We consider total well-founded orderings on monadic terms satisfying the replacement and full invariance properties. We show that any such ordering on monadic terms in one variable and two unary function symbols must have order typeω,ω2orωω. We show that a familiar construction gives rise to continuum many such orderings of order typeω. We construct a new family of such orderings of order typeω2, and show that there are continuum many of these. We show that there are only four such (...) orderings of order typeωω, the two familiar recursive path orderings and two closely related orderings. We consider also total well-founded orderings onNnwhich are preserved under vector addition. We show that any such ordering must have order typeωkfor some 1 ≤k≤n. We show that ifkshrink)

We describe a sequent calculus, based on work of Herbelin, of which the cut-free derivations are in 1-1 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cut-elimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.

We gather the following miscellaneous results in proof theory from the attic.1. 1. A provably well-founded elementary ordering admits an elementary order preserving map.2. 2. A simple proof of an elementary bound for cut elimination in propositional calculus and its applications to separation problem in relativized bounded arithmetic below S21.3. 3. Equivalents for Bar Induction, e.g., reflection schema for ω logic.4. 4. Direct computations in an equational calculus PRE and a decidability problem for provable inequations in PRE.5. 5. Intuitionistic (...) fixed point theories which are conservative extensions of HA.6. 6. Proof theoretic strengths of classical fixed points theories.7. 7. An equivalence between transfinite induction rule and iterated reflection schema over IΣn.8. 8. Derivation lengths of finite rewrite rules reducing under lexicographic path orders and multiply recursive functions.Each section can be read separately in principle. (shrink)

We re-express our theorem on the strong-normalisation of display calculi as a theorem about the well-foundedness of a certain ordering on first-order terms, thereby allowing us to prove the termination of systems of rewrite rules. We first show how to use our theorem to prove the well-foundedness of the lexicographic ordering, the multiset ordering and the recursive path ordering. Next, we give examples of systems of rewrite rules which cannot be handled by these methods (...) but which can be handled by ours. Finally, we show that our method can also prove the termination of the Knuth-Bendix ordering and of dependency pairs. Keywords: rewriting, termination, well-founded ordering, recursive path ordering.. (shrink)

We introduce a new formulation of the axiom of dependent choice, which can be viewed as an abstract termination principle that in particular generalises recursive path orderings, the latter being fundamental tools used to establish termination of rewrite systems. We consider several variants of our termination principle, and relate them to general termination theorems in the literature.

A successivity in a linear order is a pair of elements with no other elements between them. A recursive linear order with recursive successivities U is recursively categorical if every recursive linear order with recursive successivities isomorphic to U is in fact recursively isomorphic to U . We characterize those recursive linear orders with recursive successivities that are recursively categorical as precisely those with order type k 1 + g 1 + k 2 + (...) g 2 +…+ g n -1 + k n where each k n is a finite order type, non-empty for i ϵ{2,…, n -1} and each g i is an order type from among {ω,ω * ,ω+ω * }∪{ k ·η: k. (shrink)

§1. Iterated Gödelian extensions of theories. The idea of iterating ad infinitum the operation of extending a theory T by adding as a new axiom a Gödel sentence for T, or equivalently a formalization of “T is consistent”, thus obtaining an infinite sequence of theories, arose naturally when Godel's incompleteness theorem first appeared, and occurs today to many non-specialists when they ponder the theorem. In the logical literature this idea has been thoroughly explored through two main approaches. One is that (...) initiated by Turing in his “ordinal logics” and taken very much further in Feferman's work on transfinite progressions, which also introduced the more general study of extensions by reflection principles, of which consistency statements are a special case. This approach starts from an assignment of theories to ordinal notations, and extracts sequences of theories through a suitable choice of a path in the set of ordinal notations. The second approach, illustrated in particular by the work of Schmerl and Beklemishev, starts instead from a suitably well-behaved primitive recursive well-ordering, which is used to define a sequence of theories. This second approach has led to precise results about the relative proof-theoretical strength of sequences of theories obtained by iterating different reflection principles. The Turing-Feferman approach, on the other hand, lends itself well to an investigation in qualitative and philosophical terms of the relevance of such iterated reflection extensions to mathematical knowledge, in particular because of two developments associated with this approach. (shrink)

Drawing on biosemiotic theory and the Peircean idea of ‘abduction’, I shall propose the idea of a layered structure of bio / semiotic evolution, in which humanknowledge is systemic and recursive — and thus emergent both from what is forgotten and from earlier evolutionary strata. I will argue that abductions are those processes by which we move creatively between often unacknowledged types of knowledge which are rooted in our natural and cultural evolutionary past (e.g., unconscious, preconscious, or tacit knowledge; (...) knowledge that is experienced affectively) and the more familiar types of knowledge associated with self-conscious deductive and inductive reasoning. I shall suggest that these processes of ‘hooking back’ into the past in order to make new sense in the light of subsequent experience, are characteristic of all human inventiveness in both the arts and the sciences, and are facilitated by what Peirce called ‘The Play of Musement’.My reasons for attempting this task are that I hope to offer a semiotic and biosemiotic corrective to the widespread and culturally dominant idea that the progress ofhuman knowledge and cultural evolution depends on self rational efficiency, conceived of in terms of self-conscious deduction and induction alone — a conception which runs the risk of excluding from the account what is actually the most creative part of human knowing. Second, I will suggest that a properly semiotically informed understanding of human creativity — i.e., one which understands the Peircean semiotic as triadic and which draws on the post-Peircean theory that the semiotic drive in nature and in culture derives from the need to model the world as accurately as possible — should provide a very stern warning against the dangers of confusing the map with the territory. For creative artists and scientists (and life-livers, in general) progress, I shall suggest, inasmuch as they ignore the utilitarian dogma in practice. A biosemiotic understanding of human reasoning as an evolutionary semiotic process should thus contribute to a removal of the impediments of modernity which lie in the failure to properly grasp both what language (and semiosis in general) is, as well as the historical and prehistorical evolution that makes such semiosis possible. (shrink)

We re-express a previous general result in a way which seems easier to remember, using the terminology of infinite games. We show how this can be applied to construct recursive linear orderings, showing, for example, that if there is a ▵ 0 2β + 1 linear ordering of type τ, then there is a recursive ordering of type ω β · τ.

The complexity of aII 4 set of natural numbers is encoded into a linear order to show that the finite condensation of a recursive linear order can beII 2–II 1. A priority argument establishes the same result, and is extended to a complete classification of finite condensations iterated finitely many times.

It is shown that for every nonzero r.e. degree c there is a linear ordering of degree c which is not isomorphic to any recursive linear ordering. It follows that there is a linear ordering of low degree which is not isomorphic to any recursive linear ordering. It is shown further that there is a linear ordering L such that L is not isomorphic to any recursive linear ordering, and L together (...) with its ‘infinitely far apart’ relation is of low degree. Finally, an analogue of the recursion theorem for recursive linear orderings is refuted. (shrink)

We consider intuitionistic number theory with recursive infinitary rules . Any primitive recursive binary relation for which transfinite induction schema is provable is in fact well founded. Its ordinal is less than ε 0 if the transfinite induction schema is intuitionistically provable in elementary number theory. These results are provable intuitionistically. In fact, it suffices to consider transfinite induction with respect to one particular number-theoretic property.

We prove a model theoretic generalization of an extension of the Keisler-Morley theorem for countable recursively saturated models of theories having a K-like model, where K is an inaccessible cardinal.

This paper is concerned with implicit computational complexity of the exptime computable functions. Modifying the lexicographic path order, we introduce a path order EPO. It is shown that a termination proof for a term rewriting system via EPO implies an exponential bound on the lengths of derivations. The path order EPO is designed so that every exptime function is representable as a term rewrite system compatible with EPO (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

This article examines central tensions in cybernetics, defined as the study of self-organization, communication, automated feedback in organisms, and other distributed informational networks, from its wartime beginnings to its contemporary adaptations. By examining aspects of both first- and second-order cybernetics, the article introduces an epistemological standpoint that highlights the tension between its definition as a theory of recursion and a theory of control, prediction, and actionability. I begin by examining the historical outcomes of the Macy Conferences to provide a context (...) for cybernetics’ initial development for scientific epistemology, ethics, and socio-political thought. I draw extensively from Norbert Wiener, Heinz von Foerster, Ross Ashby, and Gregory Bateson, key figures of this movement. I then elaborate upon certain premises of cybernetics to further elucidate an intellectual history from which to begin to construct a cybernetic epistemology. I conclude by offering the second-order cybernetic concept of recursivity as a model and method for ethico-epistemological questioning that can account for both the constructive potential and the limitations of cybernetics in science and society. (shrink)

In this paper first order theories for nonmonotone inductive definitions are introduced, and a proof-theoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given.

The paper characterizes the second order arithmetic theorems of a set theory that features a recursively Mahlo universe; thereby complementing prior proof-theoretic investigations on this notion. It is shown that the property of being recursively Mahlo corresponds to a certain kind of β-model reflection in second order arithmetic. Further, this leads to a characterization of the reals recursively computable in the superjump functional.

In [6], Metakides and Nerode introduced the study of the lattice of recursively enumerable substructures of a recursively presented model as a means to understand the recursive content of certain algebraic constructions. For example, the lattice of recursively enumerable subspaces,, of a recursively presented vector spaceV∞has been studied by Kalantari, Metakides and Nerode, Retzlaff, Remmel and Shore. Similar studies have been done by Remmel [12], [13] for Boolean algebras and by Metakides and Nerode [9] for algebraically closed fields. In (...) all of these models, the algebraic closure of a set is nontrivial., is given in §1, however in vector spaces, cl is just the subspace generated byS, in Boolean algebras, cl is just the subalgebra generated byS, and in algebraically closed fields, cl is just the algebraically closed subfield generated byS.)In this paper, we give a general model theoretic setting in which we are able to give constructions which generalize many of the constructions of classical recursion theory. One of the main features of the modelswhich we study is that the algebraic closure of setis just itself, i.e., cl = S. Examples of such models include the natural numbers under equality 〈N, = 〉, the rational numbers under the usual ordering 〈Q, ≤〉, and a large class ofn-dimensional partial orderings. (shrink)

The idea of this paper is to approach linear orderings as generalized ordinals and to study how they are made from their initial segments. First we look at how the equality of two linear orderings can be expressed in terms of equality of their initial segments. Then we shall use similar methods to define functions by recursion with respect to the initial segment relation. Our method is based on the use of a game where smaller and smaller initial segments of (...) linear orderings are considered. The length of the game is assumed to exceed that of the descending sequences of elements of the linear orderings considered. By use of such game-theoretical methods we can for example extend the recursive definitions of the operations of sum, product and exponentiation of ordinals in a unique and natural way for arbitrary linear orderings. Extensions coming from direct limits do not satisfy our game-theoretic requirements in general. We also show how our recursive definitions allow very simple constructions for fixed points of functions, giving rise to certain interesting linear orderings. (shrink)

By , we denote the system of second-order arithmetic based on recursive comprehension axioms and Σ10 induction. is defined to be plus weak König's lemma: every infinite tree of sequences of 0's and 1's has an infinite path. In this paper, we first show that for any countable model M of , there exists a countable model M′ of whose first-order part is the same as that of M, and whose second-order part consists of the M-recursive sets (...) and sets not in the second-order part of M. By combining this fact with a certain forcing argument over universal trees, we obtain the following result : if proves X!Y with arithmetical, so does . We also discuss several improvements of this results. (shrink)

The so-called weak König's lemma WKL asserts the existence of an infinite path b in any infinite binary tree . Based on this principle one can formulate subsystems of higher-order arithmetic which allow to carry out very substantial parts of classical mathematics but are Π 2 0 -conservative over primitive recursive arithmetic PRA . In Kohlenbach 1239–1273) we established such conservation results relative to finite type extensions PRA ω of PRA . In this setting one can consider also (...) a uniform version UWKL of WKL which asserts the existence of a functional Φ which selects uniformly in a given infinite binary tree f an infinite path Φf of that tree. This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in Kohlenbach [10] actually can be used to eliminate even this uniform weak König's lemma provided that PRA ω only has a quantifier-free rule of extensionality QF-ER instead of the full axioms of extensionality for all finite types. In this paper we show that in the presence of , UWKL is much stronger than WKL: whereas WKL remains to be Π 2 0 -conservative over PRA, PRA ω ++ UWKL contains full Peano arithmetic PA. We also investigate the proof–theoretic as well as the computational strength of UWKL relative to the intuitionistic variant of PRA ω both with and without the Markov principle. (shrink)

We consider first order modal logic C firstly defined by Carnap in “Meaning and Necessity” [1]. We prove elimination of nested modalities for this logic, which gives additionally the Skolem-Löwenheim theorem for C. We also evaluate the degree of unsolvability for C, by showing that it is exactly 0′. We compare this logic with the logics of Henkin quantifiers, Σ11 logic, and SO. We also shortly discuss properties of the logic C in finite models.

This article discusses approaches to the problem of the number of preference orderings that can be constructed from a given set of alternatives. After briefly reviewing the prevalent approach to this problem, which involves determining a partitioning of the alternatives and then a permutation of the partitions, this article explains a recursive approach and shows it to have certain advantages over the partitioning one.

continent. 2.3 (2012): 218–223 Vijay Prashad. Arab Spring, Libyan Winter . Oakland: AK Press. 2012. 271pp, pbk. $14.95 ISBN-13: 978-1849351126. Nearly a decade ago, I sat in a class entitled, quite simply, “Corporations,” taught by Vijay Prashad at Trinity College. Over the course of the semester, I was amazed at the extent of Prashad’s knowledge, and the complexity and erudition of his style. He has since authored a number of classic books that have gained recognition throughout the world. The Darker (...) Nations , a peoples’ history of the Third World, sent defibrillating shockwaves through an academic world that had almost forgotten the epic scope and historic dignity of the non-aligned movement and post-colonial struggles. In his recent, award-winning work, Arab Spring, Libyan Winter , Prashad delves into his capacious knowledge of the Third World to excavate the discourses and narratives surrounding the upheavals of 2011. “Revolutions have no specific timetable,” states Prashad in the opening line of this exciting and provocative look at contemporary events. Instantly, an atmosphere of suspense emerges. The reader is alerted to the problem of history. Does the making of history then involve a suspension of historical time, or is it a continuous narrative structure into which events must eventually be integrated? Arab Spring, Libyan Winter shows that the answer to the question, “How is history made?” lies just as easily in the asking. The first half of the book works through the events that transpired to bring about the explosive popular uprisings of Arab Spring: Tunisia and Egypt, Yemen and Bahrain. But the events are not put together in a cohesive, chronological fashion. Using an uncommonly gripping style more akin to the folk-story motif of the djeli than to traditionally Orientalist academia, Prashad suspends a given situation, points out its components, and traces back the characters and genealogies that define each link before resuming a narrative. Every moment is an end to itself, and an origin of something different. Thus, the making of history becomes the suspension of its own progress, it’s catastrophic 'stability,' in a process of differentiation through inclusion. Empirically, one might suggest that history is a condition of time, and by extension, of the subject, but history is also an eminent producer of the Subject and her concept of time through memory and narrative. Hence, history is often overdetermined by a dominant narrative of the sovereign. The apparently chaotic composition of Prashad’s historicity is, then, an interstitial morphology of resistance. It illustrates that the time to act for the revolutionary lies in the gaps within the dominant historical realities. It points out the sorties of signifiers constituting nothing in the obliterated ruins of history. Through such illuminations, Arab Spring, Libyan Winter breaks through “the surface of history” to elaborate revolution’s snapping synapses, exploding interruptions and breaches, connecting flows. Messianic Politics It is tempting to think, 'History is either singular, or it is diffuse. It is either one coherent factual narrative, or it is comprised of the aletheia of the multitude.' Yet the reader finds Prashad examining the structure of history on multiple levels. For Prashad, world history is shaped by geographically defined political movements, such as communism or national liberation, with historical agents like the working class for the former and the nationalist militant for the latter. Religion, however, is another matter: “Religion has an unshakable eschatology which a post-utopian secular politics lacks.” Within the telos of religion, there appears on the horizon the image of Benjamin’s thoughts on messianic time—time as “a small fissure in the continuous catastrophe,” a break with history, may contain an ultimate redemption of the human beyond the political power struggle. Succoring the split between geo-political and utopian-religious (messianic) time in the context of Arab Spring, Prashad indicates that the base of “the deep desire and commitment to some form of democracy” is forged by the affinity between eschatological Islamist politics with the People. To think about the Subject of messianic time in the context of the lack of a “coherent timetable” for emancipatory politics, it is perhaps best to return to the modern tradition of Martin Luther King, Jr., whose prefigurative blending of liberation theology and emancipatory politics became most important during his works of 1963, the climactic year of “The Letter from a Birmingham Jail,” the March on Washington, and the bombing of the 16th Street Baptist Church. In “The Letter,” King states the point bluntly: Frankly, I have yet to engage in a direct-action campaign that was “well timed” in view of those who have not suffered unduly from the disease of segregation. For years now I have heard the word “wait!” It rings in the ear of every Negro with piercing familiarity. This “Wait” has almost always meant ‘Never.” We must come to see, with one of our distinguished jurists, that “justice too long delayed is justice denied.” The timing of rebellious action must not be a part of history. It must change history. The kairos appears spontaneous and untimely in its convulsive, shocking presence, yet it is, deeper still, a path, a longue durée , forged through diligent and rigorous praxis. King put a finer point on the path of historic liberation in his declaration in his 1963 speech at Western Michigan University: “[T]ime is neutral.... Somewhere along the way we must see that time will never solve the problem alone but that we must help time. Somewhere we must see that human progress never rolls in on the wheels on inevitability. It comes through the tireless efforts and the persistent work of dedicated individuals who are willing to be co-workers with God. Without this hard work, time itself becomes an ally of the insurgent and primitive forces of irrational emotionalism and social stagnation. We must always help time and realize that the time is always right to do right.” Time, then, exists in “the neutral,” while action must turn to the kairos , not only of taking time, but taking the right time. By turning our relation to time into kairos , we move time from its context within history to a duration between histories—the longue dure?e of Messianic time. The implementation of neutrality in this context suggests a broad space of time to extend through the valley of positive and negative. In Roland Barthes’s lectures from 1978, the neutral takes place in this minimal distance of non-conflict that exists outside of two extremes. It is, in the words of Maurice Blanchot, “the non-general, the non-generic, as well as the non-specific.” For Blanchot, the neutrality of time permeates life as death, impassive and beyond control; such is the neutrality of messianic time, which permeates the charge of history, changing the "I" into the "one." The neutrality of time therefore implicates history in a case against extremism, showing that the work of the radical is not to tend to any particular side, but to have the time to navigate through the terrible terrain of history by altering its course, its patterns and rhythms of movement, connection, sociality. Hence, one does not simply “make history,” one alters the course of history in accordance to “the right time.” The historical path toward emerging power is transmitted as a gesture to the outside of history, against history; a gesture as simple as reaching out. Thus a counterhistory, to use Foucault’s term, is brought forward as the guide of time past a point of no return, a Rubicon, the point where the normal path of history falls to the past. The subject, who must be the only true agent of history, facilitates time through helpfulness, and both the Subject and her history become decentered in relation to one another. Yet history negates the neutrality of time. As Barthes discloses, although the neutral is the “thought and practice of the nonconflictual, it is nevertheless bound to assertion, to conflict, in order to make itself heard.” Engaged with history, time is charged with a destiny, and becomes an opposition. As Prashad insists, “For Arab lands, the events of early 2011 were not the inauguration of a new history, but the continuation of an unfinished struggle that is a hundred years old.” The emergence of the Subject during Arab Spring did not conceive of a new history, but changed the path of history toward a different destiny. “Historical grievances combined with inflationary pressures now met with the subjective sense that victory might be at hand—this was not simply a protest to scream into the wind, but a protest to actually remove autocrats from their positions of authority. The facts of resistance had given way to the expectation of revolutionary change.” The negativity of this charge toward “revolutionary destiny” rendered control, as a positive force, seriously lacking. The lack of control lies in the problem that the Subject is not totally detemporalized or timeless, but untimely in her presence. The Subject is untimely, because her work is visionary, and it is only through such visionary work that the Rubicon can be crossed. Still, the crossing of this border is haunted by anxiety over an impending disaster that lays in wait. It is only through the form of what Benjamin calls divine violence, captivated not with the justice of the means, but the ends—the transformation of history—that a revelation of history’s traumatic foundations can be liberated, and patterns and rhythms of time developed throughout obscured traditions awakened. In this situation, the Subject appears to be outside of right, but setting the state to rights. Because her position is correct, in-so-far as the rebelling subject rebels due to a lack of recognition, her representation appears outside of the norm, which is mistaken as right. Therefore, such visionary work must be carried out through obscured traditions, underground, away from the surveillance of empire. Explaining his method from the start along the lines of Marx’s metaphor of the mole, Prashad insists, “It is the burrowing that is essential, not simply the emergence onto the surface of history.” Arab Spring, Libyan Winter is a book on uncharted networks of time, spread out over the 252 pages like an elaborate spider’s web of passages intertwining with and overwhelming the machinery of the state. Although untimely, Arab Spring was not a flash in the pan, or a Facebook or Twitter revolution. Prashad notes that the government’s suspension of these tools led to further radicalism by furthering communications through face-to-face encounters. In existential terms, Arab Spring might be thought of as a revolution of Being over techne , an uprising of the unchartable, infinite potential of the Other. The name of this Other is found in Chance. The faith of the revolution lies in the proper decentering of the subject, its giving to the Other of time, for only with respect to time does history actually appear on the horizon of the subject, rather than as an imposition. This time of historical agency appears as a moment when anything can happen—a revolutionary truth event where everything comes into question while being realized in its Otherness as a community of the people begins anew in the streets amidst discourse, friendship, reconfigurations of hegemony, and a becoming of a constituent power. Throwing a wrench into the gears of the “cobwebbed tradition” of Orientalism, the decentering of history in Arab Spring, Libyan Winter is a defining quality of its impassioned revolutionary cry. But perhaps the most paradigmatic points of the book emerge from the unexpected voices. To make the connection between the particular manifestations of revolutionary demands to the general historic terms of revolution, Prashad quotes an anonymous young Egyptian in Tahrir Square, who reminds us, “[T]he French Revolution took a very long time so the people could eventually get their rights.” Far from timelessness, the historicization of Arab Spring must exist precisely within the most uncanny appearance of time, as something that does not appear to come from the natural state of time as we know it, but in arriving has completely transformed the way that we understand time. Strange Monsters Revolution is never as simple as a revolt from below against a rusting structure of elites trying to remain in power. As with the French Revolution, Arab Spring consisted of complex familial ties, outside interests, and religious factions fighting alongside, often in awkward juxtaposition to, liberals, working class parties, farmers, and students. It is perhaps because of this historically difficult and incongruous composition that the Arabic word for revolution is thawra , referring to the image of the bull, or thawr , which has religious significance as a pagan deity for the Ancient Semitic tribes and Cartheginians. If Daniel Guérin was correct in saying, “Anarchism and Marxism drink from the same spring,” then it is quite a different oasis that drove the thirsty beast of revolution through what Prashad calls, “the Libyan labyrinth.” Recalling Bataille’s Acéphale, the intestinal labyrinth gains significance as the mode of metabolism and rumination in which, “(the Acephale) has lost himself, loses me with him, and in which I discover myself as him, in other words, as a monster.” Unraveling the discourse of the revolution, the flows of power and hegemony—from Qaddafi’s nationalist coup in 1969 to the wars with Chad from 1978 to 1987, the neoliberal reforms of the 1990s to, finally, the War on Terror—we find that the metabolic process of Libyan Winter ends in the production of oil. In a process of what Prashad calls “involution,” Qaddafi’s bellicose policies, along with his attempts to nationalize Islam together with oil, mutated the source of his power, turning his house against itself. Already a divided nation, with its two major cities on opposite geographic sides of the country, Libya became split between two opposed political powers—liberal reformists and staunch nationalists—as Qaddafi’s political disengagement manifested through what can only be described in psychoanalytic terms as a recursive passage a l’act (mysteriously calling Wikipedia “Kleenex,” declaring that opponents drank hallucinogens with their instant coffee, and so on). Prashad declares, “The mercurial style that Qaddafi adopted was not about his personality alone, but also a leader’s natural response to a system that relied upon power brokers whose own loyalty… did not have any ideological commitment to the system.” As in the case of other nations during Arab Spring, the true expression of revolt was an outward exposition of what had inwardly been happening in microcosmic societies throughout the realm—from indigenous tribes to political parties, for a century, the people, partly motivated by (and in resistance to) economic impositions of development, had been changing the traditional social compositions and participating in what Ruth Wilson Gilmore calls, “the relative autonomy of the movement from the leaders.” It was this empowerment of political reconfigurations taking place under the context of new democratic assemblages that shocked the elites and evoked such a powerful response. Unlike Qaddafi’s awkward policy choices and rants, real acts of power from the West were clear and decisive. Prashad sets the stages of war and diplomacy, far removed from the deserts, mountains, and cities that forged the backdrop of popular politics. Taking place under the now-classic architecture of the “four pillars” of US interests—oil, the War on Terror, Israel, and the circumvention of Iranian hegemony—we find elites rubbing elbows in “Heliopolist cocktail parties and hushed conferences in Kasr al-Ittihadiya” as well as the Concorde-Lafayette hotel in Paris, which provides the setting for a meeting of anti-Qaddafi figures consolidating their power. These scenes are buttressed with the careful portraiture of key historical actors. As Prashad brings the stage of history to life, we find the diplomats and liberals like Frank Wisner, whose career has brought him from Enron in the late 1990s to the Obama Administration, under the aegis of which he was meeting with Mubarak about military support during Arab Spring. We spy the provocateurs, for instance, gauche cavalier , Bernard Henri-Levy, who telephones Sarkozy from Benghazi about the need for more NATO air strikes. We follow the rebel military establishment as it suffers mysterious deaths and even more mysterious assents (like that of apparent CIA cohort, Khalifa Hifter). In each of these intriguing characters, we find different representations of the security state biopolitique: an oil-injected reification that drives Arab Spring from the resentment of rising food prices to the brink of implosion in Libyan Winter. Reminiscent of the scenes of Cold War soirées represented in old Bond films, the aristocratic fight for oil against democracy that was Libyan Winter presents a harsh truth that the new global crisis is simply the continuation of the old history: the global exploitation of capitalism waged against the imagination of the people. As Benjamin laments, “The labyrinth is the right path for the person who always arrives early enough at his destination. This destination is the marketplace.” Yet, “(t)he labyrinth is the habitat of… a humanity (a class) which does not want to know where its destiny is taking it.” Thus, the labyrinth leads, like the desert, only further into itself. As Blanchot explains, “The desert is even less certain than the world; it is never anything but the approach to the desert.” Only the visionary can take time out of the involution toward oblivion. It is here, in this approach to and escape from history, that we find ourselves within Benjamin’s Golgotha, Blanchot’s desert: A space of indeterminate uncertainty where we become familiar only with our own exile. Blanchot writes, “For the moderated and moderate man, the room, the desert, and the world are strictly determined places. For the man of the desert and the labyrinth, devoted to the error of a journey necessarily a little longer than his life, the same space will be truly infinite, even if he knows that it is not, all the more so since he knows it.” The desert, as allegory, provides an eschatology, a mortality in the destiny of Arab Spring. But any allegory of nature might lead to a labyrinthine eschatology (for example, in Bachelard, the forest presents "a limitless world"); the prescience of the untimely is always an uncanny acceptance of the infinite within the finite—the outside of what is understood. Transgenerational and occupied with the image of the future, the untimely makes otherness its home as it proceeds toward liberation. In the work of helping time navigate this dangerous passage of history, Prashad illustrates that the more violent break with history may have occurred in the peaceable struggles of Tahrir Square, and not in the military clashes of Qaddafi with his former Generals who defected to the CIA and NATO countries. It might be possible to suggest, then, that in the case of Tunisia and Egypt, the historical subject of the people was drawn together in a Dionysian dance of different rhythms in the marvelous realm of the ancients, while we fear that the case of Libya suggests an Apollonian future where time, itself, may lay dying under the machinic hand of history. The properly Nietzschean inversion would follow: time is dead, for history has killed it. Yet, if time is dead, struck down in Golgotha, exposed in the desert, mauled by the thawra , it is only in the present sense that it is rendered impossible outside of the context into which the untimely has thrown us. Thus, if history becomes an art of revolution, life as being-towards-death (death as the provocation of the neutrality of time) becomes what Benjamin calls “the allegory of resurrection” through the glorious ruins of history. Prashad ends his work with a rousing finale: “The time of the impossible has presented itself. In Egypt, where the appetite for the possibilities of the future are greatest, the people continue to assert themselves into Tahrir Square and other places, pushing to reinvigorate a Revolution that must not die… For them the slogan is simple: Down with the Present. Long live the Future. May it be so.&rdquo. (shrink)

We provide three new results about interpolating 2-r.e. or 2-REA degrees between given r.e. degrees: Proposition 1.13. If c h are r.e. , c is low and h is high, then there is an a h which is REA in c but not r.e. Theorem 2.1. For all high r.e. degrees h g there is a properly d-r.e. degree a such that h a g and a is r.e. in h . Theorem 3.1. There is an incomplete nonrecursive r.e. A (...) such that every set REA in A and recursive in 0′ is of r. e. degree. The first proof is a variation on the construction of Soare and Stob . The second combines highness with a modified version of the proof strategy of Cooper et al. . The third theorem is a rather surprising result with a somewhat unusual proof strategy. Its proof is a 0‴ argument that at times moves left in the tree so that the accessible nodes are not linearly ordered at each stage. Thus the construction lacks a true path in the usual sense. Two substitute notions fill this role: The true nodes are the leftmost ones accessible infinitely often; the semitrue nodes are the leftmost ones such that there are infinitely many stages at which some extension is accessible. Another unusual feature of the construction is that it involves using distinct priority orderings to control the interactions of different parts of the construction. (shrink)

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditionalω. Namely, the equivalence between normal transfinite recursion scheme and newdependent transfinite recursionscheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universeVof sets is treated as the given (...) totality. (shrink)