Essays in commutative harmonic analysis

Logic is -- according to Poincaré -- the study of properties which are common to all classifications. There are two different kinds of classifications: predicative classifications, which are not modified by the introduction of new elements; and impredicative classifications, which are modified by new elements. Definitions as well as classifications are divided into predicative and impredicative . A set is defined by a law according to which every element is generated. In the case of an infinite set, the process of generating elements is unfinished; thus there are always new elements. If their introduction changes the classification of already generated objects, then the definition is impredicative. For example, look at phrases containing a finite number of words and defining a point of space. These phrases are arranged in alphabetical order and each of them is associated with a natural number: the first is associated with number 1, the second with 2, etc. Hence every point defined by such phrases is associated with a natural number. Now suppose that a new point is defined by a new phrase. To determine the corresponding number it is necessary to insert this phrase in alphabetical order; but such an operation modifies the number associated with the already classified points whose defining phrase follows, in alphabetical order, the new phrase. Thus this new definition is impredicative.

CaSO 4 ⋅ 1 2 H 2 O + 1 1 2 H 2 O ⟶ CaSO 4 ⋅ 2 H 2 O {\displaystyle {\ce {{/2H2O}+1\!1/2H2O->}}}