# Calculi of Lambda Conversion by Alonzo Church

By Alonzo Church

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U(par n)nU. Then f'orm 1 conv 1, fem ' conv J, and f'om If conv fom 1¥1 (fem 112 ) If represents an even positive integer. From this it follows that form has the property ascribed to it above; for 11' If represents the Gadel number of a combination A 1 belonging to a f'ormula A, containing no free variables, then fom If conv A', and A' conv A. 6]] + ['. :. 5. Noting that the Gadel numbers of JI, T, JT, JTT are respectively 12, 156, 24812, 623375746, the reader may verify that: al¥ conv 1, 2, 3, or 4 11' tr represents a positive integer; " TV.

Take each of these occurrences of ~ in order and perfonn a terminating sequence or contractions on the residuals of the ((,\"'j~)Wj) occurring in it. Let us call such a terminating sequence of contractions on the parts ((i'. "J~),,j) of A. ~1+cb contractions. Consider now any sequence of contractions, µ, on the parts ( (h"j~)Wj) or A. xp"'p)"P) or A will have just one residual (which will always be the entire fonnula) up to the point that a contraction of its residual occurs, and thereafter will have no residual; moreover, if the sequence of contractions ·is continued, a contraction of the residual of ( (,\~),,p) must occur within at most a+b+l contractions.

O• .... 1''1 l1.. 1 o. PROPOSITIONAL FUNCTIONS; THE KT ;;;'j;"'iZ )t-FUNCTION. , truth and falsehood. al function of one variable; a relation is a propositional function of two variables. function is falsehood, and which has no value otherwise. \-definable function. functions: III. -b]. • exc ab + exc ba. From this :follows the i\-definability o:r a great variety o:r properties and relations o:r positive integers which are expressible by means o:r equations and inequalities; conjunction, disjunction, and negation of equations and inequalities can be provided :for by USL"lg min, max, and i\a.

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