Several Stories about California: #4

The Black Oak is a dangerous place. I walked out with six books, of which three fit into the category of books I was required to buy on the spot because the chance might very well not come around again. I scored a copy of Sheri Tepper's The Bones, published in the 1980s and long out of print. Given that it's a sequel (to Blood Heritage), it would seem that the horror thing sold well enough for her for her publisher to print a second installement. Having read Blood Heritage, this fact surprises me -- and if The Bones was sufficiently worse to kill off the series, one might legitimately ask what I'm doing buying it. But don't you see, this means I only have three Tepper novels left (excluding the pseudonymous mysteries, which I've never been able to get into) and I'll have read her complete works! And as for the overall quality issue, as Monty Python might say, "she got better."

There was also an advance pre-publication reviewer's copy of Alan Lightman's new novel, The Diagnosis. This is huge. Einstein's Dreams was awesome, and Good Benito was an overlooked little gem of a novel. I have high hopes, and getting my grubby little paws on on a copy months before the rest of the world gets the multi-city author tour and national publicity campaign has made me one selfishly happy little camper.

But to top it all off, back on their quite remarkable back-room shelf of mathematical texts and monographs, I picked up a copy of Paul Cohen's Set Theory and the Continuum Hypothesis, a book whose main contents consist of a course-notes-style exposition of Cohen's own proof that the Axiom of Choice (and the Generalized Continuum Hypothesis also) is logically independent of the standard axioms of set theory, and can neither be proven nor disproven from them. The book is also an exposition of the proof technique he used to show the result, indeed, invented for that very purpose, called "forcing." Forcing is a concept I understand only very vaguely -- in fact, I've never seen it actually presented for real anywhere, which is why I was so gung-ho to find this book -- but it's way cool. The next couple of paragraphs are going to be somewhat hand-wavey explanation, so people not interested in the foundations of mathematics may want to skip the next two paragraphs. Come to think of it, since the paragraph after is going to consist of mathematical gossip, you might just want to skip on down to the next entry entirely.

Okay. There's a (by now) standard construction in mathematical logic, by which we can pass from semantic concerns back to purely syntactical ones. The various completeness theorems for propositional and first-order logic use this kind of argument: in order to show that a system has a reasonable interpretation, we construct what appears to be a highly artificial interpretation for the system, one which treats all the system's statements as referring only to objects in a jury-rigged universe set up specifically for the purpose of being referred to by the system. That is, if the system has a statement of the form "three is prime," we throw objects called "three" and "prime" into our bucket, and also remember that the "three" object and the "prime" object are related in the "is" way. It's a kind of reification of the abstraction of a formal system, and at the end of the day, this magical alternate universe of discourse has the extremely useful property of not reflecting any features other than ones that actually sprung from the system we wanted to work with. Cohen's forcing technique -- in sketchy essence -- consists in noting that in certain circumstances, these syntactic models can be a bit underdetermined, so that there are different paths down which the construction could go.

In the old-school proofs, one would throw out the extra paths, since what mattered was the existence of a path at all. In Cohen's new-school proof, one gathers together this flexibility and makes a series of judicious choices along the way, with the result that the final model has a couple of additional properties -- properties not fully specified by the original system -- that we've hand-selected for. Our flexibility is not absolute -- we can't wind up contradicting anything the system we're tampering with actually was definitive on, nor can we stir in arbitrary results or unrealistic superpowers -- but Cohen noticed that if one started with Zermelo-Fraenkel set theory, there was enough wiggle room to "force" the Axiom of Choice to be false. Combined with Godel's earlier result that the Axiom of Choice was consistent with the Zermelo-Fraenkel axioms, this was enough to garner Cohen a Fields Medal. It's sort of a homeopathic construction, or maybe akin to the making of a golem: a finite sequence of individual binary decisions about individual integers and a single set winds up being, under the right circumstances, magnified into a much more sweeping property of the system and its claims about uncountably infinite sets. And the whole thing proceeds by a structural induction on the syntax of formulae of the system. Cool, no?

So, anyway, the other cool thing about this book is the acknowledgements section. The book is basically the transcript of a graduate seminar given at Harvard in 1965. Cohen thanks the various students who scribed and took the notes that were presumably later turned into the book: L. Corwin, D. Pincus, T. Scanlon, J. Xenakis, and R. Walton. A bit of reflection made me realize that this list may very well include Larry Corwin, late professor of mathematics at Rutgers and co-author of a notoriously unclear (and poorly typeset) textbook on serious calculus; Tom Scanlon, professor of philosophy at Harvard and among the most articulate and clearly-spoken people alive; and Bob Walton. I suspect that anyone familiar with (any) two out of the these three will find the fact that they once took a class together very funny.