomatizations are quite beside the point.
(2) I've read Russel's arguments about impredicative definitions, and
I'm talking about something quite different.  (3) There certainly _is_
a problem with recursive definitions as any logician will tell you.
You can't assume that a recursive definition actually defines anything
without a seperate proof of some sort.  Otherwise, I could recursively
define "x = 1 + x" as a number that is one greater than itself. (4) It
may very well be that I _am_ confused about something, but you won't
be able to point it out until you understand what I'm saying in the
first place.

In article  <1992Oct31.045821.6028@tamsun.tamu.edu> Chris Menzel writes:
]In article <24780@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:
]DG:
]>It looks like you are assuming that all first order formulas express
]>properties.  
]
]As clearly noted, the property talk is informal and intuitive, and can
]be dispensed with--though I do believe that every first-order formulas
]expresses a property, relation, or proposition (but would want to be
]clear about what the primitives of the language are, and what it's
]logic is).

It isn't clear to me that property talk can be dispensed with.  To a
realist who beleives that no language can express all properties, it
certainly cannot be dispensed with.

]DG:
]>This is clearly false since Russel's Paradox can be
]>expressed in such formulas without reference to sets.
]
]Not so clear; there are of course property theoretic versions of the
]paradox, but don't forget that it takes more to generate the paradox
]than the assumption that every FO formula with a single free variable
]expresses a property.  In particular, one needs the property theoretic
]equivalent of lambda conversion

No, all one needs is the ability to define properties recursively.
Lambda abstraction is only one way to do this.  Since I claim that the
paradox is caused by recursive definition, this just proves my point.

]CM:
]>]Say that a *condition* is a first-order
]>]formula A(x) (in an interpreted language) with at most a single free
]>]variable x, and that an object m *satisifies* A(x) just in case A(x)
]>]is true when x is assigned m.  Then one can say that RP showed that
]>]there are conditions A(x) such that there is no set consisting of
]>]exactly the things that satisfy A(x).  
]
]DG:
]>If A(x) is not a property at all, but merely a syntactic well-formed
]>formula that does not express a proposition, this is merely a
]>syntactic curiosity.
]
]Your assertion is groundless, and, as far as I can see, confused.
]This is very straightforward, standard stuff.  Is it also a syntactic
]curiousity that in ZF set theory theory, every condition of the form x
]\in a & A(x) *does* pick out a set?  What is the difference?

That assertion is an assertion about the theory ZF, and is therefore
an assertion about syntax.  I wouldn't call this one a "curiosity"
since it is more important.  The other one is a curiosity because it
is just one of many examples of how recursive definitions can cause
problems.

]Well, you may not like that as an explanation of RP, but mysterious it
]is not; this is very basic set theory.  The iterative hierarchy, the
]notion of rank, the set/class distinction, these are all perfectly
]well defined and (modulo the intrinsic complexities of the hierarchy)
]well understood.

I know basic set theory, thank you.  My objection is not about whether
these notions are well defined, but whether they are useful.  Or, more
specifically, whether they are the best solution to RP.  They are
mysterious from a metaphysical standpoint, not from the standpoint of
the syntactic theories.

]DG:
]>In the first place, I have no
]>reason to supose that there is any metaphysical significance to the
]>iterative hierarchy.  
]
]What ARE you talking about?

I'm talking about an intuitive, metaphysical set theory, as I said in
the very first article.  Do you know the difference between realism
and formalism?  Do you know the difference between a theory and a
formal theory?  Have you ever read anything by Frege?  If not, then
you have no idea what I'm talking about, and we are not communicating
at all.

]DG:
]>]>Take the (naive) axiom of comprehension
]>]>
]>]>  y elem {x : p(x)} <=> p(y)...
]
]This is not comprehension; naive comprehension is the principle that
]every condition determines a set.

This is a very common way to express comprehension, I did nothing
unusual.  However if you don't like it, you can divide it up into two
different statements and still carry the argument through.

]DG:
]>[Lengthy musings on definitions deleted]
]>
]>This (or so I claim) is the single, simple, obvious cause of Russel's
]>paradox.  
]
]Two l's in "Russell".

This is another trivial and unimportant syntactic argument.

]DG:
]>In particular, R = {x : x not elem x} is a definition...
]>that contains an implicit reference to R because x quantifies
]>over a set that contains R.  
]
]So, after all that, your problem is with the impredicativity of the
]definition of the Russell class, just like Poincare and the early
]Russell.  Why didn't you say so in the first place?

If you had read my "lengthy musings on definitions" you would have
seen where my argument is different.  In particular, I'm not concerned
with the fact that the quantifer quantifies over things which are not
known to exist, I'm concerned that something is being defined by an
equation that may not have a unique solution.  The impredicativity is
important only for the way it allows recursion to sneak in even though
the name being defined occurs on only one side of the equation.
-- 
					David Gudeman
gudeman@cs.arizona.edu