Editing Indicator function (section)

==Characteristic function in recursion theory, Gödel's and Kleene's representing function==
[[Kurt Gödel]] described the ''representing function'' in his 1934 paper "On undecidable propositions of formal mathematical systems" (the symbol "{{math|¬}}" indicates logical inversion, i.e. "NOT"):<ref name=Martin-1965>{{cite book |pages=41–74 |editor-link=Martin Davis (mathematician) |editor-first=Martin |editor-last=Davis |year=1965 |title=The Undecidable |publisher=Raven Press Books |place=New York, NY}}</ref>{{rp|page=42}} 

{{blockquote|1=There shall correspond to each class or relation {{mvar|R}} a representing function <math>\phi(x_1, \ldots x_n) = 0</math> if <math>R(x_1,\ldots x_n)</math> and <math>\phi(x_1,\ldots x_n) = 1</math> if <math>\neg R(x_1,\ldots x_n).</math>}}

[[Stephen Kleene|Kleene]] offers up the same definition in the context of the [[primitive recursive function]]s as a function {{mvar|φ}} of a predicate {{mvar|P}} takes on values {{math|0}} if the predicate is true and {{math|1}} if the predicate is false.<ref name=Kleene1952>{{cite book |last=Kleene |first=Stephen |author-link=Stephen Kleene |year=1971 |orig-year=1952 |title=Introduction to Metamathematics |page=227 |publisher=Wolters-Noordhoff Publishing and North Holland Publishing Company |location=Netherlands |edition=Sixth reprint, with corrections}}</ref>

For example, because the product of characteristic functions <math>\phi_1 * \phi_2 * \cdots * \phi_n = 0</math> whenever any one of the functions equals {{math|0}}, it plays the role of logical OR: IF <math>\phi_1 = 0\ </math> OR <math>\ \phi_2 = 0</math> OR ... OR <math>\phi_n = 0</math> THEN their product is {{math|0}}. What appears to the modern reader as the representing function's logical inversion, i.e. the representing function is {{math|0}} when the function {{mvar|R}} is "true" or satisfied", plays a useful role in Kleene's definition of the logical functions OR, AND, and IMPLY,<ref name=Kleene1952 />{{rp|228}} the bounded-<ref name=Kleene1952 />{{rp|228}} and unbounded-<ref name=Kleene1952 />{{rp|279 ff}} [[mu operator]]s and the CASE function.<ref name=Kleene1952 />{{rp|229}}