\(\Sigma _{1}\)-elementarity and Skolem hull operators (Q866561)

This article is reviewed together with the author's paper [``Ordinal arithmetic based on Skolem hulling'', Ann. Pure Appl. Logic 145, No. 2, 130--161 (2007; Zbl 1117.03063)] above. These two articles together with the earlier one [``The Bachmann-Howard structure in terms of \(\Sigma_1\)-elementarity'', Arch. Math. Logic 45, No. 7, 807--829 (2006; Zbl 1110.03053)] and the later one [``Assignment of ordinals to patterns of resemblance'', J. Symb. Log. 72, No. 2, 704--720 (2007; Zbl 1121.03081)] constitute an extensively revised version of the author's Dissertation [\(\Sigma_1\)-elementarity and Skolem hull operators. Münster: Univ. Münster, Fachbereich Mathematik und Informatik (2004; Zbl 1075.03028)]. In it, he develops and investigates ordinal notation systems based on two different approaches. The first approach is based on ``\(\Sigma_1\)-elementarity'', which T. J. Carlson started, and the second uses the more classical idea of ``Skolem hulling''. The earliest of the four papers applies the elementarity approach to the Bachmann-Howard ordinal. The first one of the two papers reviewed here, in the following referred to as [I], presents a toolkit for the construction of a notation system by means of Skolem hulling, and the remaining two papers offer comparison und mutual influences of the two approaches. Elementarity is a semantic notion: start from (the graph of) the ordinal addition \(+\) and the usual \(\leq\). Define a new relation \(\leq_1\) on ordinals by induction so that \[ \alpha \leq_1 \beta \quad \hbox{iff} \quad (\alpha; 0, +, \leq, \leq_1) \leq_{\Sigma_1} (\beta; 0, +, \leq, \leq_1). \] Here, \(\leq_{\Sigma_1}\) is the usual relation of \(\Sigma_1\) substructures. [\(\leq_1\) is a partial order coarser than \(\leq\), and has interesting properties. For instance, \(\varepsilon_0\) is the least \(\alpha\) such that \(\alpha \leq_1 \alpha + \alpha\). For any \(\alpha\), \(\{\beta \mid \alpha \leq_1 \beta \}\) is a closed interval. This interval may be endless, i.e., \(\alpha \leq_1 \beta\) if \(\alpha \leq \beta\). In this case it is written \(\alpha <_1 \infty\).] In the second paper under review, the author characterizes those \(\alpha <_1 \infty\) to be \(\tau_{\rho}\) for some \(\rho\) that were defined in [I]. The first such number is shown to be a proof-theoretic ordinal of the set theory \(\text{KP}\ell_0\). This paper offers exemplary applications of tools in [I]. The toolkit in [I] begins with the selection of \(\Omega_0\) to be 1 or an epsilon number \(\tau\), \(\Omega_1\) to be an uncountable regular cardinal \(>\tau\), and \(\Omega_i\), \(i>2\), to be successive cardinals. Using ordinals \(<\tau\) and \(\Omega_i\)'s as parameters, the author defines collapsing functions \(\varphi_m\), \(m<\omega\), and a notation system \(T^{\tau}\). The \(\varphi_m\)'s are defined by `bits and pieces', unlike the usual way of using a long transfinite induction. Namely, some \(\varphi_m\)'s are defined to sufficiently large extents, then \(\varphi_n\)'s with larger \(n\) start being defined which, in turn, help extending domains of lower \(\varphi_m\)'s. \{The reviewer is reminded of the construction of a `skyscraper'.\} \(\varphi_k\) sends 0 to \(\Omega_k\) and \(\xi \in T^{\tau} \cap \Omega_{k+2}\) in \(T^{\tau}\). An ordinal in \(T^{\tau}\) is denoted by a unique term built up from parameters using \(+\) and \(\varphi_k\)'s. Structures of these terms are investigated, and relations among them when different \(\tau\)'s are used are clarified. In the final section, the author gives new normal forms by means of 0, 1, \(+\) and \(\varphi_k\)'s and appropriate epsilon numbers \(\tau\). All four articles are written very clearly. However, due to the nature of the subject, particularly in [I], a huge number of definitions and symbols are mobilized, so much so that the one-page index of symbols which the author provides is a necessity.