Infinity in ethics

2. Infinitely many options

Suppose that an agent has infinitely many options (possible choices). To say that there are infinitely many options is just to say that there are more options than any finite number. The presence of infinitely many options does not automatically generate problems, but it can where agents are required to perform an option that is maximally good (at least as good as any other option). Suppose, for example, that the options are numbered, that o₁ has a value of 1/2, that o₂ has a value of 2/3 and that, in general, o_n has a value of n/(n+1). In this case, there is no option with a maximal value. The values are all finite and less than one, but for any option, o_n say, there is another option with greater value (for example, o_n+1). No option is maximally good and thus no option is permissible according to a value-optimizing theory (see Teleological ethics; Utilitarianism). The result that nothing is permissible is puzzling, but it can be avoided by replacing the optimization requirement with a requirement that a chosen option be at least as good as ‘trivially less’ (on some specified criterion) than the best one can do. For example, if one-billionth of a unit of goodness is the cut-off for being trivial, then, in the above example, there are infinitely many options that satisfy this requirement (and they are all ‘almost’ maximal).

In the above case there are infinitely many options, each option has a finite value and the values of the options are bounded (that is, there is some finite value – for example, 1 in this case – such that no option has a greater value). Things are not so simple when the values are not bounded. Suppose, for example, that the value of o₁ is 1, o₂ is 2 and, in general, o_n is n. Given that there are infinitely many options, there is no finite limit on how high the values can be (even though each option has a finite value). In this case, optimizing and ‘almost optimizing’ theories say that no option is permissible. Absolute satisficing theories – that is, theories that judge an option permissible just in case its value is ‘good enough’ on some specified absolute sense – have no problem with this case. Whatever the criterion for being good enough, there are infinitely many options that are permissible. People who are inclined to defend an optimizing, or almost optimizing, theory in the finite case thus either have to accept that nothing is permissible in such infinite cases (a strange claim) or to explain why satisficing is acceptable in the infinite case but not in the finite case. (One possibility is to hold that one should maximize when possible; that, when this is not possible, one should almost-maximize; and that, when this is not possible, one should satisfice.)