Computable Diagonalizations and Turing’s Cardinality Paradox

A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a Turing machine, computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing’s discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing’s two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers.

Graft preservation solutions in cardiovascular surgery.

Vein grafts are still the most commonly used graft material in cardiovascular surgery and much effort has been spent in recent years on investigating the optimal harvesting technique. One other related topic of similar importance remained more or less an incidental one. The storage solutions of vein grafts following procurement and prior to implantation are, despite their assumed impact, a relatively neglected theme. There is no doubt that the endothelium plays a key role in long-term patency of vein grafts, but the effects of the different storage solutions on the endothelium remain unclear : In a review of the literature, we could find 20 specific papers that addressed the question, of which the currently available preservation solutions are superior, harmless, damaging or ineffective. The focus lies on saline and autologous whole blood. Besides these two storage media, novel or alternative solutions have been investigated with surprising findings. In addition, a few words will be spent on potential alternatives and novel solutions on the market. As there is currently no randomized clinical trial regarding saline versus autologous whole blood available, this review compares all previous studies and methods of analysis to provide a certain level of evidence on this topic. In summary, saline has negative effects on the endothelial layers and therefore may compromise graft patency. Related factors, such as distension pressure, may outbalance the initial benefit of autologous whole blood or storage solutions and intensify the harmful effects of warm saline. In addition, there is no uniform consent on the superiority of autologous whole blood for vein graft storage. This may open the door to alternatives such as the University of Wisconsin solution or one of the specific designed storage solutions like TiProtec™ or Somaluthion™. Whether these preservation solutions are superior or advantageous remains the subject of further studies.