Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

94 the realm of the infinitethat concerns the compression achieved by the informal style in which mathematicalarguments are actually written. This is explored a little bit further in Woodin (1998).With proper inputs and global determination, one could verify with current technologythat a given sequence of length at most 10 24 is a proof of (¬) from ZFC 0 .However, we obviously do not expect to be able to find a sequence of length less than10 24 that is a proof of (¬) from ZFC 0 . This actually gives a prediction about thephysical universe because one can code any candidate for such a sequence by a binarysequence of length at most 10 26 . The point is that, assuming the validity of the quantumview of the world, it is possible to build an actual physical device that must have anonzero chance of finding such a sequence if such a sequence can exist. The devicesimply contains (a suitably large number of independent) modules, each of which performsan independent series of measurements that in effect flips a quantum coin. Thepoint, of course, is that by quantum law any outcome is possible. The prediction issimply that any such device must fail to find a sequence of length less than 10 24 that isa proof of (¬) from ZFC 0 . One may object that the belief that any binary sequenceof length 10 26 is really a possible outcome of such a device requires an extraordinaryfaith in quantum law; but any attempt to build a quantum computer that is useful (forfactoring) requires the analogous claim where 10 26 is replaced by numbers at least aslarge as 10 5 .This, of course, requires something like quantum theory. In the universe as describedby Newtonian laws, the argument just described does not apply because truly randomprocesses would not exist. One could imagine proving that for a large class of chaotic(but deterministic) processes (“mechanical coin flippers”), no binary sequence of length10 24 that actually codes a formal proof can possibly be generated. In other words, forthe nonquantum world, the prediction that no such sequence (as just presented) can begenerated may not require that the conception of V n is meaningful where n =|V 1000 |.Granting quantum law, and based only on our collective experience in mathematicsto date, how can one account for the prediction (that one cannot find a sequence oflength less than 10 24 that is a proof of (¬) from ZFC 0 ) unless one believes that theconception of V n is meaningful where n =|V 1000 |?Arguably (given current physical theory) this is already a conception of a nonphysicalrealm.4.3 Beyond the Finite RealmIn this section we briefly summarize the basic argument of Woodin (2009), althoughhere our use of this argument is for a different purpose.Skeptic’s Attack: The mathematical conception of infinity is meaningless and withoutconsequence because the entire conception of the universe of sets is a complete fiction.Further, all the theorems of set theory are merely finitistic truths, a reflection of themathematician and not of any genuine mathematical “reality.”Throughout this section, the “Skeptic” simply refers to the metamathematical positionthat denies any genuine meaning to a conception of uncountable sets. The counterviewis that of the “Set Theorist.”