We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-model potentialism, and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, (...) and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account. (shrink)

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...) view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for. (shrink)

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal κ, for example, becomes fully indestructible by <κ-directed closed forcing; a strong cardinal κ becomes indestructible by κ-strategically closed forcing; and a strongly compact cardinal κ becomes indestructible by, among others, the forcing to (...) add a Cohen subset to κ, the forcing to shoot a club Cκ avoiding the measurable cardinals and the forcing to add various long Prikry sequences. The lottery preparation works best when performed after fast function forcing, which adds a new completely general kind of Laver function for any large cardinal, thereby freeing the Laver function concept from the supercompact cardinal context. (shrink)

We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Everyset. for example, is decidable by such machines, and the semi-decidable sets form a portion of thesets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators.

The Lévy-Solovay Theorem [8] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.

The dream solution of the continuum hypothesis would be a solution by which we settle the continuum hypothesis on the basis of a newly discovered fundamental principle of set theory, a missing axiom, widely regarded as true. Such a dream solution would indeed be a solution, since we would all accept the new axiom along with its consequences. In this article, however, I argue that such a dream solution to $\mathrm {CH}$ is unattainable.

The literature has shown that different types of moral dilemmas elicit discrepant decision patterns. The present research investigated the role of uncertainty in contributing to these decision patterns. Two studies were conducted to examine participants' choices in commonly used dilemmas. Study 1 showed that participants’ perceived outcome probabilities were significantly associated with their moral choices, and that these associations were independent from the dilemma type. Study 2 revealed that participants had significantly less preference for killing the individual when the outcome (...) probabilities were stated using the modal verb ‘will’ than when they were stated using the numerical phrasing of ‘100%’. Our findings illustrate a discord between experimenter and participant in the interpretation of task instructions. (shrink)

We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.

We introduce the $\Sigma _1$ -definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, the sequence is $\Sigma _1$ -definable and provably finite; the sequence is empty in transitive models; and if M is a countable model of set theory in which the sequence is s and t is any finite extension of s in this model, then there is an end-extension of M to a (...) model in which the sequence is t. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it. (shrink)

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

In this paper, following an idea of Christophe Chalons. I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence varphi holding in some forcing extension $V^P$ and all subsequent extensions $V^{P\ast Q}$ holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme $(\lozenge \square \varphi) \Rightarrow \square \varphi$ , and is equivalent to (...) the modal theory S5. In this article. I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in φ, is equiconsistent with the scheme asserting that $V_\delta \prec V$ for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is $\square MP\!_{\!\!\!\!\!\!_{\!\!_\sim}}$ , which asserts that $MP\!_{\!\!\!\!\!\!_{\!\!_\sim}}$ holds in V and all forcing extensions. From this, it follows that $0^\#$ exists, that $x^\#$ exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain. (shrink)

Biologists and historians often present natural history and molecular biology as distinct, perhaps conflicting, fields in biological research. Such accounts, although supported by abundant evidence, overlook important areas of overlap between these areas. Focusing upon examples drawn particularly from systematics and molecular evolution, I argue that naturalists and molecular biologists often share questions, methods, and forms of explanation. Acknowledging these interdisciplinary efforts provides a more balanced account of the development of biology during the post-World War II era.

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

Biological evolution provides a creative fount of complex and subtle adaptations, often surprising the scientists who discover them. However, because evolution is an algorithmic process that transcends the substrate in which it occurs, evolution’s creativity is not limited to nature. Indeed, many researchers in the field of digital evolution have observed their evolving algorithms and organisms subverting their intentions, exposing unrecognized bugs in their code, producing unexpected adaptations, or exhibiting outcomes uncannily convergent with ones in nature. Such stories routinely reveal (...) creativity by evolution in these digital worlds, but they rarely fit into the standard scientific narrative. Instead they are often treated as mere obstacles to be overcome, rather than results that warrant study in their own right. The stories themselves are traded among researchers through oral tradition, but that mode of information transmission is inefficient and prone to error and outright loss. Moreover, the fact that these stories tend to be shared only among practitioners means that many natural scientists do not realize how interesting and lifelike digital organisms are and how natural their evolution can be. To our knowledge, no collection of such anecdotes has been published before. This paper is the crowd-sourced product of researchers in the fields of artificial life and evolutionary computation who have provided first-hand accounts of such cases. It thus serves as a written, fact-checked collection of scientifically important and even entertaining stories. In doing so we also present here substantial evidence that the existence and importance of evolutionary surprises extends beyond the natural world, and may indeed be a universal property of all complex evolving systems. (shrink)

We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one set-forcing ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed (...) from any definable class to V, among many other possibilities we consider, including generic embeddings, definable embeddings and results not requiring the axiom of choice. We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions. (shrink)

A cardinal κ is tall if for every ordinal θ there is an embedding j: V → M with critical point κ such that j > θ and Mκ ⊆ M. Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a strong cardinal. Any tall cardinal (...) κ can be made indestructible by a variety of forcing notions, including forcing that pumps up the value of 2κ as high as desired. (shrink)

The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call (...) exact preservation theorems, I use this new technology to perform a kind of partial Laver preparation, and thereby finely control the class of posets which preserve a supercompact cardinal. Eventually, I prove the ‘As You Like It’ Theorem, which asserts that the class of κ -directed closed posets which preserve a supercompact cardinal κ can be made by forcing to conform with any pre-selected local definition which respects the equivalence of forcing. Along the way I separate completely the levels of the superdestructibility hierarchy, and, in an epilogue, prove that the notions of fragility and superdestructibility are orthogonal — all four combinations are possible. (shrink)

The twentieth century witnessed a dramatic increase in the use of statistics by biologists, including systematists. The modern synthesis and new systematics stimulated this development, particularly after World War II. The rise of "the statistical frame of mind " resulted in a rethinking of the relationship between biological and mathematical points of view, the roles of objectivity and subjectivity in systematic research, the implications of new computing technologies, and the place of systematics among the biological disciplines.

Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal κ becomes superdestructible--any further <κ--closed forcing which adds a subset to κ will destroy the measurability, even the weak compactness, of κ. Nevertheless, after small forcing indestructible cardinals remain resurrectible, but never strongly resurrectible.

The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a submodel of N (...) or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω1+ 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most OrdM; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈 HFM, ∈M〉 of M. Indeed, 〈 HFM, ∈M〉 is universal for all countable acyclic binary relations. (shrink)

A pointwise definable model is one in which every object is \loos definable without parameters. In a model of set theory, this property strengthens $V=\HOD$, but is not first-order expressible. Nevertheless, if \ZFC\ is consistent, then there are continuum many pointwise definable models of \ZFC. If there is a transitive model of \ZFC, then there are continuum many pointwise definable transitive models of \ZFC. What is more, every countable model of \ZFC\ has a class forcing extension that is pointwise definable. (...) Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. (shrink)

Technological development plays a critical role in society’s ability to address environmental issues. Building on Teece’s profiting from innovation framework, we articulate how a double-externality problem weakens the appropriability regime surrounding pollution reduction technology (PRT). We then develop a theoretical framework articulating how weak appropriability induces firms to modify their innovation strategies for PRT development by increasing the extent to which they engage in organizational exploration (rather than exploitation) and emphasizing incremental (rather than radical) technologies. Noting that the effects of (...) weak appropriability are unlikely to be static, we detail how the accumulation of organizational capabilities for reducing pollution diminishes the extent to which firms modify their innovation strategies for PRT. That is, firms become less exploratory and less focused on incremental technology as they gain experience with pollution reduction. Results of broad-sample empirical analyses of 206,277 patents held by 203 U.S. manufacturing firms over a 20-year period provide support for our framework. (shrink)

Using the lottery preparation, we prove that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all.

Laver [L] and others [G-S] have shown how to make the supercompactness or strongness of κ indestructible by a wide class of forcing notions. We show, alternatively, how to make these properties fragile. Specifically, we prove that it is relatively consistent that any forcing which preserves $\kappa^{<\kappa}$ and κ+, but not P(κ), destroys the measurability of κ, even if κ is initially supercompact, strong, or if I1(κ) holds. Obtained as an application of some general lifting theorems, this result is an (...) "inner model" type of theorem proved instead by forcing. (shrink)

After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal κ is supercompact if and only if every Π11 sentence true in a structure M (of any size) containing κ in (...) a language of size less than κ is also true in a substructure m≺M of size less than κ with m∩κ∈κ. (shrink)

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence φ holding in some forcing extension $V\P$ and all subsequent extensions V\P*\Qdot holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme $\implies\necessaryφ$, and is equivalent to the modal theory S5. In this article, (...) I prove that the Maximality Principle is relatively consistent with \ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in φ, is equiconsistent with the scheme asserting that $Vδ\elesub V$ for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that $\ORD$ is Mahlo. The strongest principle along these lines is $\necessary\MPtilde$, which asserts that $\MPtilde$ holds in V and all forcing extensions. From this, it follows that 0# exists, that x# exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain. (shrink)

This paper investigates the compatibility of Hegel’s analyses of the mechanization of work in industrial society with Hegel’s notion of freedom as rational self-determination. Work as such is for Hegel a crucial moment on the way to a more complete realization of human freedom, but, as I maintain with Adorno, the technological developments of the last two centuries raise the question of whether the nature of work itself has changed since the industrial revolution. In his Jena lectures, Hegel recognized significant (...) problems that the mechanization of work poses for human freedom that—on my assessment—his later, mature thought does not sufficiently address. One finds, however, a more comprehensive depiction of the same worries that Hegel mentioned in his Jena lectures in Adorno’s critique of the dominant forms of labor in contemporary society and his understanding of technological rationality, which according to Adorno, dominates the contemporary world. I suggest, furthermore, that juxtaposing Hegel’s and Adorno’s thoughts on technology and the mechanization of work reveals Adorno’s philosophy of technology to be an extension, rather than a critique, of the spirit of Hegel’s own philosophical project as immanent critique of the necessary entailments of any given concept. (shrink)

Beginning in the mid-1930s the comparative physiologists Laurence Irving and Per Fredrik Scholander pioneered the study of diving mammals, particularly harbor seals. Although resting on earlier work dating back to the late nineteenth century, their research was distinctive in several ways. In contrast to medically oriented physiology, the approaches of Irving and Scholander were strongly influenced by natural history, zoology, ecology, and evolutionary biology. Diving mammals, they argued, shared the cardiopulmonary physiology of terrestrial mammals, but evolution had modified these basic (...) adaptive processes in extreme ways. In particular, seals’ remarkable ability to hold breath, lower metabolism, produce energy anaerobically, and resist asphyxiation, provided a sharp contrast with terrestrial mammals, including humans. This diving physiology was an extreme elaboration of a general regulatory mechanism that allowed seals and other diving mammals to remain active underwater for extended periods. The decrease in heart rate referred to as bradycardia or the “diving reflex” was highly developed in diving mammals, but also found in less developed form in many other organisms faced by asphyxia. It therefore served as a kind of “master switch” for lowering metabolism in diving, hibernation, parturition, drowning, and other physiological responses involving lack of oxygen. Studying bradycardia unified a wide diversity of physiological phenomena, while also providing a context for contrasting the physiological responses of various species, including humans. Conducted in the laboratory and the field, this research served as a bridge between a comparative physiological ecology focused on non-human species and a human-centered general physiology. (shrink)

We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets; that is, $\mathrm{HOA}=\mathrm{HOD}$. Moreover, we show that every algebraic model of $\mathrm{ZF}$ is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp—an algebraic analogue of the constructible universe—which is obtained by iteratively adding not only (...) the sets that are definable over what has been built so far, but also those that are algebraic over the existing structure. While we know that $\mathrm{Imp}$ can differ from $L$, the subtler properties of this new inner model are just now coming to light. Many questions remain open. (shrink)

Applying the seed concept to Prikry tree forcing P μ , I investigate how well P μ preserves the maximality property of ordinary Prikry forcing and prove that P μ Prikry sequences are maximal exactly when μ admits no non-canonical seeds via a finite iteration. In particular, I conclude that if μ is a strongly normal supercompactness measure, then P μ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. Hugh Woodin's.

We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost-hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.

The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

If the Wholeness Axiom wa $_0$ is itself consistent, then it is consistent with v=hod. A consequence of the proof is that the various Wholeness Axioms are not all equivalent. Additionally, the theory zfc+wa $_0$ is finitely axiomatizable.

What inductive biases must be incorporated into multi-agent artificial intelligence models to get them to capture high-fidelity imitation? We think very little is needed. In the right environments, both instrumental- and ritual-stance imitation can emerge from generic learning mechanisms operating on non-deliberative decision architectures. In this view, imitation emerges from trial-and-error learning and does not require explicit deliberation.

The content of Christian spirituality that made waves since the inception of the early church soon took on different contours as the faith got adapted to different gentile contexts. The expression of this faith, along with its liturgical symbolism and sacramental observances, is still gaining momentum in African Christianity. The emerging practice of the use of ‘anointing oil’ in its religious expression is receiving more attention than the Christ of the Gospel. In this article, we argue that against its primitive (...) intent, the use of the ‘anointing oil’ by the African Church is a mere display of fetish ancestral religion that expresses its unique African traditional religious root rather than a true expression of Christian spirituality. Our thesis is framed on the basis that the manner in which some African churches apply the purported ‘anointing oil’ is discriminatory vis-à-vis its ancient understanding and purpose. In our attempt to address this damaging practice to true Christian spirituality, also standing as a huge challenge for pastoral theology, we undertook a careful historical–theological analysis of the extant biblical data and its contextual interpretation vis-à-vis its distortion today. We concluded that what pastoral theologians have to deal with within the Christian community in Africa is offering the right biblical perspective against the distorted mode of the application of the contemporary purported ‘anointing oil’ that is falsely projecting the Christian faith and belief in a bad light. Contribution: The application of the anointing oil in contemporary Christian religion in Africa is, to say the least, not an inherent textually prescribed requisite criterion for Christian spirituality, but merely an outburst of fetish ancestral religious worldview that stands contra the hermeneutics of the biblical text and its ancient tradition. That no Old Testament prophet, not even Jesus nor Paul, mentioned the subject, makes its contemporary application textually and theologically suspect, and therefore, heretically infectious for the spiritual health of the community of faith. These insights sit quite well with the textual hermeneutics within the mainline transdisciplinary religious and multidisplinary theological perspective of this journal. (shrink)

The 2019 coronavirus (COVID‐19), also known as SARS‐CoV‐2, is highly pathogenic and virulent, and it spreads very quickly through human‐to‐human contact. In response to the growing number of cases, governments across the spectrum of affected countries have adopted different strategies in implementing control measures, in a hope to reduce the number of new cases. However, 5 months after the first confirmed case, countries like the United States of America (US) seems to be heading towards a trajectory that indicates a health (...) care crisis. This is in stark contrast to the downward trajectory in Europe, China, and elsewhere in Asia, where the number of new cases has seen a decline ahead of an anticipated second wave. A data‐driven approach reveals three key strategies in tackling COVID‐19. Our work here has definitively evaluated these strategies and serves as a warning to the US, and more importantly, a guide for tackling future pandemics. Also see the video abstract here https://youtu.be/gPkCi2_7tWo. (shrink)

The following observations reflect on the moral-psychological value of the form of reading and writing that Nietzsche referred to as the human, all too human, particularly as he saw it embodied in Goethe, and as taken up by Hans Blumenberg in his collection Goethe zum Beispiel.

ABSTRACTAlexithymia is characterised by a lack of words for emotional experiences and it has been implicated in deficits in emotion processing. Research in this area has typically focused on judgem...

"This foundational text was one of the first books to integrate work from moral philosophy, developmental/moral psychology, applied psychology, political and social economy, and political science, as well as business scholarship. The third edition utilizes ideas from the first two to provide readers with a practical model for ethical decision making and includes examples from I-O research and practice, as well as current business events"--.

This collection of 44 stunning black-and-white photos of the marble quarries in northern Italy is preceded by an Introduction that describes Leivick's goals and experiences as a photographer working at Carrara and provides a brief outline ...