I once argued (in ‘What Numbers Could Not Be’) that an adequate account of the numbers and our arithmetic practice must satisfy not only the conditions usually recognized to be necessary: (a) identify some w-sequence as the numbers, and (b) correctly characterize the cardinality relation that relates a set to a member of that sequence as its cardinal number—it must also satisfy a third condition: the ‘<’ of the sequence must be recursive. This paper argues that adding this further condition was a mistake—any w-sequence would do, no matter how undecidable its ‘<’ relation.

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