A note on Quine's treatment of transfinite recursion

Charles Parsons

doi:10.2307/2270371

A note on Quine's treatment of transfinite recursion

Published online by Cambridge University Press: 12 March 2014

Charles Parsons

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Charles Parsons*: Affiliation:
Harvard University

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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.

Type: Research Article
Information: The Journal of Symbolic Logic , Volume 29 , Issue 4 , December 1964 , pp. 179 - 182

DOI: https://doi.org/10.2307/2270371 [Opens in a new window]

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² Cambridge, Mass.: Harvard University Press, 1963.

³ (Editor's note: the symbols “and ‘of Quine's book are rendered as ″ and ′, respectively, in the present paper.)

⁴ This could be proved from ‘Fune γ’ if we had the axiom schema of replacement. For then transfinite induction could be derived without the hypothesis of 24.7. Now suppose z ∈ NO and Then by the argument of 26.16, Func Aγ↾z. Since Then we can conclude by transfinite induction that for all z ∈ NO. The consequences of ‘C₁’ needed for this argument can be proved by the axiom schema of replacement.