# The calculi of lambda-conversion by Alonzo Church PDF By Alonzo Church

The description for this publication, The Calculi of Lambda Conversion. (AM-6), might be forthcoming.

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15. A COMBINATORY EQUrl/ALENT OF CONVERSION. It is desiI-able to have a set of' operations (upon combinations) which have the property that they always change a combination into a combination and which constitute an equivalent of' conversion in the sense that a combination X can be changed into a combination " NUMBERS 'N. COMBINATIONS, GODEL Y by a sequence ~ ( o or more of) these operations if and only if X conv Y. Such a set of operations is the following ( OI OXXXVIII) -- where F, A, B, C, O are arbitrary combinations, I- is ~.

Xl 2. v(par x 1 )x111(1J(par x 2 )x 2 v), (these formulas D and U can be explicitly 1'1'1tten down by ref'erring to the proofs of 14 I and 14 II). 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]] + ['.

T- F(f3ww). "Y - JT(JT)(JT) • ~ - -y(J I-y)(J /) • 111 - -y(-y(f3-y{-y(f3JT)T)}T). , given the combination operated on and the particular one of the thirty-eight operations which is applied, the combination resulting is uniquely determined; (2) they do not involve the idea of' substitution at an arbitrarr place, but only that of' substitution at a specif'ied place. This has the effect of' rendering some of' the developments in §16 much simpler than they othezwise might be. The proof' of' the equivalence of' OI-OXXXVIII to conversion is too long to be included here.