Re: Contradiction or paradox

On May 18, 8:33 am, Charlie-Boo <shymath...@xxxxxxxxx> wrote:
Defining Q to be Q={Q} no more constructs a Quine atom than defining
N to be a counter-example to Goldbach's Conjecture constructs a
counterexample.

How would you construct such a set? My solution is the first Google
match on Quine Atom.

http://cs.nyu.edu/pipermail/fom/2002-September/005844.html

If you've ever been curious about the fact that no one on FOM replied
to your request for comments, I'd suggest you have a look at Peter
Aczel's well-known 1983 study Non-Well-Founded Sets, where the
mathematics of structures such as those in your little example is
studied in depth.

A. You're chomping

"champing" (Sorry, grammar geek here.)

at the bit to prove me wrong. (You imply that repeatedly.)

Not at all. What I have on several occasions tried to point out is,
not that you are wrong, but that what you are doing doesn't qualify as
mathematics -- not, as you like to imply, because it runs contrary to
received wisdom. Rather, the problem is simply that your efforts do
not satisfy even the most minimal standards of rigor required for
something to *count* as mathematics. The problem with your little
example of a Quine atom, in particular, is not so much that it is
wrong but that is underspecified. You don't provide enough background
for it to count as a piece of mathematics. Additionally, you tend
then to present such underspecified results as if they were something
new. Notably, set theoretic structures containing Quine atoms are --
and had been for many years at the time of your FOM post -- extremely
well known and well understood; the two references I provided for you
study them in depth. (Aczel's study bases non-well-founded sets on
the theory of labeled graphs and uses that as a basis for proving what
he calls the Solution Lemma that characterizes precisely when a set is
a solution to a system of equations. Barwise and Moss go in the other
direction and start with the Solution Lemma as an axiom and derive the
correlations between non-wf sets and labeled graphs.)

To give just a bit more substance to my point about your example being
underspecified, here's what you say (I restrict attention to your
function s1):

...
s1(x) = { x(x) }

...s1 takes in function x, applies x to x, and returns
the set containing the value of x applied to x.

1. Consider the following specific set:

s1(s1) = { s1(s1) }

Thus s1(s1) is a Quine atom, a set that contains only itself.

Ok, that might be. What's missing is the specification of a framework
in which the claim can be assessed. Currently it is about as much an
example of a Quine atom as "Let n be an even number that is not the
sum of two primes" is a refutation of Goldbach's Conjecture ;-).
There are several ways to approach the problem here. First, you need
to specify a framework on which the notion of a self-applicable
function makes sense. In standard ZF set theory, of course, as well
as in the typed lambda calculus, such functions are impossible. So
the question is, what framework are you using? Since you say that
s1(s1) is a set, the appropriate framework would appear to be some non-
well-founded set theory, such as ZFA -- ZFC - foundation + AFA, where
AFA is perhaps the most widely-adopted of several possible anti-
foundation axioms. Is that the one you are using? Your claim simply
has no purchase until we know.

Now let's look more closely at s1. You define s1 to be a function
such that s1(x) = { x(x) }. So, a few questions. First, what is x(x)
when x is not a function? What is, e.g., 0(0)? You could of course
dispense with that via some sort of convention, e.g., that x(x) = x
when x is not a function; or you could figure out some other way of
specifying s1 that doesn't use functional notation. Or perhaps you
intend your universe to consist entirely of functions, as in the
untyped lambda calculus. But you, not your reader, should be the one
to figure that out.

Second, and more seriously, how do we know that s1 exists at all? And
if it does, how do we know it is a set and not a proper class? In the
context of non-well-founded set theories, it is obviously NOT a set,
as there are proper class many self-applicable functions. (Proof: for
every set s, consider the function f_s = {<f_s,f_s>,<s,s>}, i.e., the
function that returns itself given itself and s given s, and is
undefined otherwise -- note I said *the* function f_s, which itself
embodies an assumption (or theorem) that needs to be made explicit
regarding the *uniqueness* of the solution of a system of equations.)
Hence, since (in standard non-well-founded set theories) a proper
class cannot be a member of a set or of itself, s1 cannot be in its
own domain. So your specification "s1(s1) = {s1(s1)}" is simply ill-
defined.

So, the point is, not that your little example is wrong or incoherent;
rather, it is that it is not framed in the context of enough
mathematics for the claim even to be assessed. This is exactly the
problem that is pervasive in your Arxiv paper where you attempt to say
what CBL is. Every single one of your "theorems" take a form similar
to your "demonstration" above of the existence of a Quine atom. The
claims are not so much false as simply unverifiable, due to huge gaps
that need to be filled with concrete mathematics.

B. If you were to give a construction of a Quine Atom already published:

1. You would prove me wrong.
2. Everyone would be able to see it and see that you're right.
3. You will have proved your point.
4. It would be an easy thing to do.

Easy yes in the sense that the mathematics is not difficult. Not easy
to do in a usenet post, and also entirely pointless, because the
mathematics in question would require one to copy quite a few pages
out of an easily accessible text. But I will be happy to point
everyone to the construction you seek in Barwise and Moss's Vivious
Circles: see Chapter 10, pages 119-121. The Quine atom in question is
called Omega. In the context of ZFA, Omega is the unique solution to
the equation x = {x}. (BTW, do you have a proof of uniqueness? Or
non-uniqueness? Both are possible depending on the framework.) And,
wonder of wonders, it appears that Aczel's book is available on the
web: http://standish.stanford.edu/pdf/00000056.pdf. It appears to
have been scanned in and is a rather huge 35MB. The mathematics
needed for the construction of Omega is given in the first 5 pages
(and more rigorously in later pages) and the construction of Omega
itself is found on page 6, with discussion on the following pages that
motivate the introduction of his axiom AFA. (BTW, the pictures of
graphs in Aczel's exposition are not "flowcharts", as you seem to
characterize such pictures in another post. They are just
representations of mathematical graphs that could just as easily have
been represented explicitly as ordered pairs consisting of a set of
nodes and a binary relation (the set of arcs) over the nodes. Such
pictures are standard fare in texts on mathematical graph theory.)

C. You decline to give a construction for a Quine Atom.

This being the web and all, I take it that links to explicit
constructions of the sort you are looking for will suffice. I hope
you, or at least others, will find them useful and illuminating.

.

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