LOGIC OF INFINITE QUANTUM-SYSTEMS

Limits of sequences of finite-dimensional (AF) C-algebras, such as th e CAR algebra for the ideal Fermi gas, are a standard mathematical too l to describe quantum statistical systems arising as thermodynamic lim its of finite spin systems. Only in the infinite-volume limit one can, for instance, describe phase transitions as singularities in the ther modynamic potentials, and handle the proliferation of physically inequ ivalent Hilbert space representations of a system with infinitely many degrees of freedom. As is well known, commutative AF C-algebras corr espond to countable Boolean algebras, i.e., algebras of propositions i n the classical two-valued calculus. We investigate the noncommutative logic properties of general AF C-algebras, and their corresponding s ystems. We stress the interplay between Godel incompleteness and quoti ent structures - in the light of the ''nature does not have ideals'' p rogram, stating that there are no quotient structures in physics. We i nterpret AF C-algebras as algebras of the infinite-valued calculus of Lukasiewicz, i.e., algebras of propositions in Ulam's ''twenty questi ons'' game with lies.