ABSTRACT
This is a synthetic descriptive sentence. We can then, further, define the ptb ‘ LogSatz * so that ‘ LogSatz (x> u)y means: “ In the series of positions extending from x to x + u, an S i occurs.” Then the sentence: “ Every expression of the form 3 = 331 is an S i ” will be rendered in I by
‘ (Var (#) • Id (#*) • LogZz (#n))D LogSatz (,x, 2)’ ;
this is an analytic sentence which follows from the definition of * LogSatz \
As we have already mentioned, it is always possible to replace any ptb by an fu^ Several different ptb may be called homogeneous if at most one of them can appertain to any position. Then it is always possible to replace a class of homogeneous ptb by one fiib, by correlating one value of the fUb, either systematically or arbi trarily, to each one of the individual ptb-\Example: Let the class of colours which are to be expressed be finite. We can ex press every colour by a ptb, ‘ Blue 4 Red ’, and so on. These ptb are then homogeneous and therefore we can replace them by a single fiib, say ‘ cor, by numbering the colours in some way, and
stipulating that ‘ col (a) = b ’ shall mean: ‘ The position a has the colour No. b.’ ] Similarly, in the formulation of the syntax of I in I, we shall not designate the different kinds of symbols by different ptt> (as, for instance, in the example given in § 18 by ‘ Id ’, etc.) but by one fUb, namely ‘ zei\ We shall correlate the values of ‘ zei’ to the different symbols (symbol-designs), partly arbitrarily and partly in accordance with certain rules. These values are called the term-numbers of the symbols. For instance, we shall co-ordinate the term-number 15 to the symbol of identity. This means that (instead of * Id (a)’) we shall write ‘ zei (a) = 15* when we wish to express the fact that the symbol of identity occurs at the position a. Not only the economy in primitive syntactical concepts, but other reasons which will be discussed later, justify the choice of this method of the arithmetization of syntax, (in this arithmetization, we make use of the method which Godel [Unentscheidbare] has applied with such success in meta mathematics or the syntax of mathematics.)
In general, the establishment of term-numbers for the different symbols can be effected arbitrarily. All that must be provided for is the fact that, for the variables, of which the number is un limited, an unlimited number of term-numbers must be available — likewise for the 33, pr, and fu. We will now specify infinite classes of numbers for the term-numbers of these kinds of symbols in the following way. Let p run through all the prime numbers greater than 2. Stipulations: the term -num ber of a 3 shall be ap (that is, a prime number greater than 2); the term-number of a defined 33 shall be a p2 (that is, the second power of some prime number greater than 2); the term-number of an undefined pr shall be a^ >3; that of a defined pr, a p^ \ that of an undefined fu, a p5 (and specifically, the term-number of ‘ zei’ shall be 3s, which is 243); and that of a defined fu, a pQ. But not all the numbers of the classes determined in this way will be used as termnumbers: the choice of them will be determined later. To the remaining symbols-namely, the undefined logical constants-we assign (arbitrarily) other numbers, namely:
to the symbol: 0 ( ) , 1 = ] K ^ V * d the term-number: 4 6 10 12 14 15 18 20 21 22 24 26 30 33 34.
