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

62 marcus giaquintoof every such sequence; hence that the validity of induction up to ω 2 can bemade immediately evident.What I think Gödel had in mind here is that, using the ω-square representation,there is a visuo-spatial way of telling that every decreasing sequenceof members of ω 2 terminates. From this fact the validity of induction upto ω 2 quickly follows, a fact which Gödel presumably took as backgroundknowledge for his readers. But how can we tell that every decreasing sequenceof members of ω 2 terminates? Thinking of ω 2 in terms of an ω-square asdescribed earlier, it is obvious that for any decreasing sequence, among rowscontaining members of that sequence, there will be an uppermost row; andthat, of the members of the sequence in that row there will be a leftmostmember—call it α. Then, recalling the ordering of ω 2 , it is clear that α isthe least member of the sequence: so the decreasing sequence terminates. Thisway of acquiring the knowledge is relatively immediate, and is concrete in thesense that it has an experiential element. This may have been what Gödel hadin mind, but we do not know. Either way, it does substantiate all but one ofhis claims. It does not support Gödel’s implied claim that one can grasp at oneglance all the structural possibilities for decreasing sequences in ω 2 ,forthereare decreasing sequences with elements that occur arbitrarily far to the right inan ω-square.¹⁹ But Gödel’s other claims are untouched by this.It is credible that we can have the same kind of grasp of the structure of ω 2as we can have of N, and that this plays a role in actual mathematical thinking.Investigation is needed to determine how much further into the transfinite wemay go before this kind of cognition of ordinal structure becomes impossible.At present I do not see how to extend the kind of account I have suggestedfor ω 2 to ω ω . But I think that we can form a visual category specification forω-many layers of ω-squares, an ω-cube, thus forming a representation of ω 3 .Therewouldbeatopω-square, and beneath each ω-square another one. Theordering within ω-squares is unchanged; for elements α and β in differentω-squares, α precedes β if and only if α’s ω-square is above β’s ω-square.This uses our natural representation of space as extending infinitely in each ofthree dimensions. As our natural represention of space lacks a fourth dimension,it is clear that we cannot get to ω 4 from ω 3 in the same way as we got to ω 3from ω 2 . What we can do is to take each element of an ω-cube to representan ω-string, an ω-square or an ω-cube, thus getting a way of thinking of ω 4 ,ω 5 or ω 6 . But this way of using a visual category specification for ω-cubesis not a way of extending it to get another visual category specification. Canwe imagine putting an ω-cube in each position in an ω-cube, as opposed¹⁹ Robert Black pointed this out in discussion.