  The most basic judgments about finitary numerals are those about equality and inequality. In addition, the finite standpoint allows operations on finitary objects. Here the most basic is that of concatenation. The concatenation of the numerals 11 and 111 is communicated as “\(2 + 3\),” and the statement that 11 concatenated with 111 results in the same numeral as 111 concatenated with 11 by “\(2 + 3 = 3 + 2\).” In actual proof-theoretic practice, as well as explicitly  in  (Hilbert and Bernays, 1934;  Bernays, 1930),  these basic operations are generalized to operations defined by recursion, paradigmatically, primitive recursion, e.g., multiplication and exponentiation (see  Parsons 1998 for philosophical difficulties in relation to exponentiation and 2007 for an extended discussion of intuitive mathematics and finitism). Similarly, finitary judgments may involve not just equality or inequality but also basic decidable properties, such as “is a prime.” This is finitarily acceptable as long as the characteristic function of such a property is itself finitary: For instance, the operation which transforms a numeral to 1 if it is prime and 11 otherwise can be defined by primitive recursion and is hence finitary. Such finitary propositions may be combined by the usual logical operations of conjunction, disjunction, negation, but also bounded quantification.  (Hilbert, 1926)  gives the example of the proposition that “there is a prime number between \(p + 1\) and \(p! + 1\)” where \(p\) is a certain large prime. This statement is finitarily acceptable since it “serves merely to abbreviate the proposition” that either \(p + 1\) or \(p + 2\) or \(p + 3\) or … or \(p! + 1\) is a prime.