Infinity in ethics

1. Lexicographic priority

Can the value of some events or states (for example, the pleasure of hearing music that one absolutely loves) be infinitely greater in relative terms than the value of some other event or state (for example, the pleasure of eating a carrot that one is not especially excited about) (see Good, theories of the; Infinity)? Is it possible, that is, that there is no finite number of the latter events such that the value of all those events together is at least as great as the value of the former event? This is impossible if all states and events have some standard finite value. One can, however, coherently reject this assumption.

First, value need not be representable by numbers. It may simply be ordinally representable by a ranking relation (that is, as more, less or equally valuable, but with no assignment of specific numbers for value). All else being equal, more of the infinitely less valuable sources of value makes the world more valuable, but no finite number of such sources can ever compensate for the loss of one of the former sources of value. The infinitely more valuable sources of value are simply lexicographically prior in the generation of overall value to the infinitely less valuable sources of value. Thus we can make perfect sense of the idea, if we do not require that value be numerically representable.

Second, even if one requires that numbers be assigned to the value of states of the world, this can be done using infinitesimal numbers of non-standard arithmetic. In standard mathematics, there are no numbers that are infinitesimally small. In the 1960s, however, Abraham Robinson (1966) proved that one can make perfect mathematical sense of such infinitesimals and thus that it is legitimate to posit them. The addition of a positive infinitesimal to a given number produces a larger number, but the sum of finitely many infinitesimals is still infinitesimally small – and hence smaller than any finite number (although greater than each of the original infinitesimals). If infinitesimals are recognized, then some sources of value may generate only infinitesimal value relative to other sources of value.

Thus, the idea of infinite (or infinitesimal) relative value is not problematic.