But there seem to be little to choose among the [possible set-theoretic models for the natural numbers]. Relative to our purposes in giving an account of these matters, one will do as well as another, stylistic preferences aside. There is no way connected with the reference of number words that will allow us to choose among them, /for the accounts differ at places where there is no connection whatever between features of the accounts and our uses of the words in question/. If all the above is cogent, then there is little to conclude except that any feature of an account that identified 3 with a set is a superflous one – and that therefore 3, and its fellow numbers, could not be sets at all.


There is another reason to deny that it would be legitimate to use the reducibility of arithmetic to set theory as a reason to assert that numbers are really sets after all. Gaisi Takeuti has shown that the Gödel–von Neumann–Bernays set theory is in a strong sense reducible to the theory of ordinal numbers less than the least inaccessible number (1954). No wonder numbers are sets; sets are really (ordinal) numbers, after all. But now, which is really which?