Explanation of Notations

Explanation of Notations In general we follow the same conventions as in volume l. But it is necessary to make a few changes as follows: Since the supply of block letters and capital Greek letters is limited, we form symbols for new constants of the same kind by means of special brackets [, ]. An expression consisting of a pair of such brackets with an ordinary expression between them functions as the name of an ob. If the enclosed expression has already an intuitive meaning, the bracketed expression will denote a corresponding formal ob according to definitions made in each case; in particular if the enclosed expression designates a number in ordinary arithmetic, the bracketed expression will denote the corresponding formal numeral (§ 13A). The enclosed expression may contain U'-variables ; if so these variables play the same role as variable affixes attached to a block letter. Certain changes will be made in Table I of Appendix A. The use of' ']' for the Rosser combinator is abandoned, thus leaving '1' free for the category of individuals. Likewise we shall use T' for negation, and 'N' for natural number. Several of the letters in parentheses in Appendix A are introduced formally in this volume. For the definitions see the index. As an addition to Appendix B we shall adjoin the property (n), This is defined in § HAS. The use of German letters is considerably restricted in this volume. We have found it convenient to use ordinary italic letters in most cases where we used German letters in volume l. In the case of expressions like '£-ob', '''-ob' etc., where we used script letters in volume I, we now use roman letters, thus 'H-ob', 'J-ob'. This practice was used in vol. I in the case of 'F-ob', 'F-deduction' etc., and that practice is continued here. In connection with the use of letters 'x', 'y', 'z' etc. for variables it is to be understood once and for all that distinct letters denote distinct variables unless the contrary is indicated. Reference to volume I by chapter and section will not need a special indication, since the numbering of chapters here continues that of volume I. But references by page numbers will be made in the following form: pp. I 203205. In references to volume I as a whole, we use the form - vol. I. In referring to sets of formulas with a common number, we shall use subscripts to indicate the first, second, etc. formula in the set. Thus (3 2 ) is the second formula in (3).

XIV

EXPLANATION OF NOTATIONS

In condensed proofs we use the following abbreviations: Th. Theorem Df. Definition hp. hypothesis hp.ind. hypothesis of induction hp.ind. (n) hypothesis of induction on n. Notations of the form [YIX]M to denote the result of substituting Y for X in M will now be used when Y, X, M stand for names of arbitrary formal or informal notions - obs, formulas, natural numbers, definitions, etc. Thus now [n + lin] (3) will stand for the result of substituting n + I for n in formula (3).