We do not believe in abstract entities. No one supposes that abstract entities—classes, relations, properties, etc.— exist in space-time; but we mean more than this. We renounce them altogether.

We shall not forego all use of predicates and other words that are often taken to name abstract objects. We may still write ‘x is a dog,’ or ‘x is between y and z’; for here ‘is a dog’ and ‘is between … and’ can be construed as syncate-gorematic: significant in context but naming nothing. But we cannot use variables that call for abstract objects as values. In ‘x is a dog,’ only concrete objects are appropriate values of the variable. In contrast, the variable in ‘x is a zoölogical species’ calls for abstract objects as values (unless, of course, we can somehow identify the various zoological species with certain concrete objects). Any system that countenances abstract entities we deem unsatisfactory as a final philosophy.

The possibility of constructing a numerical equivalent of a system of trans-finite ordinals, in recursive number theory, was briefly indicated in a previous paper, where consideration was confined to ordinals less than ε (the first to satisfy ε = ω). In the present paper we construct a representation, by functions of number-theoretic variables, for ordinals of any type.

In addition to definite numerals, and numeral variables, we introduce majorant variables σ, ω, ωr , r ≧ 1. A relation containing a single majorant variable σ is required to hold, not necessarily for all non-negative integral values of σ, but for all values greater than some assigned constant.

In his Mathematical logic Quine characterizes the theorems of quantification theory by six principles: