Ordering Anything: Rejiggering Linnebo’s Ordinal Abstraction

Eileen Nutting

Øystein Linnebo develops an abstractionist account of the natural numbers as ordinals. On this account, the natural numbers are abstracted from orderings of concrete numerals. But Linnebo also gestures towards an alternative version of his account, on which the restriction to concrete numerals is lifted. I develop something like this alternative account, show how it avoids the Burali-Forti paradox, and show how it guarantees that every number has a successor. Given these and other good features, I claim that Linnebo should prefer this alternative account to the one he develops.