Some Pictures Are Worth 2^(N)o Sentences
According to the cliché a picture is worth a thousand words. But this is a canard, for it vastly underestimates the expressive power of many pictures and diagrams. In this note we show that even a simple map such as the outline of Manhattan Island, accompanied by a pointer marking North, implies a vast infinity of statements—including a vast infinity of true statements. Back to the outline map of Manhattan. What are the reading conventions for it? First, we have to decide on the syntactic elements. To this end, let a division be any partition of the geometrical line into connected intervals. Then we take as syntactic elements the elements of any division. Since the set of connected intervals of the real line has cardinality 2^(N)o, it follows that there are that many syntactic elements.
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- November 14, 2014