Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

cognition of structure 61ordinal numbers, because the concretely evident steps, such as α → α 2 ,aresosmall that they would have to be repeated ɛ 0 times in order to reach ɛ 0 .¹⁸The significant implications of this passage for present concerns are that the stepfrom ω to ω 2 is ‘concretely evident’; that, as one can ‘grasp at one glance’ thestructural possibilities for decreasing sequences in ω 2 , one can have ‘immediateconcrete knowledge’ that all such sequences terminate; hence that the validityof recursion (induction) for ω 2 can be made ‘immediately evident’, whereasthe same is not true for ɛ 0 in place of ω 2 .Isω 2 really knowable in the impliedway? Let us first step back. The ordinal ω under the membership relation hasthe structure N; in fact this is normally what set theory uses to represent theset of natural numbers under ‘