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In article <26788@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:
>A "recursive definition" of a name X is a sentence of the form
>
>  X := E
>
>where E is an expression that contains X as a free variable.  Naively we
>might associate with such definitions the axiom
>
>  X := E => X = E
>
>To express what a definition means.  (We also need some syntactic rules
>to allow the introduction of definitions so that the same name is not
>used for different purposes.)
>
>Here are some examples of recursive definitions:
>
>(1) x := x + 1
>(2) x := x * 2
>(3) x := 1/x
>(4) x := x * 1

It seems to me, David, that you cause unnecessary confusion by calling
these and, as you do later in your post, the definition of the Russell
set recursive definitions.  By my lights, the usual understanding is
that a recursive definition is one in which a function f is defined on
a well-founded relation R such that for each object b in the field of
R, f(b) is defined in terms of f(x) for x such that xRb.  (The
legitimacy of such definitions in set theory is of course a theorem to
be proved; I believe von Neumann was both the first to see this and
the first to provide the requisite proof.  Levy's Basic Set Theory
has a nice, detailed proof in ZF.)  Unless I'm really missing
something, the above are not recursive definitions in this sense, nor
is the definition of the Russell set.

Since you say at the beginning of your note what *you* mean by a
recursive definition, my comment here is essentially a quibble that
doesn't address the substance of your remarks.  But if you are in fact
using such a deeply entrenched term in a nonstandard fashion, you
would seem to be a lot better off calling the notion you've defined
something else.

--Chris Menzel
  Philosophy Department
  Texas A&M University