Subject: Proof of nonexistence of membership

From: categories <[email-masked]>
Date: Wed, 20 Mar 1996 14:48:23 -0400 (AST)

Date: Wed, 20 Mar 1996 10:25:56 -0800
From: Vaughan Pratt <[email-masked]>

It occurs to me that the arguments that have been advanced against
membership can be strengthened to a mathematical proof that the usual
notion of membership as a 2-valued binary relation is an inconsistent
notion.  That is,

Theorem.  Membership does not exist.

Proof.  Clearly the universe exists, or we're in serious trouble.  But
if membership also exists then the Cantor-Russell argument leads to a
contradiction.  QED

My real point here is that the Cantor-Russell argument doesn't *really*
prove there is no universal set, or no 2-valued membership relation, or
that some sets we can name are not in the universe, or that the
universe we exist in must be different from the one in which
mathematical objects exist, or that we are arguing with an unreliable
logic.

What it proves is the sentence "false."

Anything powerful enough to prove false is a theorem of the universe
dual to ours.  Such a theorem is a gedanken wavefunction.  To bring it
into our universe it has to collapse to a Gedankeneigenfunction of the
operator by which we observe it, that is, our logic.

When you are young and unobserved you are just some gedanken
wavefunction.  When you become observed, whether for the purpose of
influencing future generations or getting tenure, you collapse to one
of the schools of thought constituting the Gedankeneigenfunctions of
the observation operator, whether set theorist, or category theorist,
or intuitionist, etc.  That is, you have to take a stand or risk
failure to communicate.

I have tried to communicate without taking a stand.  I may have
underestimated the disadvantages of not collapsing to a
Gedankeneigenfunction.  There's a lot to be said in favor of collapse.

Vaughan Pratt

---

From: categories <[email-masked]>
Date: Wed, 20 Mar 1996 21:02:02 -0400 (AST)

Date: Wed, 20 Mar 1996 15:58:58 -0500 (EST)
From: [email-masked]

Dear QED,
   Your proof of nonexistence of the universe is short and convincing. But why
do you call it membership?
    John Isbell

---

From: categories <[email-masked]>
Date: Thu, 21 Mar 1996 11:08:33 -0400 (AST)

Date: Wed, 20 Mar 1996 18:48:59 -0800
From: Vaughan Pratt <[email-masked]>

	From: [email-masked]
	Your proof of nonexistence of the universe is short and convincing.
	But why do you call it membership?

Bill Lawvere visited Stanford in 1988 to give a talk and share ideas.
I vaguely registered that his (cream-colored?) jacket had a Members
Only label, and I found myself wondering why I was noticing such a
trivial detail, and why it kept coming back to haunt me in the
intervening years.

As I stared at your message, John, trying to decide which of its ten
meanings you most likely intended, free associating like crazy,
suddenly the irony hit me.  Thanks!  God knows how much longer it would
have taken otherwise for my subconscious to deliver this particular
message.

I should explain what gave rise to my very off-the-wall posting.  I'd
asked a well-known set theorist where set theorists preferred to set
the boundary between the ordinals that existed and those that didn't,
or whether they didn't try, and the exchange that followed was about
what you'd expect of two explorers from different solar systems meeting
and trying to find a common alphabet, lexicon, syntax, and semantics in
that order.

But we got there, and I thought, ah, *this* must be what a wavefunction
feels like when its pushed out of one eigenstate of an operator into
another.  (Nothing contradictory there when you analyze it in terms of
a second operator whose eigenstates are different from the other one,
applied for the purpose of temporarily getting out of an eigenstate of
the other operator.)

The metaphor doesn't have to be quantum mechanics.  Instead of two
operators you could have two drains and one plug in your bathtub.  The
water will pick a direction to swirl as one of the two eigenstates of
the open drain, and will then get nudged out of that eigenstate when
you move the plug over to the other drain.  The first open drain
represents conferences and journal publication in some discipline, and
its eigenstates represent schools of thought in that discipline.  The
other represents a method of getting out of the rut so that you have a
chance when you do go back to the first drain of finding yourself in
the other eigenstate.  Provided the operators are sufficiently
orthogonal you can expect this method to succeed after two tries on
average (1/2 + 1/4 + 1/8 + ...).

I only know of analog metaphors for this phenomenon, which it seems to
me nicely describes the relationship between the competing schools of
foundations and the Cantor-Russell-Goedel paradox.  (To which some
people these days add Heisenberg, I'm on the side that likes this
connection, but there's plenty on the other side too.)

In the absence of discrete metaphors I'm not sure I can add anything
helpful to this.  If the above doesn't do it, well, it was just a silly
idea then.

Vaughan