Home

# A system of abstract constructive ordinals

## Extract

As Gödel [6] has pointed out, there is a certain interchangeability between the intuitionistic notion of proof and the notion of constructive functional of finite type. He achieves this interchange in the direction from logic to functionals by his functional interpretation of Heyting arithmetic H in a free variable theory T of primitive recursive functionals of finite type. In the present paper we shall extend Gödel's functional interpretation to the case in which H and T are extended by adding an abstract notion of constructive ordinal. In other words, we obtain the Gödel functional interpretation of an intuitionistic theory U of numbers (i.e., nonnegative integers) and constructive ordinals in a free variable theory V of finite type over both numbers and constructive ordinals. This allows us to obtain an analysis of noniterated positive inductive definitions [8].

The notion of constructive ordinal to be treated is as follows. There is given a function J which embeds the nonnegative integers in the constructive ordinals. A constructive ordinal of the form Jn is said to be minimal. There is also given a function δ which associates to each constructive ordinal Z and number n a constructive ordinal δZn which we denote by Zn. When Z is nonminimal, each Zn is called an immediate predecessor of Z. The basic principle for forming constructive ordinals says: for every function f from numbers n to constructive ordinals, there exists a constructive ordinal Z such that Zn = fn for all n. The principle of transfinite induction with respect to constructive ordinals says: if a property Q(Z) of constructive ordinals Z holds for minimal Z, and if ∀nQ(Zn) → Q(Z) holds for all Z, then Q(Z) holds for all Z.

## References

Hide All
[1]Bachmann, H., Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungzahlen, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, vol. 95 (1950), pp. 115147.
[2]Brouwer, L. E. J., Zur Begründung der intuitionistischen Mathematik. III, Mathematische Annalen, vol. 96 (1927), pp. 451489.
[3]Curry, H. and Feys, R., Combinatory logic, Vol. 1, North-Holland, Amsterdam, 1958.
[4]Gerber, H., An extension of Schütte's Klammersymbols, Mathematische Annalen, vol. 174 (1967), pp. 203216.
[5]Gerber, H., Brouwer's bar theorem and a system of ordinal notations, Proceedings of the Summer Conference on Intuitionism and Proof Theory, North-Holland, Amsterdam, 1970.
[6]Gödel, K., Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica, vol. 12 (1958), pp. 280287.
[7]Howard, W. A., Functional interpretation of bar induction by bar recursion, Compositio Matkematica, vol. 20 (1968), pp. 107124.
[8]Kreisel, G. and Troelstra, A. S., Formal systems for some branches of intuitionistic analysis, Annals of Mathematical Logic, vol. 1 (1970), pp. 229387.
[9]Tait, W. W., Infinitely long terms of transfinite type, Formal systems and recursive functions, ed. Crossley, J. and Dummett, M., North-Holland, Amsterdam, 1965, pp. 176185.
[10]Tait, W. W., Intensional interpretations of functionals of finite type. I, this Journal, vol. 32 (1967), pp. 198212.
[11]Zucker, J., Proof-theoretic studies of systems of iterated inductive definitions and subsystems of analysis, Ph.D. dissertation, Stanford University, 1971.

# A system of abstract constructive ordinals

## Metrics

### Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *