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The irreducibility of impredicative principles

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Literatur

  1. SeePaul Bernays, Journal of Symbolic Logic,2, 65 (1937).

  2. Related results were announced in our note on p. 479 of Proceedings of the National Academy of Sciences of USA,36 (1950). However, the proofs sketched there are different and contain an erroneus assertion (namely, the sentence which begins at the bottom of p. 482).J. B. Rosser has made on p. 143 of Journal of Symbolic Logic16 (1951) adverse criticisms of our note which do not seem justified; compare our brief “Reply to ProfessorRosser” to appear in that same journal.

  3. Compare, e. g., the formulation on p. 150 of Proceedings of the National Academy of Sciences of USA,35 (1949).

  4. We assume the formulation given inQuine's book (Mathematical Logic, 3rd printing, 1951). In this formulation, there are infinitely many axioms and no other variables except those for the objects (or “individuals”).

  5. SeeI. L. Novak: Fundamenta Mathematicae37, 87 (1950) orJ. B. Rosser andHao Wang, Journal of Symbolic Logic15, 113 (1950).

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  6. The Consistency of the Continuum Hypothesis, 1940, Princeton University Press.

  7. We emphasize that if we are patient, we can actually write out, with the constructions inGödel's monograph, the sentenceF (x, y) required in 2.2.

  8. Alternatively, we could also include ι as a primitive symbol; see, e. g., Journal of Symbolic Logic4, 18 (1939). Note that in the definition 2. 4. 2, we should actually add the case where an ι-expression occurs in place of the variablez.

  9. See, e. g.:Church, Introduction to Mathematical Logic, 1944, Princeton University Press, p. 60 and p. 84. The symbol | in 2. 6 is for joint negation from which, as is well-known, all the other truth-functions can be defined.

  10. See, e. g.:Church, op. cit.

  11. Cf.Skolem, on p. 26 of Les entretiens de Zurich, Zurich 1941.

  12. We omit the proofs here. Details of some similar proofs are included in our paper “Certain Predicates Defined by Recursion Schemata”, to appear in Journal of Symbolic Logic.

  13. For a more detailed discussion of the general notion of “translation”, see Transactions of the American Mathematical Society71, 283 (1951).

  14. See respectively, Journal of Symbolic Logic15, 241 (1950);6, 135 (1941) and Quine's book, op. cit.

  15. Viz. the principle *202 on p. 162 ofQuine's book, op. cit.

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  1. Cambridge, Mass., USA

    Hao Wang

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  1. Hao Wang
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Wang, H. The irreducibility of impredicative principles. Math. Ann. 125, 56–66 (1952). https://doi.org/10.1007/BF01343107

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