Richard's paradox

Richard's paradox is a falsidical paradox of mathematical mapping first described by the French mathematician Jules Richard in 1905. Today, it is ordinarily used in order to show the importance of carefully distinguishing between mathematics and metamathematics.

Resolving the paradox

Richard's Paradox is falsidical; it is but a magic trick, and can be easily explained away. An essential but tacit assumption concerning the ordering of definitions was ignored while setting up the paradox.

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It was agreed to consider the arithmetical properties of integers, i.e., properties that can be spoken about using additions, multiplication, etc. But then later in the paradox a definition was added to the series which involves reference to the notation used in arithmetical properties. This is obviously not allowed. The definition of being Richardian does not belong to the series initially intended, because this definition involves meta-mathematical notions such as the number of letters occurring in expressions.

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Explaining away Richard's Paradox is as easy as being careful to distinguish between statements within arithmetic (which make no reference to any system of notation) and statements about some system of notation in which arithmetic is codified.

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