A note on Quine's treatment of transfinite recursion
Charles D. Parsons · Journal of Symbolic Logic · 1964
In this note we point out a limitation of the treatment of transfinite recursion in Quine's Set Theory and its Logic2 and develop a method for overcoming it. We shall also mention and indicate how to overcome a second more trivial limitation. We assume familiarity with Set Theory and its Logic and use its notation and terminology. Numbered references are to Quine's numbered formulae. In our work we continue the numbering of § 26. The first difficulty is that the theorems which show that his device for defining functions by transfinite recursion (§ 25) accomplishes its purpose do not apply to functions defined by recursion over all the ordinals. Let γ be the function which gives the value of the function we want for an ordinal y from the sequence of its values for ordinals less than y.