Ptykes in Gödels T und Definierbarkeit von Ordinalzahlen. (Ptykes in Gödel's T and definability of ordinal numbers)
type: Item
instance of: scholarly article
title: Ptykes in Gödels T und Definierbarkeit von Ordinalzahlen. (Ptykes in Gödel's T and definability of ordinal numbers)
zbMATH Open document ID: 0685.03038
DOI: 10.1007/BF01633986
published in: Archive for Mathematical Logic
publication date: 1989-01-01T00:00:00Z
review text: In this paper two of the inequalities are proved which are needed to obtain a result on the ordinal values of ptykes of type 2, which are definable in Gödel's T. The paper is written in German, but contains an extensive survey in English. Ptykes have been introduced by Girard as a generalization of dilators to (finite) higher types. Gödel's T is an equational calculus containing as axioms the defining equations of Gödel's primitive recursive functionals of finite types. Girard has used the ptykes of finite types to construct a natural model of a subsystem resp. variant of T and suggests to obtain from this model the following intrinsic ordinal assignment to closed T-terms of pure types. The pure types are defined inductively by \(n+1:=n\to o\). ``The'' ptyx \(\Xi_ n\) is defined to be the sum of all recursive ptykes of type n in an arbitrary enumeration. For a closed T-term of type \(n+1\) the ordinal assigned to t is \(o(t):=t(\Xi_ n).\) The main result of this paper yields the least upper bounds of \(Spec_ 1\) and of \(Spec_ 2\), where the spectra of T are defined as follows. For \(n\geq 1:\) \[ Spec_ n:=\{t(\Xi_{n-1})|\text{ t closed T-term of type }n\}. \] Rough upper bounds for these spectra follow from work of Girard and Ressayre, who study the ptykes \(\Xi_ n\) and their relation to analytic complexity [\textit{J.-Y. Girard} and \textit{J. P. Ressayre}, Recursion theory, Proc. Symp. Pure Math. 42, 389-445 (1985; Zbl 0573.03029)]. In particular they show that \(\Xi_{n+1}(\Xi_ n)\)- the supremum of the ordinals which can be described recursively in \(\Xi_ n\) as a parameter - equals the supremum of the \(\Pi^ 1_{n+1}\)-definable well-orderings of subsets of \(\omega\). The supremum of \(Spec_ 1\) is \(\Phi_{\epsilon_{\Omega +1}}(0)\), the so-called Bachmann-Howard ordinal. This ordinal also equals the supremum of the ordinals which are provably \(\Sigma_ 1\)-definable in KPu, Kripke-Platek set theory extended by an axiom of infinity. The supremum of \(Spec_ 2\)…
Mathematics Subject Classification ID: 03F10
Mathematics Subject Classification ID: 03F15
Mathematics Subject Classification ID: 03D60
zbMATH DE Number: 4121989
zbMATH Keywords: ptykes of type 2
zbMATH Keywords: dilators
zbMATH Keywords: Gödel's T
zbMATH Keywords: Kripke-Platek set theory
zbMATH Keywords: Bachmann-Howard ordinal
zbMATH Keywords: Bachmann hierarchy
author: Peter Paeppinghaus
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4121989