We introduce a class of set-theoretic operators on a tolerance space that models the process of minimal belief contraction, and therefore a natural process of iterated contraction can be defined. We characterize the class of contraction operators and study the properties of the associated iterated belief contraction.

We describe a model of iterated belief revision that extends the AGM theory of revision to account for the effect of a revision on the conditional beliefs of an agent. In particular, this model ensures that an agent makes as few changes as possible to the conditional component of its belief set. Adopting the Ramsey test, minimal conditional revision provides acceptance conditions for arbitrary right-nested conditionals. We show that problem of determining acceptance of any such nested conditional can be reduced (...) to acceptance tests for unnested conditionals. Thus, iterated revision can be accomplished in a "virtual" manner, using uniterated revision. (shrink)

Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead. Contrasts are drawn to accounts of induction by Poincaré, Heyting, and Kreisel. On the phenomenological side, the analysis provides an (...) alternative to that in Tieszen’s Mathematical Intuition, and confirms a view of Gödel on his Dialectica Interpretation. (shrink)

In 'Intuition, iteration, induction', Mark van Atten argues that iteration is an object of intuition for Brouwer and explains the intuitive character of the act of iteration drawing from Husserl’s phenomenology. I find the arguments for this reading of Brouwer unconvincing. In this note I set out some issues with his claim that iteration is an object of intuition and his Husserlian explication of iteration. In particular, I argue that van Atten does not accomplish his (...) goals due to tensions with Brouwer’s comments on second-order mathematics and because Husserl does not understand the experience of succession as Brouwer does. (shrink)

Two of the major problems in AGM-style belief revision, namely the difficulties in accounting for iterated change and for Ramsey test conditionals, have satisfactory solutions in descriptor revision. In descriptor revision, the input is a metalinguistic sentence specifying the success condition of the operation. The choice mechanism selects one of the potential outcomes in which the success condition is satisfied. Iteration of this operation is unproblematic. Ramsey test conditionals can be introduced without giving rise to the paradoxical results that (...) they generate in other systems. In addition to standard Ramsey test conditionals, a more general variant of epistemic conditionals is defined, representing statements of the form ”if the belief state is changed to satisfy condition A then it will satisfy condition B”. An axiomatic characterization of such descriptor conditionals is presented. It is related in intricate ways to the KLM postulates for cumulative reasoning. (shrink)

The axiom of recovery, while capturing a central intuition regarding belief change, has been the source of much controversy. We argue briefly against putative counterexamples to the axiom—while agreeing that some of their insight deserves to be preserved—and present additional recovery-like axioms in a framework that uses epistemic states, which encode preferences, as the object of revisions. This makes iterated revision possible and renders explicit the connection between iterated belief change and the axiom of recovery. We provide a representation theorem (...) that connects the semantic conditions we impose on iterated revision and our additional syntactical properties. We show interesting similarities between our framework and that of Darwiche–Pearl (Artificial Intelligence 89:1–29 1997). In particular, we show that intuitions underlying the controversial (C2) postulate are captured by the recovery axiom and our recovery-like postulates (the latter can be seen as weakenings of (C2)). We present postulates for contraction, in the same spirit as the Darwiche–Pearl postulates for revision, and provide a theorem that connects our syntactic postulates with a set of semantic conditions. Lastly, we show a connection between the contraction postulates and a generalisation of the recovery axiom. (shrink)

Using matrix iterations of ccc posets, we prove the consistency with ZFC of some cases where the cardinals on the right hand side of Cichon’s diagram take two or three arbitrary values (two regular values, the third one with uncountable cofinality). Also, mixing this with the techniques in J Symb Log 56(3):795–810, 1991, we can prove that it is consistent with ZFC to assign, at the same time, several arbitrary regular values on the left hand side of Cichon’s diagram.

Recent work has considered the problem of extending to the case of iterated belief change the so-called `Harper Identity' (HI), which defines single-shot contraction in terms of single-shot revision. The present paper considers the prospects of providing a similar extension of the Levi Identity (LI), in which the direction of definition runs the other way. We restrict our attention here to the three classic iterated revision operators--natural, restrained and lexicographic, for which we provide here the first collective characterisation in the (...) literature, under the appellation of `elementary' operators. We consider two prima facie plausible ways of extending (LI). The first proposal involves the use of the rational closure operator to offer a `reductive' account of iterated revision in terms of iterated contraction. The second, which doesn't commit to reductionism, was put forward some years ago by Nayak et al. We establish that, for elementary revision operators and under mild assumptions regarding contraction, Nayak's proposal is equivalent to a new set of postulates formalising the claim that contraction by ¬A should be considered to be a kind of `mild' revision by A. We then show that these, in turn, under slightly weaker assumptions, jointly amount to the conjunction of a pair of constraints on the extension of (HI) that were recently proposed in the literature. Finally, we consider the consequences of endorsing both suggestions and show that this would yield an identification of rational revision with natural revision. We close the paper by discussing the general prospects for defining iterated revision in terms of iterated contraction. (shrink)

Iterative characterizations of computable unary functions are useful patterns for the definition of programming languages based on iterative constructs. The features of such a characterization depend on the pairing producing it: this paper offers an infinite class of pairings involving very nice features.

This is a discussion of the problem of extending the basic AGM belief revision theory to iterated belief revision: the problem of formulating rules, not only for revising a basic belief state in response to potential new information, but also for revising one’s revision rules in response to potential new information. The emphasis in the paper is on foundational questions about the nature of and motivation for various constraints, and about the methodology of the evaluation of putative counterexamples to proposed (...) constraints. Some specific constraints that have been proposed are criticized. The paper emphasizes the importance of meta-information—information about one’s sources of information—and argues that little of substance can be said about constraints on iterated belief revision at a level of abstraction that lacks the resources for explicit representation of meta-information. (shrink)

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). A (...) modal stage theory, MST, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that MST interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory. (shrink)

Let N be a transitive model of ZFC such that ωN ⊂ N and P(R) ⊂ N. Assume that both V and N satisfy "the core model K exists." Then KN is an iterate of K. i.e., there exists an iteration tree J on K such that J has successor length and $\mathit{M}_{\infty}^{\mathit{J}}=K^{N}$. Moreover, if there exists an elementary embedding π: V → N then the iteration map associated to the main branch of J equals π ↾ K. (...) (This answers a question of W. H. Woodin, M. Gitik, and others.) The hypothesis that P(R) ⊂ N is not needed if there does not exist a transitive model of ZFC with infinitely many Woodin cardinals. (shrink)

The semantic automata framework, developed originally in the 1980s, provides computational interpretations of generalized quantifiers. While recent experimental results have associated structural features of these automata with neuroanatomical demands in processing sentences with quantifiers, the theoretical framework has remained largely unexplored. In this paper, after presenting some classic results on semantic automata in a modern style, we present the first application of semantic automata to polyadic quantification, exhibiting automata for iterated quantifiers. We also discuss the role of semantic automata in (...) linguistic theory and offer new empirical predictions for sentence processing with embedded quantifiers. (shrink)

Iterated conditionals of the form If p, then if q, r are an important topic in philosophical logic. In recent years, psychologists have gained much knowledge about how people understand simple conditionals, but there are virtually no published psychological studies of iterated conditionals. This paper presents experimental evidence from a study comparing the iterated form, If p, then if q, r with the “imported,” noniterated form, If p and q, then r, using a probability evaluation task and a truth-table task, (...) and taking into account qualitative individual differences. This allows us to critically contrast philosophical and psychological approaches that make diverging predictions regarding the interpretation of these forms. The results strongly support the probabilistic Adams conditional and the “new paradigm” that takes this conditional as a starting point. (shrink)

In this paper it is argued that, in order to solve the problem of iterated belief change, both the belief state and its input should be represented as epistemic entrenchment (EE) relations. A belief revision operation is constructed that updates a given EE relation to a new one in light of an evidential EE relation. It is shown that the operation in question satisfies generalized versions of the Gärdenfors revision postulates. The account offered is motivated by Spohn's ordinal conditionalization functions, (...) and can be seen as the Jeffrization of a proposal considered by Rott. (shrink)

The numerous systems designed to facilitate animals’ use of computers often are specific to the animals involved, their unique context, and the applications – enrichment among them. Hence, several development methods have arisen in parallel, largely transposed from the human-computer interaction (HCI) domain. In light of that prior work, the paper presents a step-by-step guide for iteratively designing and constructing interactive computers for animals, informed by the rich history of HCI yet applying animal-centred principles, to enrich animal-computer interaction. For each (...) stage in the iterative design (requirements, ideation, prototyping, and testing), the author reflects on real-world experience of building interactive devices for various animals. The paper concludes with overarching considerations vital for future practice of developing interactive computers for animals. Thus, it serves as a valuable reference and information source for researchers designing novel computer systems for animals. (shrink)

Epistemic iteration principles are principles according to which some or another epistemic operator automatically iterates---e.g., if it is known that P, then it is known that P, or there is evidence that P, then there is evidence that there is evidence that P. This article provides a survey of various arguments for and against epistemic iteration principles, with a focus on arguments relevant to a wide range of such principles.

We investigate iterating the construction of $C^{*}$, the L-like inner model constructed using first order logic augmented with the “cofinality $\omega $ ” quantifier. We first show that $\left (C^{*}\right )^{C^{*}}=C^{*}\ne L$ is equiconsistent with $\mathrm {ZFC}$, as well as having finite strictly decreasing sequences of iterated $C^{*}$ s. We then show that in models of the form $L[U]$ we get infinite decreasing sequences of length $\omega $, and that an inner model with a measurable cardinal is required for that.

Machine awareness is a disputed research topic, in some circles considered a crucial step in realising Artificial General Intelligence. Understanding what that is, under which conditions such feature could arise and how it can be controlled is still a matter of speculation. A more concrete object of theoretical analysis is algorithmic iteration for computational intelligence, intended as the theoretical and practical ability of algorithms to design other algorithms for actions aimed at solving well-specified tasks. We know this ability is (...) already shown by current AIs, and understanding its limits is an essential step in qualifying claims about machine awareness and Super-AI. We propose a formal translation of algorithmic iteration in a fragment of modal logic, formulate principles of transparency and faithfulness across human and machine intelligence, and consider the relevance to theoretical research on -AI as well as the practical import of our results. (shrink)

Iterated admissibility embodies a minimal criterion of rationality in interactions. The epistemic characterization of this solution has been actively investigated in recent times: it has been shown that strategies surviving \ rounds of iterated admissibility may be identified as those that are obtained under a condition called rationality and m assumption of rationality in complete lexicographic type structures. On the other hand, it has been shown that its limit condition, with an infinity assumption of rationality ), might not be satisfied (...) by any state in the epistemic structure, if the class of types is complete and the types are continuous. In this paper we analyze the problem in a different framework. We redefine the notion of type as well as the epistemic notion of assumption. These new definitions are sufficient for the characterization of iterated admissibility as the class of strategies that indeed satisfy \. One of the key methodological innovations in our approach involves defining a new notion of generic types and employing these in conjunction with Cohen’s technique of forcing. (shrink)

We define infinitary extensions to classical epistemic logic systems, and add also a common belief modality, axiomatized in a finitary, fixed-point manner. In the infinitary K system, common belief turns to be provably equivalent to the conjunction of all the finite levels of mutual belief. In contrast, in the infinitary monotonic system, common belief implies every transfinite level of mutual belief but is never implied by it. We conclude that the fixed-point notion of common belief is more powerful than the (...) iterative notion of common belief. (shrink)

Discusses the metaphysics of the iterative conception of set.

The iterative conception of set commonly is regarded as supporting the axioms of Zermelo-Fraenkel set theory (ZF). This paper presents a modified version of the iterative conception of set and explores the consequences of that modified version for set theory. The modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets. It is suggested that this modified iterative conception of set supports the axioms of Quine's set theory NF.

The set of unary functions of complexity classes defined by using bounded primitive recursion is inductively characterized by means of bounded iteration. Elementary unary functions, linear space computable unary functions and polynomial space computable unary functions are then inductively characterized using only composition and bounded iteration.

We show that \, the basic theory of bounded arithmetic corresponding to the complexity class \, proves the \ axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the \ iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, \ can also prove the integer division axiom, and the \-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories \ and \. As a side (...) result, we also prove that there is a well-behaved \ definition of modular powering in \\). (shrink)

This paper contrasts two influential theoretical accounts of language change and evolution – Iterated Learning and Social Coordination. The contrast is based on an experiment that compares drawings produced with Garrod et al’s ‘pictionary’ task with those produced in an Iterated Learning version of the same task. The main finding is that Iterated Learning does not lead to the systematic simplification and increased symbolicity of graphical signs produced in the standard interactive version of the task. A second finding is that (...) Iterated Learning leads to less conceptual and structural alignment between participants than observed for those in the interactive condition. The paper concludes with a comparison of the two accounts in relation to how each promotes signs that are effi cient, systematic and learnable. (shrink)

We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let Ω be a regular uncountable cardinal. Let m < ω and M be an m-sound premouse and Σ be...

Let T be an $\omega_{1}-Souslin$ tree. We show the property of forcing notions; "is $\lbrace\omega_{1}\rbrace-semi-proper$ and preserves T" is preserved by a new kind of revised countable support iteration of arbitrary length. As an application we have a forcing axiom which is compatible with the existence of an $\omega_{1}-Souslin$ tree for preorders as wide as possible.

An important approach to game theory is to examine the consequences of beliefs that agents may have about each other. This paper investigates respect for public preferences. Consider an agent A who believes that B strictly prefers an option a to an option b. Then A respects B’s preference if A assigns probability 1 to the choice of a given that B chooses a or b. Respect for public preferences requires that if it is common belief that B prefers a (...) to b, then it is common belief that all other agents respect that preference. Along the lines of Blume, Brandenburger and Dekel [3] and Asheim [1], I treat respect for public preferences as a constraint on lexicographic probability systems. The main result is that given respect for public preferences and perfect recall, players choose in accordance with Iterated Backward Inference. Iterated Backward Inference is a procedure that generalizes standard backward induction reasoning for games of both perfect and imperfect information. From Asheim’s characterization of proper rationalizability [1] it follows that properly rationalizable strategies are consistent with respect for public preferences; hence strategies eliminated by Iterated Backward Inference are not properly rationalizable. (shrink)

This study first investigates robust iterative learning control issue for a class of two-dimensional linear discrete singular Fornasini–Marchesini systems under iteration-varying boundary states. Initially, using the singular value decomposition theory, an equivalent dynamical decomposition form of 2-D LDSFM is derived. A simple P-type ILC law is proposed such that the ILC tracking error can be driven into a residual range, the bound of which is relevant to the bound parameters of boundary states. Specially, while the boundary states of 2-D (...) LDSFM satisfy iteration-invariant boundary states, accurate tracking on 2-D desired surface trajectory can be accomplished by using 2-D linear inequality theory. In addition, extension to 2-D LDSFM without direct transmission from inputs to outputs is presented. A numerical example is used to illustrate the effectiveness and feasibility of the designed ILC law. (shrink)

A tracking control algorithm of nonlinear multiple agents with undirected communication is studied for each multiagent system affected by external interference and input saturation. A control design scheme combining iterative learning and adaptive control is proposed to perform parameter adaptive time-varying adjustment and prove the effectiveness of the control protocol by designing Lyapunov functions. Simulation results show that the high-precision tracking control problem of the nonlinear multiagent system based on adaptive iterative learning control can be well realized even when the (...) input is saturated. Finally, the validity of the proposed algorithm is verified by numerical analysis. (shrink)

We show that $$ VTC ^0$$, the basic theory of bounded arithmetic corresponding to the complexity class $$\mathrm {TC}^0$$, proves the $$ IMUL $$ axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the $$\mathrm {TC}^0$$ iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, $$ VTC ^0$$ can also prove the integer division axiom, and (by our previous results) the $$ RSUV $$ -translation of induction and minimization for sharply (...) bounded formulas. Similar consequences hold for the related theories $$\Delta ^b_1\text{- } CR $$ and $$C^0_2$$. As a side result, we also prove that there is a well-behaved $$\Delta _0$$ definition of modular powering in $$I\Delta _0+ WPHP (\Delta _0)$$. (shrink)

In this paper we propose a conditional logic called IBC to represent iterated belief revision systems. We propose a set of postulates for iterated revision which are a small variant of Darwiche and Pearl''s ones. The conditional logic IBC has a standard semantics in terms of selection function models and provides a natural representation of epistemic states. We establish a correspondence between iterated belief revision systems and IBC-models. Our representation theorem does not entail Gärdenfors'' Triviality Result.

Relative to a hyperstrong cardinal, it is consistent that measure one covering fails relative to HOD. In fact it is consistent that there is a superstrong cardinal and for every regular cardinal κ, κ + is greater than κ + of HOD. The proof uses a very general lemma showing that homogeneity is preserved through certain reverse Easton iterations.

For “natural enough” systems of ordinal notation we show that α times iterated local reflection schema over a sufficiently strong arithmetic T proves the same Π 1 0 -sentences as ω α times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at ε-numbers. We also derive the following more general “mixed” formulas estimating the consistency strength of iterated local reflection: for all ordinals α ⩾ 1 and all β, β ≡ Π 1 0 (...) T ω α · , α ≡ Π 1 0 T β + ω α . Here T α stands for α times iterated local reflection over T , T β stands for β times iterated consistency, and ≡ Π 1 0 denotes mutual Π 1 0 -conservativity. In an appendix to this paper we develop our notion of “natural enough” system of ordinal notation and show that such systems do exist for every recursive ordinal. (shrink)

Iterated conditionals of the form If p, then if q, r are an important topic in philosophical logic. In recent years, psychologists have gained much knowledge about how people understand simple conditionals, but there are virtually no published psychological studies of iterated conditionals. This paper presents experimental evidence from a study comparing the iterated form, If p, then if q, r with the “imported,” noniterated form, If p and q, then r, using a probability evaluation task and a truth‐table task, (...) and taking into account qualitative individual differences. This allows us to critically contrast philosophical and psychological approaches that make diverging predictions regarding the interpretation of these forms. The results strongly support the probabilistic Adams conditional and the “new paradigm” that takes this conditional as a starting point. (shrink)

We consider a classM of Boolean algebras with strictly positive, finitely additive measures. It is shown thatM is closed under iterations with finite support and that the forcing via such an algebra does not destroy the Lebesgue measure structure from the ground model. Also, we deduce a simple characterization of Martin's Axiom reduced to the classM.

Languages are transmitted from person to person and generation to generation via a process of iterated learning: people learn a language from other people who once learned that language themselves. We analyze the consequences of iterated learning for learning algorithms based on the principles of Bayesian inference, assuming that learners compute a posterior distribution over languages by combining a prior (representing their inductive biases) with the evidence provided by linguistic data. We show that when learners sample languages from this posterior (...) distribution, iterated learning converges to a distribution over languages that is determined entirely by the prior. Under these conditions, iterated learning is a form of Gibbs sampling, a widely-used Markov chain Monte Carlo algorithm. The consequences of iterated learning are more complicated when learners choose the language with maximum posterior probability, being affected by both the prior of the learners and the amount of information transmitted between generations. We show that in this case, iterated learning corresponds to another statistical inference algorithm, a variant of the expectation-maximization (EM) algorithm. These results clarify the role of iterated learning in explanations of linguistic universals and provide a formal connection between constraints on language acquisition and the languages that come to be spoken, suggesting that information transmitted via iterated learning will ultimately come to mirror the minds of the learners. (shrink)

We show in this paper that the AGM postulates are too weak to ensure the rational preservation of conditional beliefs during belief revision, thus permitting improper responses to sequences of observations. We remedy this weakness by proposing four additional postulates, which are sound relative to a qualitative version of probabilistic conditioning. Contrary to the AGM framework, the proposed postulates characterize belief revision as a process which may depend on elements of an epistemic state that are not necessarily captured by a (...) belief set. We also show that a simple modification to the AGM framework can allow belief revision to be a function of epistemic states. We establish a model-based representation theorem which characterizes the proposed postulates and constrains, in turn, the way in which entrenchment orderings may be transformed under iterated belief revision. (shrink)

It is known that there is a close relation between Prikry forcing and the iteration of ultrapowers: If U is a normal ultrafilter on a measurable cardinal κ and 〈Mn, jm,n | m ≤ n ≤ ω〉 is the iteration of ultrapowers of V by U, then the sequence of critical points 〈j0,n | n ∈ ω〉 is a Prikry generic sequence over Mω. In this paper we generalize this for normal precipitous filters.

This paper offers a comparative analysis of the uses and formulations of speech-act theory in Derrida's and Deleuze/guattari's work. It begins by juxtaposing Derrida's concept/nonconcept of ‘iterability’ to Deleuze and Guattari's conception of the ‘order-word’ and then examines these theories of the speech act in terms of their implications and consequences for a politics of resistance. Whereas Deleuze and Guattari generate a detailed material stratum — an order-word plateau — for exploring the performative in socio-political contexts, Derrida attends to the (...) singular, wholly unique acts that occur when a repeatable formula such as a political declaration is uttered or otherwise takes place. The paper then concludes by drawing a tentative link between iterability and Deleuze and Guattari's notion of ‘deterritorialisation’ via the order-word plateau. (shrink)

We give a model for iterated belief change in multi-agent systems. The formal tool we use for this is a combination of modal and dynamic logic. Two core notions in our model are the expansion of the knowledge and beliefs of an agent, and the processing of new information. An expansion is defined as the change in the knowledge and beliefs of an agent when it decides to believe an incoming formula while holding on to its current propositional beliefs. To (...) prevent our agents from forming inconsistent beliefs they do not expand with every piece of information they receive. Instead, our agents remember their original beliefs and every piece of information they receive. After every receipt of information they decide which subset of the received information should be incorporated into their original beliefs. This procedure is called the processing of new information. We show that our model of belief update behaves in an intuitive way and that it is not sensitive to criticism on comparable models. (shrink)

In this paper we investigate three approaches to iterated contraction, namely: the Moderate (or Priority) contraction, the Natural (or Conservative) contraction, and the Lexicographic contraction. We characterise these three contraction functions using certain, arguably plausible, properties of an iterated contraction function. While we provide the characterisation of the first two contraction operations using rationality postulates of the standard variety for iterated contraction, we found doing the same for the Lexicographic contraction more challenging. We provide its characterisation using a variation of (...) Epistemic ranking function instead. (shrink)

The interpretation of the utilitarianism of John Stuart Mill has been a matter of controversy at least since J.O. Urmson published his well known paper over twenty-five years ago. Urmson attributed to Mill a form of “rule-utilitarianism”, contrasting his reading with the “received view” on which Mill held a form of “act-utilitarianism”. Since then, the interpretive problem has typically been seen to be that of determining which of these two types of theory should be attributed to Mill, or, at least (...) of determining whether Mill was a “rule-utilitarian”. However, as the distinction is typically made, it is possible to have a utilitarian theory which is neither an act- nor a rule-utilitarian theory. 1n particular, as I will attemptto show, Mill's theory is of neither type but is an example of a sophisticated type of utilitarianism which we might call “iterated-utilitarianism”, for reasons which will become clear in what follows. (shrink)

At first thought, iteration seems banal. It is about repeating the existing; nothing is changing. But this special issue shows that, in an era obsessed with the new, it is often the repetition of the old that creates social change. Iteration fosters persuasion. It affords opportunities for critical and creative engagement with meaning, values and knowledge. It invites collaboration, though its apparent simplicity often belies a tremendous amount of individual and collective labour involved in the practices of (...) class='Hi'>iteration. Through its repetition of the existing, however, iteration also can be a mechanism for reproducing the status quo. Its pervasiveness and banality naturalises power, and its mimetic qualities shrink spaces for critical distance and care. The editors of this special issue have brought together a delightful and fascinating diversity of articles focussed on iteration in cultural production in the digital age. We hop across geographies to examine lockdown diaries, artists’ books, socialist memes, fake news, the design of social media platforms and artificial intelligence, activism, film, social media forum moderation, news website reader comments and more. Iterating through the collection as a whole, across its many disciplines, is a commitment to theorising through empirical evidence, to explaining with critique, and to providing pathways to praxis. These characteristics of this special issue, and the many concepts and arguments it puts forward, make this collection of work exceptionally rich material for seeing iteration and how it shapes the world we live in today, as well as the world we want it to be. In this preface, I take a media sociology approach to show how iteration can be usefully understood as collaborative communication for change. I see this understanding of iteration, whose ascendancy is related to the ascendancy of computer science, as baked into the form of communication technologies—and thus as shaping the kinds of iteration that are possible when we use these technologies. This understanding also prompts us to focus on the connection between iteration and social change. To explore how this works, I analytically slow down the practice of iteration to show that it is a communication practice of transmission. That transmission practice is itself constituted of cognate communication practices—the reception, evaluation and production of knowledge—in which visibility and persuasion are key. In the latter parts of the preface, I illustrate this through the example of witnessing as iteration, as the high-stakes nature of witnessing make it a canary in the coalmine, more generally, for mediated communication in the digital age. I show how breaking the witnessing practice down into its various parts allows us to see how power enters and inflects who and what are iterated, when—and who and what are not. Thinking critically with iteration and against unequal power relations, the praxis this preface suggests is one—much in line with the rest of this special issue—of explaining how iteration might move the grassroots towards their goals. (shrink)

Why are natural theories pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. This approach was inspired by Martin's Conjecture, one of the most prominent conjectures in recursion theory. Fixing a reasonable subsystem $T$ of arithmetic, the goal was to classify the recursive functions that are monotone with respect to the Lindenbaum algebra of $T$. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate $\mathsf{Con}_T^\alpha$ of the consistency operator ``in (...) the limit'' within the ultrafilter of sentences that are true in the standard model. In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as $\mathsf{Con}_T$ in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as $\mathsf{Con}_T$ in the limit nor as strong as $\mathsf{Con}_T^2$ in the limit. In fact, for every $\alpha$, we produce a function that is cofinally equivalent to $\mathsf{Con}^\alpha_T$ yet cofinally equivalent to $\mathsf{Con}_T$. (shrink)

Sous ce titre qui rend hommage à la pensée de Derrida, sont rassemblées des conférences tenues par le biologiste Hans-Jörg Rheinberger. Six textes aussi brefs que denses, nourris de vie de laboratoire et de philosophie franco-allemande - de Gaston Bachelard à Bruno Latour en passant par Martin Heidegger et Jacques Lacan, qui donnent à voir un mouvement épistémologique sur le temps long. Tentant de faire dialoguer cet héritage philosophique avec la biologie moléculaire contemporaine, Itérations propose des variations sur l'inconnu et (...) l'insoupçonné dans la recherche scientifique, et sur les systèmes expérimentaux comme contextes déterminants de la production des savoirs - cernant toujours les manifestations de l'Etre comme écriture. Excellente introduction à l'épistémologie historique, Itérations présente également l'un des prolongements les plus féconds de la pensée de Derrida. (shrink)

In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We showed that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a non-reflecting (...) stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds. (shrink)