There are various different strategies that friends of large and small denumerable, physical infinities can pursue in the face of the problem cases presented in the first chapter of this book. Perhaps the single most important strategy that friends of large and small denumerable, physical infinities can pursue is simply to outsmart those who present these problem cases for consideration: In many cases, these allegedly absurd situations are just what one ought to expect if there were large and small denumerable, physical infinities. Another important strategy that crops up repeatedly is consideration of the effects of altering the order of quantifiers in an apparently problematic claim. However, it is important to recognise that there are other strategies that need to be pursued as well: Some of the details of the stories embody genuine confusions that need to be cleared up. No one – whether friend or foe of large and small denumerable, physical infinities – wants to deny that it is possible to tell inconsistent stories about large and small denumerable, physical infinities. It is, after all, possible to tell inconsistent stories about anything you please. The question at issue is whether it is possible to tell consistent stories involving large and small denumerable, physical infinities – and so the allegedly problematic cases before us may need to be amended in order to remove merely extraneous inconsistencies.

AL-GHAZALI'S PROBLEM

To discuss Al-Ghazali's objection to large, denumerable, physical infinities, we need to begin with the distinction between ordinal and cardinal numbers.