** October 2003** **Infinite Wisdom**, a piece by Erica Klarreich in the August 30 2003 *Science News*, surveys some recent work on the continuum hypothesis. Klarreich starts with a review of Cantor's proof that the set **R** of real numbers is strictly larger, in a precise sense, than the set **Z** of integers. In this connection she shows Helaman Ferguson's clever visualization of Cantor's diagonal argument: | | *Cantor's Flickering Diagonal.* The left half of this stereo pair shows the beginning of an enumeration of the real numbers between 0 and 1. The top line represents the start of the binary expansion of the first number on the list (white=0, black=1). The next line corresponds to the second number on the list, and so on. The right half is identical, except that each diagonal element (the first digit of the first number, the second digit of the second number, and so on) has been reversed: changed from white to black or from black to white. When the two images are fused, the reversed diagonal flickers in and out. The reversed diagonal is the binary expansion of a real number that cannot occur on the original list. Since this will happen for any list, the construction shows that there is no way of listing the real numbers between 0 and 1. Click here for the stereo image of a larger array. Image courtesy Helaman Ferguson, used with permisssion. | The Continuum Hypothesis is the statement that there is no intermediate size: there is no set with strictly more elements than **Z** and strictly fewer elements than **R**. The truth or falsity of this hypothesis was number one on Hilbert's 1900 list of important unsolved problems. Klarreich continues with the history of the hypothesis, and of its relation to the standard axioms of set theory. She surveys the work of Kurt Gödel (1938) and Paul Cohen (1963): "Put together, those two results indicate that it's impossible either to prove or to disprove the continuum hypothesis using the standard axioms." Which brings us to Hugh Woodin (U.C.Berkeley) and his recent work on the characterization of an axiom which could be added to the standard set and which would "answer all questions up to the level of the hierarchy that the continuum hypothesis concerns--the realm of the smallest uncountably infinite sets." Woodin calls such an axiom "elegant." Rather than try to construct such an axiom, "Woodin has proved --apart from one missing piece that must still be filled in-- that elegant axioms do exist and, crucially, that every elegant axiom would make the continuum hypothesis false." Two survey papers by Woodin are available online: The continuum hypothesis, part I and part II. **Wolfram in Washington.** "Esoteric theorist lands starring role in Senate hearing" was Geoff Brumfiel's headline for his September 11 2003 news piece in *Nature*. The story: Stephen Wolfram appeared before the subcommittee on science, technology and space of the Senate commerce committee, at the invitation of Sen. Sam Brownback (R, Kansas). Brownback, according to Brumfiel, is "one of the more conservative members of the Senate, and has clashed with scientists on issues such as embryonic-stem-cell research and the teaching of creationism in the classroom." He "seems committed to finding a home for Wolfram's research" and told *Nature* "If it's right, then the government should be investing in it." An assessment of the first year of Wolfram's "New Kind of Science" was given by Peter Weiss in the August 16 2003 *Science News*. **African Institute.** A pan-African postgraduate mathematics institute has been inaugurated in Cape Town, South Africa. This according to a dispatch from Tom Clarke in Johannesburg to the September 18 2003 *Nature.* The institute has been set up inexpensively: "It will house students and staff in a donated former hotel at the unfashionable end of Cape Town's seafront. The library is stocked with donated books, and lecturers from universities worldwide will give 4-8 weeks of their time to teach classes." Operating funds will come from the South African government. Clarke quotes Justin Bazimaziki, a student from Rwanda: "I hope to gain the knowledge to allow me to contribute to the development of my country and of Africa as a whole." **Paul Wolfowitz, "former mathematician."** *New York Times* columnist Maureen Dowd pre-reviews Midge Decter's "Rumsfield, A Personal Portrait" on the Op-Ed page for September 28, 2003. In one of her more intemperate columns ("Drunk on Rummy"), she picks up Ms. Dicter's characterization of Deputy Secretary of Defense Paul Wolfowitz as a "former mathematician." In fact, Wolfowitz was a math major at Cornell, but his academic connection with mathematics ended when he graduated. Nevertheless Dowd works the trope for all it's worth, charging the Secretary with "refusing ... to add up how much the war was going to cost ... , to multiply the exponential problems of remaking the Middle East, or even to subtract the billions that were never coming from snubbed allies." Or to calculate "the division in America his omissions would cause when we finally got a load of the bill." -*Tony Phillips* Stony Brook Math in the Media Archive |