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.

An (un) healthy poster: When environmental cues affect consumers’ food choices at vending machines

Environmental cues can affect food decisions. There is growing evidence that environmental cues influence how much one consumes. This article demonstrates that environmental cues can similarly impact the healthiness of consumers’ food choices. Two field studies examined this effect with consumers of vending machine foods who were exposed to different posters. In field study 1, consumers with a health-evoking nature poster compared to a pleasure-evoking fun fair poster or no poster in their visual sight were more likely to opt for healthy snacks. Consumers were also more likely to buy healthy snacks when primed by an activity poster than when exposed to the fun fair poster. In field study 2, this consumer pattern recurred with a poster of skinny Giacometti sculptures. Overall, the results extend the mainly laboratory-based evidence by demonstrating the health-relevant impact of environmental cues on food decisions in the field. Results are discussed in light of priming literature emphasizing the relevance of preexisting associations, mental concepts and goals.