Since the primary purpose of this book has been to achieve a view of the range of positions that might be taken concerning the concept of the infinite, it is appropriate to conclude with a reasonably full account of the positions that we have identified. If the following taxonomy is adequate, then any view about the concept of the infinite will fall into one of these categories.

First, there is a strict finitism that maintains that we have no proper use of the concept of the infinite: The kind of extrapolation from the finite to the infinite that one might suppose furnishes us with an understanding of the infinite is simply not up to the task, and there is no plausible substitute. On this approach, classical mathematics is rejected; the only mathematics that is countenanced is finite mathematics. Moreover, the only theories in other domains that are acceptable are those that posit none but finite domains and finite magnitudes. There are only finitely many possible worlds, each with a finite “frame”, and composed of finitely many mereological atoms.

Second, there is a range of weak potential infinitisms, each of which supposes, roughly, that we have proper use for some claims of the ∀◇∃-form, but no proper use for claims of the ◇∀∃-form: While we can understand the idea that, for each (putative) natural number, it is possible that some things are characterised by that (putative) number, we cannot understand the idea that it is possible that, for each (putative) number, there are some things that are characterised by that (putative) number.