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
Dear QED, Your proof of nonexistence of the universe is short and convincing. But why do you call it membership? John Isbell
From: MTHISBEL@ubvms.cc.buffalo.edu 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
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Vaughan Pratt