In this highly original study, Nuccio Ordine uses the figure of the ass as a lens through which to focus on the thought and writings of the great Renaissance humanist philosopher Giordano Bruno.

“A little masterpiece of originality and clarity.”—George Steiner “A necessary book.”—Roberto Saviano “A wonderful little book that will delight you.”—François Busnel International Best Seller / Now in English for the First Time In this thought-provoking and extremely timely work, Nuccio Ordine convincingly argues for the utility of useless knowledge and against the contemporary fixation on utilitarianism—for the fundamental importance of the liberal arts and against the damage caused by their neglect. Inspired by the reflections of great philosophers and writers (e.g., (...) Plato, Dante, Montaigne, Shakespeare, Borges, and Calvino), Ordine reveals how the obsession for material goods and the cult of utility ultimately wither the spirit, jeopardizing not only schools and universities, art, and creativity, but also our most fundamental values—human dignity, love, and truth. Also included is Abraham Flexner’s 1939 essay “The Usefulness of Useless Knowledge,” which originally prompted Ordine to write this book. Flexner—a founder and the first director of the Institute for Advanced Study in Princeton—offers an impassioned defense of curiosity-driven research and learning. (shrink)

We investigated the perception of developmental changes in timing patterns that happen in the course of second language (L2) acquisition, provided that the native and the target languages of the learner are rhythmically similar (German and English). It was found that speech rhythm in L2 English produced by German learners becomes increasingly stress-timed as acquisition progresses. This development is captured by the tempo-normalized rhythm measures of durational variability. Advanced learners also deliver speech at a faster rate. However, when native speakers (...) have to classify the timing patterns characteristic of L2 English of German learners at different proficiency levels, they attend to speech rate cues and ignore the differences in speech rhythm. (shrink)

Pensar en el cine como ocasión y potencia del pensamiento significa en efecto, en primer lugar, sustraer la imagen contemporánea del dominio de los estudios especializados y, en segundo lugar, pensar a partir de aquello que el cine crea (como recordaba Deleuze): esto es, las imágenes mismas. El cine redescubre su potencia a través de sus propias formas. Entrevistas de la revista italiana Fata Morgana con: Jacques Rancière, Roberto Esposito, Jean-luc Nancy, Slavoj Zizek, Julia Kristeva, Werner Herzog, Raúl Ruiz, Yervant (...) Gianikian, Ángela Ricci Lucchi, Edgar Reitz, Paul Schraoer, Georges Didi-Huberman, Jean-Louis Comolli, Nuccio Ordine. (shrink)

It is natural and important to have a formal representation of plain belief, according to which propositions are held true, or held false, or neither. (In the paper this is called a deterministic representation of epistemic states). And it is of great philosophical importance to have a dynamic account of plain belief. AGM belief revision theory seems to provide such an account, but it founders at the problem of iterated belief revision, since it can generally account only for one step (...) of revision. The paper discusses and rejects two solutions within the confines of AGM theory. It then introduces ranking functions (as I prefer to call them now; in the paper they are still called ordinal conditional functions) as the proper (and, I find, still the best) solution of the problem, proves that conditional independence w.r.t. ranking functions satisfies the so-called graphoid axioms, and proposes general rules of belief change (in close analogy to Jeffrey's generalized probabilistic conditionalization) that encompass revision and contraction as conceived in AGM theory. Indeed, the parallel to probability theory is amazing. Probability theory can profit from ranking theory as well since it is also plagued by the problem of iterated belief revision even if probability measures are conceived as Popper measures (see No. 11). Finally, the theory is compared with predecessors which are numerous and impressive, but somehow failed to explain the all-important conditional ranks in the appropriate way. (shrink)

Though the Random Utility Model (RUM) was conceived entirely in terms of ordinal utility, the apparatus through which it is widely practised exhibits properties of cardinal utility. The adoption of cardinal utility as a working operation of ordinal is perfectly valid, provided interpretations drawn from that operation remain faithful to ordinal utility. The article considers whether the latter requirement holds true for several measurements commonly derived from RUM. In particular it is found that measurements of consumer surplus change may depart (...) from ordinal utility, and exploit the cardinality inherent in the practical apparatus. (shrink)

The “high school algebra” laws of exponentiation fail in the ordinal arithmetic of sets that generalizes the arithmetic of the von Neumann ordinals. The situation can be remedied by using an alternative arithmetic of sets, based on the Zermelo ordinals, where the high school laws hold. In fact the Zermelo arithmetic of sets is uniquely characterized by its satisfying the high school laws together with basic properties of addition and multiplication. We also show how in both arithmetics the behavior of (...) exponentiation depends on whether the empty set is an element of the base. (shrink)

The class of all ordinal numbers can be partitioned into two subclasses in such a way that neither subclass contains an arithmetic progression of order type ω, where an arithmetic progression of order type τ means an increasing sequence of ordinal numbers γ r, δ ≠ 0.

Abstract.This paper is the first in a series of three which culminates in an ordinal analysis of Π12-comprehension. On the set-theoretic side Π12-comprehension corresponds to Kripke-Platek set theory, KP, plus Σ1-separation. The strength of the latter theory is encapsulated in the fact that it proves the existence of ordinals π such that, for all β>π, π is β-stable, i.e. Lπ is a Σ1-elementary substructure of Lβ. The objective of this paper is to give an ordinal analysis of a scenario of (...) not too complicated stability relations as experience has shown that the understanding of the ordinal analysis of Π12-comprehension is greatly facilitated by explicating certain simpler cases first.This paper introduces an ordinal representation system based on ν-indescribable cardinals which is then employed for determining an upper bound for the proof–theoretic strength of the theory KPi+ ∀ρ ∃ππ is π+ρ-stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set. (shrink)

Taking up ordinal notations derived from Skolem hull operators familiar in the field of infinitary proof theory we develop a toolkit of ordinal arithmetic that generally applies whenever ordinal structures are analyzed whose combinatorial complexity does not exceed the strength of the system of set theory. The original purpose of doing so was inspired by the analysis of ordinal structures based on elementarity invented by T.J. Carlson, see [T.J. Carlson, Elementary patterns of resemblance, Annals of Pure and Applied Logic 108 (...) 19–77], [G. Wilken, Σ1-Elementarity and Skolem hull operators, Annals of Pure and Applied Logic 145 162–175], and [G. Wilken, Assignment of ordinals to patterns of resemblance, The Journal of Symbolic Logic ]. Within the arithmetical context laid bare in this work, the “-numbers” play a role analogous to the role epsilon numbers play in the ordinal arithmetic based on the notion of Cantor normal form. (shrink)

Any set x is uniquely specified by the graph of the membership relation on the set obtained by adjoining x to the transitive closure of x. Thus any operation on sets can be looked at as an operation on these graphs. We look at the operations of ordinal arithmetic of sets in this light. This turns out to be simplest for a modified ordinal arithmetic based on the Zermelo ordinals, instead of the usual von Neumann ordinals. In this arithmetic, addition (...) of sets corresponds to concatenating graphs, and multiplication corresponds to replacing each edge of a graph by a copy of another graph. Characterizations for the von Neumann case are also given. (shrink)

Assume [Formula: see text]. Let [Formula: see text] be a [Formula: see text] equivalence relation coded in [Formula: see text]. [Formula: see text] has an ordinal definable equivalence class without any ordinal definable elements if and only if [Formula: see text] is unpinned. [Formula: see text] proves [Formula: see text]-class section uniformization when [Formula: see text] is a [Formula: see text] equivalence relation on [Formula: see text] which is pinned in every transitive model of [Formula: see text] containing the real (...) which codes [Formula: see text]: Suppose [Formula: see text] is a relation on [Formula: see text] such that each section [Formula: see text] is an [Formula: see text]-class, then there is a function [Formula: see text] such that for all [Formula: see text], [Formula: see text]. [Formula: see text] proves that [Formula: see text] is Jónsson whenever [Formula: see text] is an ordinal: For every function [Formula: see text], there is an [Formula: see text] with [Formula: see text] in bijection with [Formula: see text] and [Formula: see text]. (shrink)

Ordinal Computability discusses models of computation obtained by generalizing classical models, such as Turing machines or register machines, to transfinite working time and space. In particular, recognizability, randomness, and applications to other areas of mathematics, including set theory and model theory, are covered.

Dynamic ordinal analysis is ordinal analysis for weak arithmetics like fragments of bounded arithmetic. In this paper we will define dynamic ordinals – they will be sets of number theoretic functions measuring the amount of sΠ b 1(X) order induction available in a theory. We will compare order induction to successor induction over weak theories. We will compute dynamic ordinals of the bounded arithmetic theories sΣ b n (X)−L m IND for m=n and m=n+1, n≥0. Different dynamic ordinals lead to (...) separation. In this way we will obtain several separation results between these relativized theories. We will generalize our results to further languages extending the language of bounded arithmetic. (shrink)

We generalize ordinary register machines on natural numbers to machines whose registers contain arbitrary ordinals. Ordinal register machines are able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the sets of ordinals in a model of the Zermelo-Fraenkel axioms ZFC. This allows the following characterization of computable sets: a set of ordinals is ordinal register (...) computable if and only if it is an element of Gödel’s constructible universe L. (shrink)

This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We (...) prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable [Formula: see text]-Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD. (shrink)

It is widely held that under ordinal utility, utility differences are ill-defined. Allegedly, for these to be well-defined (without turning to choice under risk or the like), one should adopt as a new kind of primitive quaternary relations, instead of the traditional binary relations underlying ordinal utility functions. Correlatively, it is also widely held that the key structural properties of quaternary relations are entirely arbitrary from an ordinal point of view. These properties would be, in a nutshell, the hallmark of (...) cardinal utility. While much is obviously true in these two tenets, this note explains why, as stated, they should be abandoned. Any ordinal utility function induces a rich quaternary relation. There is such a thing as ordinal utility differences. Furthermore, this induced quaternary relation respects, apart from completeness, the most standard structural properties of quaternary relations. These properties are, from an ordinal point of view, anything but arbitrary; from a quaternary perspective only completeness should be considered the hallmark—if any—of cardinal utility. These facts are explained to be especially relevant to the critical appreciation of the ordinalist methodology. (shrink)

Reseña: Ordine, Nuccio. Los retratos de Gabriel García Márquez. Repetición y diferencia. Trad. Manuel Prada Londoño – Eleanor Londero. Cali: Editorial Bonaventuriana, 2018.

The paper relativizes the method of ordinal analysis developed for Kripke–Platek set theory to theories which have the power set axiom. We show that it is possible to use this technique to extract information about Power Kripke–Platek set theory, KP.As an application it is shown that whenever KP+AC proves a ΠP2 statement then it holds true in the segment Vτ of the von Neumann hierarchy, where τ stands for the Bachmann–Howard ordinal.

The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard (Ann Pure Appl Log 33:1–21, 1987). There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised the following questions. Which subsets of the Optional Axioms are consistent over the base theory? What are the proof-theoretic strengths (...) of the consistent theories? The first question was answered completely by Friedman and Sheard; all subsets of the Optional Axioms were classified as either consistent or inconsistent giving rise to nine maximal consistent theories of truth.They also determined the proof-theoretic strength of two subsets of the Optional Axioms. The aim of this paper is to continue the work begun by Friedman and Sheard. We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to well-known theories ranging from PA to the theory of ${\Sigma^1_1}$ dependent choice. (shrink)

In this paper, we are concerned with the preorderings (SS) and (BC) induced in the set of players of a simple game by the Shapley–Shubik and the Banzhaf–Coleman's indices, respectively. Our main result is a generalization of Tomiyama's 1987 result on ordinal power equivalence in simple games; more precisely, we obtain a characterization of the simple games for which the (SS) and the (BC) preorderings coincide with the desirability preordering (T), a concept introduced by Isbell (1958), and recently reconsidered by (...) Taylor (1995): this happens if and only if the game is swap robust, a concept introduced by Taylor and Zwicker (1993). Since any weighted majority game is swap robust, our result is therefore a generalization of Tomiyama's. Other results obtained in this paper say that the desirability relation keeps itself in all the veto-holder extensions of any simple game, and so does the (SS) preordering in all the veto-holder extensions of any swap robust simple game. (shrink)

We will introduce a partial ordering $\preceq_1$ on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use $\preceq_1$ to provide a new characterization of the ubiquitous ordinal $\epsilon _{0}$.

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the$\Pi ^1_1$reflection strength order. We prove that there are no descending sequences of$\Pi ^1_1$sound extensions of$\mathsf {ACA}_0$in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any$\Pi (...) ^1_1$sound extension of$\mathsf {ACA}_0$. We prove that for any$\Pi ^1_1$sound theoryTextending$\mathsf {ACA}_0^+$, the reflection rank ofTequals the$\Pi ^1_1$proof-theoretic ordinal ofT. We also prove that the$\Pi ^1_1$proof-theoretic ordinal of$\alpha $iterated$\Pi ^1_1$reflection is$\varepsilon _\alpha $. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles. (shrink)

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). While STT, understood as (...) a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

For every partial combinatory algebra, we define a hierarchy of extensionality relations using ordinals. We investigate the closure ordinals of pca’s, i.e., the smallest ordinals where these relations become equal. We show that the closure ordinal of Kleene’s first model is ${\omega _1^{\textit {CK}}}$ and that the closure ordinal of Kleene’s second model is $\omega _1$. We calculate the exact complexities of the extensionality relations in Kleene’s first model, showing that they exhaust the hyperarithmetical hierarchy. We also discuss embeddings of (...) pca’s. (shrink)

We define ordinal representations in the simply typed lambda calculus, and consider the ordinal functions representable with respect to these notations. The results of this paper have the same flavor as those of Schwichtenberg and Statman on numeric functions representable in the simply typed lambda calculus. We define four families of ordinal notations; in order of increasing generality of the type of notation, the representable functions consist of the closure under composition of successor and α ωα, addition and α ωα, (...) addition and multiplication, and addition, multiplication, and a “weak” discriminator. (shrink)

The minimal ordinal-connection axiom $$MOC$$ was introduced by the first author in R. Freire. (South Am. J. Log. 2:347–359, 2016). We observe that $$MOC$$ is equivalent to a number of statements on the existence of certain hierarchies on the universe, and that under global choice, $$MOC$$ is in fact equivalent to the $${{\,\mathrm{GCH}\,}}$$. Our main results then show that $$MOC$$ corresponds to a weak version of global choice in models of the $${{\,\mathrm{GCH}\,}}$$ : it can fail in models of the (...) $${{\,\mathrm{GCH}\,}}$$ without global choice, but also global choice can fail in models of $$MOC$$. (shrink)

The ordination of women in the Church of England has had a long hard road. Other denominations, and other parts of the Anglican Communion took the step, but it was not until the 1990s that the first women priests were ordained in the Church of England itself. Even then, Emma Percy describes the situation as an ‘ambiguous welcome’. Careful provision has been made at every stage for those who not only will not accept women as priests, but require the service (...) of bishops who have not participated in the ordination of women. The path to acceptance for women bishops has also been lengthy and subject to the same caveats and provisions. Percy argues that this reveals an underlying failure to think theologically about gender. She recognizes that there are still profound inequalities in the Church’s treatment of women in leadership. (shrink)

Modern microeconomic theory is based on a foundation of ordinal preference relations. Good textbooks stress that cardinal utility functions are artificial constructions of convenience, and that economics does not attribute any meaning to “utils.” However, we argue that despite this official position, in practice mainstream economists rely on techniques that assume the validity of cardinal utility. Doing so has turned mainstream economic theorizing into an exercise of reductionism of objects down to the preferences of ‘ideal type’ subjects.

If α and β are ordinals, α ≤ β, and $\beta \nleq \alpha$ , then α + 1 ≤ β. The first result of this paper shows that the restriction of this statement to countable well orderings is provably equivalent to ACA 0 , a subsystem of second order arithmetic introduced by Friedman. The proof of the equivalence is reminiscent of Dekker's construction of a hypersimple set. An application of the theorem yields the equivalence of the set comprehension scheme ACA (...) 0 and an arithmetical transfinite induction scheme. (shrink)

This paper uses the framework of reverse mathematics to analyze the proof theoretic content of several statements concerning multiplication of countable well-orderings. In particular, a division algorithm for ordinal arithmetic is shown to be equivalent to the subsystem ATR0.

The report of a commission set up by Pope Francis to study the question of women as deacons in the Catholic Church was issued in May 2019. Whilst it is well known that the Catholic Church refuses to ordain women, the form of the diaconate being discussed is that of the ‘permanent diaconate’ for men, which was established after the Second Vatican Council. This article first discusses how this issue has arisen, clarifies the types of deacons and reviews the reasons (...) why the Catholic Church refuses to ordain women. It then looks at Scripture and the history of the Church to assess the historical role of women deacons. The issue of women’s ordination emerged after Vatican II. Women’s ministries have grown immensely since then and this is a factor in the question about the ordination of women deacons. There are important theological issues involved in the study around women deacons. Lastly, the article raises questions about the future of this issue under Pope Francis and his successors.Contribution: The issue of the ordination of women deacons in the Roman Catholic Church is a current and contentious issue. This article reviews the historical evidence for women deacons and the views of theologians and Church leaders in order to assess whether there are grounds for hope. (shrink)

We consider two variants of transfinite induction, one with monotonicity assumption on the predicate and one with the induction hypothesis only for cofinally many below. The latter can be seen as a transfinite analogue of the successor induction, while the usual transfinite induction is that of cumulative induction. We calculate the supremum of ordinals along which these schemata for \ formulae are provable in \. It is shown to be larger than the proof-theoretic ordinal \ by power of base 2. (...) We also show a similar result for the structural transfinite induction, defined with fundamental sequences. (shrink)

The technique of using infinitary rules in an ordinal analysis has been one of the most productive developments in ordinal analysis. Unfortunately, one of the most advanced variants, the Buchholz Ωμ rule, does not apply to systems much stronger than -comprehension. In this paper, we propose a new extension of the Ω rule using game-theoretic quantifiers. We apply this to a system of inductive definitions with at least the strength of a recursively inaccessible ordinal.

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fundamental’ than the others. This paper addresses two related issues. First, we review some of (...) these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege's foundationalist epistemology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally ‘legitimate’. (shrink)

In this paper we introduce a recursive notation system O(Π 3 ) of ordinals. An element of the notation system is called an ordinal diagram. The system is designed for proof theoretic study of theories of Π 3 -reflection. We show that for each $\alpha in O(Π 3 ) a set theory KP Π 3 for Π 3 -reflection proves that the initial segment of O(Π 3 ) determined by α is a well ordering. Proof theoretic study for such theories (...) will be reported in [4]. (shrink)

The essay maintains that, despite the significant successive transformations, a ordoliberal moment operates within the neoliberal program. Ordoliberalism establishes a specific political anthropology based on the centrality of economic action and on an a-revolutionary temporality that enhances the normative continuity of the tradition. Even against the specific configurations that the State can assume, especially the democratic one, it also aims at the constant reactivation of a political as a fundamental decision in favor of the economic.

In this paper we introduce a recursive notation system $O(\mu)$ of ordinals. An element of the notation system is called an ordinal diagram following G. Takeuti [25]. The system is designed for proof theoretic study of theories of recursively Mahlo universes. We show that for each $\alpha<\Omega$ in $O(\mu)$ KPM proves that the initial segment of $O(\mu)$ determined by $\alpha$ is a well ordering. Proof theoretic study for such theories will be reported in [9].

G. Jäger gave in Arch. Math. Logik Grundlagenforsch. 24 , 49-62, a recursive notation system on a basis of a hierarchy Iαß of α-inaccessible regular ordinals using collapsing functions following W. Buchholz in Ann. Pure Appl. Logic 32 , 195-207. Jäger's system stops, when ordinals α with Iα0 = α enter. This border is now overcome by introducing additional a hierarchy Jαß of weakly inaccessible Mahlo numbers, which is defined similarly to the Jäger hierarchy. An ordinal μ is called Mahlo, (...) if every normal-function f : μ → μ has regular fixpoints. Collapsing is defined for both Mahlo and simply regular ordinals such that for every Mahlo ordinal μ out of the J-hierarchy Ψμα is a regular σ such that Iσ0 = σ. For these regular σ again collapsing functions Ψσ are defined. To get a proper systematical order into the collapsing procedure, a pair of ordinals is associated to σ and α, and the definition of Ψσα is given by recursion on a suitable well-ordering of these pairs. Thus a fairly large system of ordinal notations can be established. It seems rather straightforward, how to extend this setting further. (shrink)

Negli scritti di Gregorio Magno, lo schema dei _ tres ordines fidelium _ ( _ praedicatores _ - _ continentes _ - _ coniugati _ ) costituisce una descrizione delle forme di vita che compongono la Chiesa. Questo articolo s’interroga sulle ragioni del loro ordine gerarchico: _ praedicatores _ all’apice, _ continentes _ in una posizione intermedia e _ coniugati _ all’ultimo gradino dello schema. A partire da un’analisi di queste tre figure, l’articolo pone in evidenza la loro relazione con (...) le nozioni di vita attiva e di vita contemplativa. In particolare, le configurazioni della gerarchia tra azione e contemplazione emergono come decisive nella comprensione dell’ordine ecclesiologico dei _ tres ordines fidelium _. Tali configurazioni sono illustrate tramite i concetti di ordine e contrordine. Difatti, se da una parte Gregorio ammette generalmente la superiorità della vita contemplativa su quella attiva ( ???? ordine), dall’altra egli considera una forma di vita composita, che unisce azione e contemplazione, come la più elevata ( ???? contrordine). Queste considerazioni paiono spiegare perché, nonostante la vita contemplativa dei _ continentes _ sia giudicata superiore a quella attiva dei _ coniugati _, la vita composita dei _ praedicatores _ occupi la sommità della gerarchia. In conclusione, l’articolo pone in rilievo il ruolo trasversale di vita attiva e di vita contemplativa rispetto alla struttura dei _ tres ordines fidelium _. Rappresentando due istanze che caratterizzano le forme di vita di ciascun _ ordo _, esse rivelano come lo schema dei _ tres ordines fidelium _ esprima al contempo le componenti antropologiche della Chiesa e tre paradigmi di progresso spirituale. _ In the writings of Gregory the Great, the _ tres ordines fidelium _ scheme ( _ praedicatores-continentes-coniugati _ ) delineates the three forms of life that constitute the Church. This article investigates the reasons behind their hierarchical order: _ praedicatores _ at the pinnacle, _ continentes _ in the middle, and _ coniugati _ at the bottom of the scheme. Starting from an analysis of these three figures, the article underscores their relationship with the concepts of active life and contemplative life. Specifically, the configurations of the hierarchy between action and contemplation emerge as crucial to understanding the ecclesiological order of the _ tres ordines fidelium _. These configurations are elucidated through the concepts of order and counter-order. In fact, while Gregory generally acknowledges the superiority of the contemplative life over the active life ( __ ???? _ _ order), he regards a mixed life that integrates action and contemplation as the highest form of life ( __ ???? _ _ counter-order). These observations seem to justify why, although the contemplative life of _ continentes _ is deemed superior to the active life of _ coniugati _, the mixed life of _ praedicatores _ ascends the top of the hierarchy. In the concluding section, the article emphasizes the transversal role of active and contemplative life in the structure of the _ tres ordines fidelium _. Representing two aspects that characterize the forms of life of each _ ordo _, they reveal how the scheme of the _ tres ordines fidelium _ simultaneously expresses the anthropological components of the Church and three paradigms of spiritual progress. _. (shrink)

This article provides a detailed comparison between two systems of collapsing functions. These functions play a crucial role in proof theory, in the analysis of patterns of resemblance, and the analysis of maximal order types of well partial orders. The exact correspondence given here serves as a starting point for far reaching extensions of current results on patterns and well partial orders. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.