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Describing ordinals using functionals of transfinite type

  • Peter Aczel (a1)

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Bachmann, in [2] shows how certain ordinals <Ω(Ω = Ω1 where Ωξ is the (1 + ξ)th infinite initial ordinal) may be described from below using suitable descriptions of ordinals <Ω2. The aim of this paper is to consider another approach to describing ordinal <Ω and compare it with the Bachmann method. Our approach will use functionals of transfinite type based on Ω.

The Bachmann method consists in denning a hierarchy of normal functions ϕδ: ΩΩ (i.e. continuous and strictly increasing) for δη0 < Ω2, starting with ϕ0(λ) = ω1 + λ. The definition of depends on a suitable description of the ordinals ≤ η0. This is obtained by defining a hierarchy 〈Fδ ∣ δ ≤ Ω2〉 of normal functions Fδ: Ω2Ω2 analogously to the definition of the initial segment 〈ϕδδΩ〉 of . The ordinal η0 is .

Note. Our description of Bachmann's hierarchies will differ slightly from those in Bachmann's paper. Let and denote the hierarchies in [2]. Then as Bachmann's normal functions are not defined at 0 we let for λ, δ < Ω2. Bachmann defines for 0 < λ < Ω2 but it seems more natural to omit this so that we let . The situation is analogous for and leads to the following definitions:

where n < ω and ξ is a limit number of cofinality Ω, and

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[1]Aczel, P., Three systems of notations for ordinals (unpublished), 1969.
[2]Bachmann, H., Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen, Vierteljahrschrift der Naturforscherden Gesellschaft in Zürich, vol. 95 (1950), pp. 537.
[3]Feferman, S., Systems of predicative analysis, this Journal, vol. 29 (1964), pp. 130.
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[4]Feferman, S., Autonomous progressions and the extent of predicative mathematics, Logic, methodology and philosophy of science. III, eds. van Rootselaar, B. and Staal, J. F., North-Holland, Amsterdam, 1968, pp. 121135.
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[8]Isles, D., Regular ordinals and normal forms, Intuitionism and proof theory, North-Holland, Amsterdam, 1970, pp. 339361.
[9]Neumer, W., Zur Konstruktion von Ordnungszahlen, Mathematische Zeitschrift. I, vol. 58 (1953), pp. 319413; II, vol. 59 (1954), pp. 434–454; III, vol. 60 (1954), pp. 1–16; IV, vol. 61 (1954), pp. 47–69; V, vol. 64 (1956), pp. 435–456.
[10]Pfeiffer, H., Ausgezetchnete Folgen fur gewisse Abschnitte der zweiten und weiterer Zahlklassen, Doctoral dissertation, Technische Hochschule, Hanover, 1964.
[11]Schütte, K., Predicative well-orderings, Formal systems and recursive functions, eds. Crossley, J. N. and Dummett, M. A. E., North-Holland, Amsterdam, 1965.

Describing ordinals using functionals of transfinite type

  • Peter Aczel (a1)

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