Department of HISTORY AND PHILOSOPHY OF SCIENCE Eotvos University, BudapestPazmany P. setany 1/A Budapest Phone/Fax: (36-1) 372 2924 Department's Home Page:http://hps.elte.huhttp://hps.elte.hu PHILOSOPHY OF SCIENCE COLLOQUIUM Room 6.54 (6th floor) Monday 4:00 PM ___________________________ 11 March 4:00 PM 6th floor 6.54 Language: English, except all participants speak Hungarian G a b o r E t e s i* ** (lecturer) and I s t v a n N e m e t i** * Yukawa Institute, Kyoto University, Japan ** Alfred Renyi Institute of Mathematics, Budapest General relativistic- (and/or quantum-) computability; computing non-Turing-computable functions in Malament-Hogarth spacetimes It used to be a (meta) theorem of mathematical logic that mankind will never know that ZFC (which forms the foundation of mathematics) is consistent, assuming it is. We will argue that this meta-theorem is gone, it is no more provable. We will report on (convergent) results of various research groups at various parts of the world coming, independently, to the same conclusion which is, roughly, that Turing computability may not (after all) be the final limit of the capabilities of artificial computing devices. Some of the above mentioned researchers are e.g. Hogarth (Cambridge), Malament, Earman, ourselves, Kieu (Australia), F. Tipler, to mention only a few. We investigate the Church--Kalmar--Kreisel--Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church--Turing-type Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary, hence certain forms of the Church--Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be different.) We also look at various ``obstacles'' to computing a non-recursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the ``design'' of our future computer). We also ask ourselves, how all this reflects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions. (We note that the goal of ``computing the uncomputable'' is distincly more modest than executing so called supertasks. Indeed, we do not claim possibility of the second.) A paper advocating carefully and it detail the view we adopt here -- that developments in the background physical theory can influence profoundly the fundamentals of the theories of computability and logic -- appeared in Bull. Symbolic Logic Vol. 6 No 3 (2000), pp.265-283 by Deutsch et al. Our paper on this subject is available on the following internet address: [http://ipsapp008.lwwonline.com/ips/frames/menu.asp?J=4779&S=36698&M…] A further useful reference is Hogarth, M.: ``Predictability, Computability, and Spacetime'', pp.1-123, available from [mh10026(a)cam.ac.uk]. ____________________________ The 60-minute lecture is followed by a 5-minute break. Then we held a 30-60-minute discussion. The participants may comment the talks and initiate discussion on the Internet. The comments should be written in the language of the presentation. The organizer of the seminar: Laszlo E. Szabo -- Laszlo E. Szabo Department of Theoretical Physics Department of History and Philosophy of Science Eotvos University, Budapest H-1518 Budapest, Pf. 32, Hungary Phone/Fax: (36-1)372-2924 Home: (36-1) 200-7318 Mobil/SMS: (36) 20-366-1172 http://hps.elte.hu/~leszabo