We prove here:Theorem. LetT be a countable complete superstable non ω-stable theory with fewer than continuum many countable models. Then there is a definable groupG with locally modular regular generics, such thatG is not connected-by-finite and any type inG eq orthogonal to the generics has Morley rank.Corollary. LetT be a countable complete superstable theory in which no infinite group is definable. ThenT has either at most countably many, or exactly continuum many countable models, up to isomorphism.

Commonsense metaphysics populates the world with an enormous variety of macroscopic objects, conceived as being capable of persisting through time and undergoing various changes in their properties and relations to one another. Many of these objects fall under J. L. Austin’s memorable description, “moderate-sized specimens of dry goods.” More broadly, they include, for instance, all of those old favourites of philosophers too idle to think of more interesting examples—tables, books, rocks, apples, cats, and statues. Some of them are natural objects, (...) such as rocks, apples, and cats. Some are artefacts, such as tables, books, and statues. Some are living things, such as apples and cats. Some are inanimate things, such as tables, books, rocks, and statues. To such a list we could add much larger natural objects, such as planets and stars, as well as much smaller ones barely visible to the naked eye, such as mites and pollen grains. Telescopes and microscopes enable us to extend the scale further in both directions, to include such things as galaxies and microbes. To call into question the existence of all or any of these things would appear to be an extravagance of the sort that only a philosopher could indulge. Until fairly recently in the history of philosophy, however, the only way in which their existence would have been called into question was in the context of the debate between realism and idealism. Idealism calls their existence into question by questioning the existence of an “external world” as such, proposing instead that only minds and their experiences exist. What is much more recent is the suggestion that, although there is an external world of entities existing in space and time, it is not in fact populated by most, or perhaps even by any, of the objects recognized by common sense. Instead, it is suggested, the world is populated by submicroscopic entities of a highly esoteric kind best understood by theoretical physicists—entities which may not even qualify as “objects” in anything like the familiar sense because, quite possibly, they should not be thought of as having precise spatiotemporal locations or as persisting over time and through processes of qualitative change. Let us call these putative entities ‘atoms’, without prejudice as to whether they conform to the descriptions currently favoured by physicists in speaking of what they call ‘atoms’. In other words, let us use ‘atom’ as a place-holder for whatever kind or kinds of entities an ideal or completed physics would postulate as the “ultimate” constituents, as opposed to infinitely descending levels of ever more fine-grained physical structure, but let us set aside dispute over these alternatives for the time being, if only because we have little idea as to how to settle it. (shrink)

We generalize Solovay's unfolding technique for infinite games and use an Unfolding Theorem to give a uniform method to prove that all analytic sets are in the $\sigma$ -algebras of measurability connected with well-known forcing notions.

The theory of infinite games with slightly imperfect information has been developed for games with finitely and countably many moves. In this paper, we shift the discussion to games with uncountably many possible moves, introducing the axiom of real Blackwell determinacy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{Bl-AD}_\mathbb{R}}$$\end{document} (as an analogue of the axiom of real determinacy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{AD}_\mathbb{R}}$$\end{document}). We prove that the consistency strength of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} (...) \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{Bl-AD}_\mathbb{R}}$$\end{document} is strictly greater than that of AD. (shrink)

We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matoušek. Then we show that for every K>1 every uncountable Polish space has a perfect subset that K-bi-Lipschitz embeds into the real line. Finally we study decompositions of infinite separable metric spaces into subsets that, for some K>1, K-bi-Lipschitz embed into the real line.

We construct the set of the title, answering a question of Cholak, Jockusch, and Slaman [1], and discuss its connections with the study of the proof-theoretic strength and effective content of versions of Ramsey's Theorem. In particular, our result implies that every ω-model of RCA 0 + SRT 2 2 must contain a nonlow set.

Some philosophers believe that entities have essences. What are we to make of the view that essences are themselves entities? E.J. Lowe has put forward an infinite regress argument against it. In this paper I challenge that argument. First, drawing on work by J.W. Wieland, I give a general condition for the obtaining of a vicious infinite regress. I then argue that in Lowe’s case the condition is not met. In making my case, I mainly (but not exclusively) consider definitionalist (...) accounts of essence. I make a re￾quirement to which definitionalists such as Lowe are committed and which, I venture, should also be palatable to non-naïve modalists. I call it the Relevance Principle. The defence trades on it, as well as on the distinction, due to K. Fine, between mediate and immediate essence. (shrink)

Pascal's wager has fascinated philosophers far in excess of its reputation as effective apologetics. Very few of the wager's defenders, in fact, have retained more than an academic interest in its power to persuade. Partly this is a matter of good manners. Pascal is supposed to have pitched his wager at folks who understand only self-interested motivations, and today it is no longer fashionable for defenders of theism to disparage the character of their opponents. But partly the low-key concern with (...) apologetics expresses a philosophical judgement. Pascal's defenders have found the question of the wager's audience to be less philosophically engaging than the logic of its argument. I believe that this assessment is mistaken. The most puzzling feature of Pascal's wager is its invocation of infinite utility. What are finite human beings, theists or otherwise, supposed to make of the idea of an infinitely desirable happiness? There are, I will argue, two sorts of response to this question, and depending on which sort Pascal had in mind, the logic of his wager comes out very differently. (shrink)

We show that in infinite worlds the following three conditions are incompatible: The spatiotemporal ordering of individuals is morally irrelevant. All else being equal, the act of bringing about a good outcome with a high probability is better than the act of bringing about the same outcome with a low probability. One act is better than another only if there is a nonzero probability that it brings about a better outcome. The impossibility of combining these conditions shows that it is (...) more costly to endorse than has been previously acknowledged. (shrink)

We call A weakly low for K if there is a c such that $K^A(\sigma)\geq K(\sigma)-c$ for infinitely many σ; in other words, there are infinitely many strings that A does not help compress. We prove that A is weakly low for K if and only if Chaitin's Ω is A-random. This has consequences in the K-degrees and the low for K (i.e., low for random) degrees. Furthermore, we prove that the initial segment prefix-free complexity of 2-random reals is infinitely (...) often maximal. This had previously been proved for plain Kolmogorov complexity. (shrink)

We show that the existence of a recursively enumerable set whose Turing degree is neither low nor complete cannot be proven from the basic axioms of first order arithmetic (P -) together with Σ 2 -collection (BΣ 2 ). In contrast, a high (hence, not low) incomplete recursively enumerable set can be assembled by a standard application of the infinite injury priority method. Similarly, for each n, the existence of an incomplete recursively enumerable set that is neither low n nor (...) high n - 1 , while true, cannot be established in P - + BΣ n + 1 . Consequently, no bounded fragment of first order arithmetic establishes the facts that the high n and low n jump hierarchies are proper on the recursively enumerable degrees. (shrink)

Let B be a superatomic Boolean algebra (BA). The rank of B (rk(B)), is defined to be the Cantor Bendixon rank of the Stone space of B. If a ∈ B - {0}, then the rank of a in B (rk(a)), is defined to be the rank of the Boolean algebra $B b \upharpoonright a \overset{\mathrm{def}}{=} \{b \in B: b \leq a\}$ . The rank of 0 B is defined to be -1. An element a ∈ B - {0} is (...) a generalized atom $(a \in \widehat{At}(B))$ , if the last nonzero cardinal in the cardinal sequence of B $\upharpoonright$ a is 1. Let a,b $\in\widehat{At}$ (B). We denote a ∼ b, if rk(a) = rk(b) = rk(a · b). A subset H $\subseteq \widehat{At}$ (B) is a complete set of representatives (CSR) for B, if for every a $\in \widehat{At}$ (B) there is a unique h ∈ H such that h ∼ a. Any CSR for B generates B. We say that B is canonically well-generated (CWG), if it has a CSR H such that the sublattice of B generated by H is well-founded. We say that B is well-generated, if it has a well-founded sublattice L such that L generates B. THEOREM 1. Let B be a Boolean algebra with cardinal sequence $\langle\aleph_0: i . If B is CWG, then every subalgebra of B is CWG. A superatomic Boolean algebra B is essentially low (ESL), if it has a countable ideal I such that rk(B/I) ≤ 1. Theorem 1 follows from Theorem 2.9, which is the main result of this work. For an ESL BA B we define a set F B of partial functions from a certain countably infinite set to ω (Definition 2.8). Theorem 2.9 says that if B is an ESL Boolean algebra, then the following are equivalent. (1) Every subalgebra of B is CWG; and (2) F B is bounded. THEOREM 2. If an ESL Boolean algebra is not CWG, then it has a subalgebra which is not well-generated. (shrink)

Let an oracle be called low for prefix-free complexity on a set in case access to the oracle improves the prefix-free complexities of the members of the set at most by an additive constant. Let an oracle be called weakly low for prefix-free complexity on a set in case the oracle is low for prefix-free complexity on an infinite subset of the given set. Furthermore, let an oracle be called low and weakly for prefix-free complexity along a sequence in case (...) the oracle is low and weakly low, respectively, for prefix-free complexity on the set of initial segments of the sequence. Our two main results are the following characterizations. An oracle is low for prefix-free complexity if and only if it is low for prefix-free complexity along some sequences if and only if it is low for prefix-free complexity along all sequences. An oracle is weakly low for prefix-free complexity if and only if it is weakly low for prefix-free complexity along some sequence if and only if it is weakly low for prefix-free complexity along almost all sequences. As a tool for proving these results, we show that prefix-free complexity differs from its expected value with respect to an oracle chosen uniformly at random at most by an additive constant, and that similar results hold for related notions such as a priori probability. Furthermore, we demonstrate that on every infinite set almost all oracles are weakly low but are not low for prefix-free complexity, while by Shoenfield absoluteness there is an infinite set on which uncountably many oracles are low for prefix-free complexity. Finally, we obtain no-gap results, introduce weakly low reducibility, or WLK-reducibility for short, and show that all its degrees except the greatest one are countable. (shrink)

In this paper, we prove several results about the Turing jumps of low c.e. sets. We show that only Δ‐levels of the Ershov Hierarchy can properly contain the Turing jumps of c.e. sets and that there exists an arbitrarily large computable ordinal with a normal notation such that the corresponding Δ‐level is proper for the Turing jump of some c.e. set. Next, we generalize the notion of jump traceability to the jump traceability with ‐ and ‐bound for every infinite computable (...) ordinal α. It is known that jump traceability and superlowness coincide on the c.e. sets and we show that for every infinite computable ordinal α, jump traceability with ‐ or ‐bound of a c.e. set A is equivalent to the fact that. Finally, we consider the generalized truth‐table reducibilities and prove that for every (not necessarily the Turing jump of a c.e. set) set A and every limit computable ordinal α, iff. (shrink)

We prove the existence of a perfect Π01 set of reals such that the uniform join of any of its infinite subsets is above 0′. We explain a couple of ways in which this demonstrates the optimality of the Low Basis Theorem.

The single day conference “Ontology, Mind and Free Will. A Workshop in Memory of E.J. Lowe (1950-2014)” took place at the Department of Humanities of the University of Macerata on March, 3 rd 2014. It included as speakers Sophie Gibb (Durham University), Mario De Caro (Roma Tre University) and Michele Paolini Paoletti (University of Macerata). This event was thought by the organizers in order to honor the British philosopher Ethan Jonathan Lowe, who suddenly passed away last January with infinite regret (...) of the scientific community. The discussion of some of his major research topics, such as mental causation, free will and essence, was therefore meant as the right homage to one of the most influential philosophers who animated and deepened the debate of the last thirty years. Regarding the singular reviews of this report, Matteo Grasso took care of Sophie Gibb’s and Mario De Caro’s talks, while Mattia Sorgon reviewed Michele Paolini Paoletti’s. (shrink)

In this thesis we present three results about the ferromagnetic quantum XXZ model: 1) Existence of a spectral gap above all infinite-volume ground states in one dimension for any choice of spin S>1/2 ; 2) Characterization of one-dimensional "droplet" states as ground states of the spin-1/2 model with up spin boundary fields; 3) An upper bound for the spectral gap above the spin-1/2 interface ground states in dimension two and higher. The upper bound vanishes as 1/R^2, where R is the (...) linear size of the system. This thesis contains research performed in collboration with other authors: O. Bolina, P. Contucci, T. Koma, B. Nachtergaele. (shrink)

Let A and B be subsets of the space 2 N of sets of natural numbers. A is said to be Wadge reducible to B if there is a continuous map Φ from 2 N into 2 N such that A = Φ -1 (B); A is said to be monotone reducible to B if in addition the map Φ is monotone, that is, $a \subset b$ implies $\Phi (a) \subset \Phi(b)$ . The set A is said to be monotone (...) if a ∈ A and $a \subset b$ imply b ∈ A. For monotone sets, it is shown that, as for Wadge reducibility, sets low in the arithmetical hierarchy are nicely ordered. The ▵ 0 1 sets are all reducible to the (Σ 0 1 but not ▵ 0 1 ) sets, which are in turn all reducible to the strictly ▵ 0 2 sets, which are all in turn reducible to the strictly Σ 0 2 sets. In addition, the nontrivial Σ 0 n sets all have the same degree for n ≤ 2. For Wadge reducibility, these results extend throughout the Borel hierarchy. In contrast, we give two natural strictly Π 0 2 monotone sets which have different monotone degrees. We show that every Σ 0 2 monotone set is actually Σ 0 2 positive. We also consider reducibility for subsets of the space of compact subsets of 2 N . This leads to the result that the finitely iterated Cantor-Bendixson derivative D n is a Borel map of class exactly 2n, which answers a question of Kuratowski. (shrink)

It was shown by Cholak, Jockusch, and Slaman that every computable 2-coloring of pairs admits an infinite low₂ homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition $C\,\not \leqslant _T \,H$, where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun's cone avoidance theorem for Ramsey's theorem. We then extend the result to show that every computable 2-coloring of (...) pairs admits a pair of low₂ infinite homogeneous sets whose degrees form a minimal pair. (shrink)

Pascal’s Wager involves expected utilities. In this chapter, we examine the Wager in light of two main features of expected utility theory: utilities and probabilities. We discuss infinite and finite utilities, and zero, infinitesimal, extremely low, imprecise, and undefined probabilities. These have all come up in recent literature regarding Pascal’s Wager. We consider the problems each creates and suggest prospects for the Wager in light of these problems.

On the standard view for something to exist is for one thing to exist: in slogan form, to be is to be countable. E.J. Lowe argues something can exist without being countable as one, however. His primary example is homogenous “stuff,” i.e., qualitatively uniform and infinitely divisible matter. Lacking nonarbitrary boundaries and being everywhere the same, homogenous stuff lacks a principle of individuation that would yield countably distinct constituents. So, for Lowe, homogenous stuff is strongly uncountable. Olson rejects Lowe’s view (...) and defends the orthodox connection between existence and number. He argues that if there is any stuff, there is a number of portions of stuff. Sider also rejects a stuff ontology, claiming it is incompatible with his preferred view that the familiar quantifiers of predicate logic carve at nature’s joints. Against these arguments, I defend the uncountability of stuff and the possibility of existence without countability. If to be is to be countable, more is needed than the arguments that Olson and Sider provide. (shrink)

An infinite binary sequence X is Kolmogorov–Loveland random if there is no computable non-monotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence.One of the major open problems in the field of effective randomness is whether Martin-Löf randomness is the same as KL-randomness. Our first (...) main result states that KL-random sequences are close to Martin-Löf random sequences in so far as every KL-random sequence has arbitrarily dense subsequences that are Martin-Löf random. A key lemma in the proof of this result is that for every effective split of a KL-random sequence at least one of the halves is Martin-Löf random. However, this splitting property does not characterize KL-randomness; we construct a sequence that is not even computably random such that every effective split yields two subsequences that are 2-random. Furthermore, we show for any KL-random sequence A that is computable in the halting problem that, first, for any effective split of A both halves are Martin-Löf random and, second, for any computable, nondecreasing, and unbounded function g and almost all n, the prefix of A of length n has prefix-free Kolmogorov complexity at least n−g. Again, the latter property does not characterize KL-randomness, even when restricted to left-r.e. sequences; we construct a left-r.e. sequence that has this property but is not KL-stochastic and, in fact, is not even Mises–Wald–Church stochastic.Turning our attention to KL-stochasticity, we construct a non-empty class of KL-stochastic sequences that are not weakly 1-random; by the usual basis theorems we obtain such sequences that in addition are left-r.e., are low, or are of hyperimmune-free degree.Our second main result asserts that every KL-stochastic sequence has effective dimension 1, or equivalently, a sequence cannot be KL-stochastic if it has infinitely many prefixes that can be compressed by a factor of α<1. This improves on a result by Muchnik, who has shown that were they to exist, such compressible prefixes could not be found effectively. (shrink)

Given two infinite binary sequences A,B we say that B can compress at least as well as A if the prefix-free Kolmogorov complexity relative to B of any binary string is at most as much as the prefix-free Kolmogorov complexity relative to A, modulo a constant. This relation, introduced in Nies [14] and denoted by A≤LKB, is a measure of relative compressing power of oracles, in the same way that Turing reducibility is a measure of relative information. The equivalence classes (...) induced by ≤LK are called LK degrees and there is a least degree containing the oracles which can only compress as much as a computable oracle, also called the ‘low for K’ sets. A well-known result from Nies [14] states that these coincide with the K-trivial sets, which are the ones whose initial segments have minimal prefix-free Kolmogorov complexity. We show that with respect to ≤LK, given any non-trivial sets X,Y there is a computably enumerable set A which is not K-trivial and it is below X,Y. This shows that the local structures of and Turing degrees are not elementarily equivalent to the corresponding local structures in the LK degrees. It also shows that there is no pair of sets computable from the halting problem which forms a minimal pair in the LK degrees; this is sharp in terms of the jump, as it is known that there are sets computable from which form a minimal pair in the LK degrees. We also show that the structure of LK degrees below the LK degree of the halting problem is not elementarily equivalent to the or structures of LK degrees. The proofs introduce a new technique of permitting below a set that is not K-trivial, which is likely to have wider applications. (shrink)

Title: Revisiting Boltzmann’s Paradox: A Theoretical Framework Based on Improbable Oscillations This work presents a groundbreaking solution to Boltzmann’s Paradox by redefining "Nothingness" as a fundamental improbable structure, characterized by chaotic oscillations between states of improbability ( P = 0 ) and infinite possibility ( P → ∞ ). These improbable oscillations serve as the driving mechanism for the emergence of time, existence, and entropy. The solution posits that the universe’s initial low-entropy state is not a statistical anomaly but a (...) natural consequence of these dynamic oscillations within Nothingness. This framework unifies philosophical and physical perspectives, integrating energy conservation, relational time, and quantum principles. By addressing the relationship between order and chaotic probabilistic dynamics, the paper not only resolves one of modern physics’ most perplexing dilemmas but also offers a novel lens through which reality emerges as a probabilistic equilibrium derived from improbable oscillations. (shrink)

In spite of its infinite expectation value, the St. Petersburg game is not only a gamble without supply in the real world, but also one without demand at apparently very reasonable asking prices. We offer a rationalizing explanation of why the St. Petersburg bargain is unattractive on both sides (to both house and player) in the mid-range of prices (finite but upwards of about $4). Our analysis – featuring (1) the already-established fact that the average of finite ensembles of the (...) St. Petersburg game grows with ensemble size but is unbounded, and (2) our own simulation data showing that the debt-to-entry fee ratio rises exponentially – explains why both house and player are quite rational in abstaining from the St. Petersburg game. The house will be unavoidably (and intentionally) exposed to very large ensembles (with very high averages, and so very costly to them), while contrariwise even the well-heeled player is not sufficiently capitalized (as our simulation data reveals) to be able to capture the potential gains from large-ensemble play. (Smaller ensembles, meanwhile, enjoy low means, as others have shown, and so are not worth paying more than $4 to play, even if a merchant were to offer them at such low prices per trial.) Both sides are consequently rational in abstaining from entry into the St. Petersburg market in the mid-range of asking prices. We utilize the concept of capitalization vis-à-vis a gamble to make this case. Classical analyses of this question have paid insufficient attention to the question of the propriety of using expected values to assess the St. Petersburg gamble. And extant analyses have not noted the average-maximum-debt-before-breaking-even figures, and so are incomplete. (shrink)

Contemporary versions of natural rights libertarianism trace their locus classicus to Robert Nozick’s Anarchy, State, and Utopia. But although there have been many criticisms of the version of political libertarianism put forward by Nozick, many of these fail objections to meet basic methodological desiderata. Thus, Nozick’s libertarianism deserves to be re-examined. In this paper I develop a new argument which meets these desiderata. Specifically, I argue that the libertarian conception of self-ownership, the view’s foundation, implies what I call the Asymmetrical (...) Value Claim: a dubious claim about the importance of choice relative to other valuable capacities. I argue that this misunderstands what is really valuable in life, and show how it causes libertarianism to generate counterintuitive public policy recommendations. (shrink)

Discussions of “privilege” have become increasingly common, but it’s often unclear what exactly people mean by “privilege.” Even well-known writings about privilege rarely take the time to define the word and explain what it means. The confusion this creates is one reason why debates about privilege are often contentious and unproductive. This essay aims to demystify privilege, presupposing no prior knowledge of philosophy. With a clear definition, it is easier to discuss some of the main debates about privilege: Is there (...) such a thing as “white privilege”? What about “male privilege”? And what’s the point of talking about privilege, anyway? (shrink)

Physicalism should be characterized in a way that makes it compatible with the possibility that the physical world is infinitely decomposable. Some have proposed solving this problem by replacing a widely accepted No Fundamental Mentality requirement on physicalism with a more general No Low-Level Mentality requirement. The latter states that physicalism could be true if there is a level of decomposition beneath which nothing is mental, whereas physicalism is false otherwise. Brown argues that this solution does not work. He devises (...) an infinitely decomposable possible world in which physicalism should come out as true even though there is mentality all the way down. I propose a solution that circumvents his argument. The key is to specify the sort of mentality that physicalism cannot abide at any level, namely, mentality that does not consist solely in a structural-dynamic arrangement of entities. I also argue that the problem Brown identifies has significant implications for what is at stake in the debate over physicalism’s truth or falsity—implications he undersells. (shrink)

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)

Fine, Lowe and Hale accept the view that necessity is to be explained by essences: Necessarily p iff, and because, there is some x whose essence ensures that p. Hale, however, believes that this strategy is not universally applicable; he argues that the necessity of essentialist truths cannot itself be explained by once again appealing to essentialist truths. As a consequence, Hale holds that there are basic necessities that cannot be explained. Thus, Hale style essentialism falls short of what Wilsch (...) calls the explanation-challenge (EC) for the metaphysics of necessity. Without endorsing the EC, I argue that Hale’s argument for basic, unexplained necessities fails due to a misunderstanding of the structure of essentialist explanations. Getting clear about the structure of essentialist explanations of necessity leads to a re-evaluation of crucial circularity- and regress-arguments that have been discussed in the debate about essentialism. (shrink)

We provide a list of open problems in the research area of generalised Baire spaces, compiled with the help of the participants of two workshops held in Amsterdam (2014) and Hamburg (2015).

The paper aims at analysing the fundamental theoretical aspects of the latest book by Maria Cristina Bartolomei, La dimensione simbolica and at showing some possible paths of research it suggests in the direction of a hermeneutical perspective opened to the metaphysics. After reflecting about what the author of the book affirms regarding the vision of the symbol from ‘high’ and from ‘low’ and the relation which it has with interpretation and concept, the suggested symbolic character of thought is analyzed in (...) connection with the themes of analogy and paradox. On one hand, it is pointed out that analogy is related both to symbol (Ricoeur) and to concept (Gadamer) and it is inquired how it is possible to describe the relationship between proportion and proportionality implied with analogy. On the other hand, it is underlined how the original principle can be thought in our fallen situation not only by means of symbol and analogy (Bartolomei), but also in the light of symbol and paradox, understood as form of thought which recomposes the break between finite and infinite without removing the signs of it (Ciancio). Finally, it is inquired if analogy and paradox, conceived as critical and conceptual forms of symbol, can light up the relationship among mystical, symbolic and common experience which seems to characterize our existence. (shrink)

A set of infinite binary sequences ${\cal C} \subseteq 2$ℕ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of ${\rm{\Pi }}_1^0 $ classes. In this paper, we introduce the notion of depth for ${\rm{\Pi }}_1^0 $ classes, which is a stronger form of negligibility. Whereas a negligible ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot (...) probabilistically compute a member of ${\cal C}$ with positive probability, a deep ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute an initial segment of a member of ${\cal C}$ with high probability. That is, the probability of computing a length n initial segment of a deep ${\rm{\Pi }}_1^0 $ class converges to 0 effectively in n.We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep ${\rm{\Pi }}_1^0 $ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin’s Ω. (shrink)

Philosophers and scientists have recently been showing renewed interest in dualistic conceptions of the human mind, owing to growing acknowledgment of the failings of materialism and reductionism in contemporary philosophical and scientific thought. This book presents a state-of-the-art overview of current developments in this exciting new area of interdisciplinary collaboration, and will be indispensable reading for all researchers and students in this field.

This article examines Bonaventure’s account of the soul and its powers, which seeks to strike a middle path between the better-known identity and distinction views of the thirteenth century. Bonaventure contends that the powers of the soul are neither fully distinct from the soul nor completely identical to it. The article argues that Bonaventure’s view comprises four key theses. Bonaventure maintains that the soul’s powers are necessary features of the soul; that they depend on the soul; that they are in (...) the same category as the soul; but that they belong to this category “by reduction”. The article also considers an objection to Bonaventure’s view raised by Peter John Olivi. (shrink)

This is an ambitious book, dealing with an issue which deserves much greater attention from historians than it has received heretofore. Only the work of John Roach comes to mind as a precursor, dea...

An important question raised in the Molinist debate is, ‘Given God's access to counterfactual knowledge, could God create a world in which free creatures always refrain from evil?’ An affirmative answer suggests that God cannot possess counterfactual knowledge since such knowledge would allow God to create seemingly more desirable worlds than the actual world. However, Alvin Plantinga has argued that it is at least possible that every possible person is transworld depraved – meaning that each person would perform some wrong (...) actions if any world in which that person is morally free were actualized. I argue that, given an infinite number of possible persons, the probability that everyone is transworld depraved is exceedingly low. In addition, I investigate whether there are enough possible persons vis-à-vis the number of moral choices per person so that God could create worlds like the actual world, except lacking in moral evil. (shrink)

We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT n k denote Ramsey's theorem for k-colorings of n-element sets, and let RT $^n_{ denote (∀ k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' ≤ T 0 (n) . Let IΣ n and BΣ (...) n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + IΣ 2 + RT 2 2 is conservative over RCA 0 + IΣ 2 for Π 1 1 statements and that $RCA_0 + I\Sigma_3 + RT^2_{ , is Π 1 1 -conservative over RCA 0 + IΣ 3 . It follows that RCA 0 + RT 2 2 does not imply BΣ 3 . In contrast, J. Hirst showed that $RCA_0 + RT^2_{ does imply BΣ 3 , and we include a proof of a slightly strengthened version of this result. It follows that $RT^2_{ is strictly stronger than RT 2 2 over RCA 0. (shrink)

In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablity and show that this lies strictly between VC-minimality and dp-minimality. To do this we prove a general result about set systems with independence dimension ≤ 1. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, (...) show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density. (shrink)

Across the UK, there has been an increase in anti-abortion activism outside abortion clinics. The activism deployed includes explicitly religious activities such as ‘prayerful witnessing’ and ‘pavement counselling’, which aim to discourage women from entering clinics. This article stems from a wider ethnographic study of public activism over abortion to determine what claims about motherhood are being made within these debates. Two arguments are presented. First, how women’s role as mothers is central and essentialised in anti-abortion discourses, with the body (...) of the mother often disappearing as activists seek to erode the distinction between a foetus and a baby by constructing pregnancy as a foetal environment. Motherhood is constructed as ‘natural’ and sacred, therefore abortion must be damaging because it destroys women’s ‘natural’ position. Second, the article argues that although the activists’ arguments are always religiously framed, their activism takes place in a largely secular context, meaning that they have to find ways of appealing to secular audiences. This leads to a complex interrelationship between secular and religious discourses, where theological viewpoints sit alongside ‘scientific’ claims to buttress activists’ views. This article explores how the presence and absence of mothers within activists’ narratives is due to the tensions between religiously based understandings of motherhood, and the need to appeal to a secular audience. (shrink)

Human beings are peculiar. In laboratory experiments, they often cooperate in one-shot prisoners’ dilemmas, they frequently offer 1/2 and reject low offers in the ultimatum game, and they often bid 1/2 in the game of divide-the-cake All these behaviors are puzzling from the point of view of game theory. The first two are irrational, if utility is measured in a certain way.1 The last isn’t positively irrational, but it is no more rational than other possible actions, since there are infinitely (...) many other Nash equilibria besides the one in which both players bid 1/2. At the same time, these behaviors seem to indicate that people are sometimes inclined to be cooperative, fair, and just. In his stimulating new book, Brian Skyrms sets himself the task of showing why these inclinations evolved, or how they might have evolved, under the pressure of natural selection. The goal is not to justify our ethical intuitions, but to explain why we have them.2.. (shrink)

The Kolmogorov complexity of a finite binary string is the length of the shortest description of the string. This gives rise to some ‘standard’ lowness notions for reals: A isK-trivial if its initial segments have the lowest possible complexity and A is low forKif using A as an oracle does not decrease the complexity of strings by more than a constant factor. We weaken these notions by requiring the defining inequalities to hold only up to all${\rm{\Delta }}_2^0$orders, and call the (...) new notions${\rm{\Delta }}_2^0$-bounded K-trivialand${\rm{\Delta }}_2^0$-bounded low for K. Several of the ‘nice’ properties ofK-triviality are lost with this weakening. For instance, the new weaker definitions both give uncountable set of reals. In this paper we show that the weaker definitions are no longer equivalent, and that the${\rm{\Delta }}_2^0$-boundedK-trivials are cofinal in the Turing degrees. We then compare them to other previously studied weakenings, namelyinfinitely-often K-trivialityandweak lowness for K. We show that${\rm{\Delta }}_2^0$-boundedK-trivial implies infinitely-oftenK-trivial, but no implication holds between${\rm{\Delta }}_2^0$-bounded low forKand weakly low forK. (shrink)

We prove various results connected together by the common thread of computability theory.First, we investigate a new notion of algorithmic dimension, the inescapable dimension, which lies between the effective Hausdorff and packing dimensions. We also study its generalizations, obtaining an embedding of the Turing degrees into notions of dimension.We then investigate a new notion of computability theoretic immunity that arose in the course of the previous study, that of a set of natural numbers with no co-enumerable subsets. We demonstrate how (...) this notion of $\Pi ^0_1$ -immunity is connected to other immunity notions, and construct $\Pi ^0_1$ -immune reals throughout the high/low and Ershov hierarchies. We also study those degrees that cannot compute or cannot co-enumerate a $\Pi ^0_1$ -immune set.Finally, we discuss a recently discovered truth-table reduction for transforming a Kolmogorov–Loveland random input into a Martin-Löf random output by exploiting the fact that at least one half of such a KL-random is itself ML-random. We show that there is no better algorithm relying on this fact, in the sense that there is no positive, linear, or bounded truth-table reduction which does this. We also generalize these results to the problem of outputting randomness from infinitely many inputs, only some of which are random.Abstract prepared by David J. Webb.E-mail: [email protected]: https://arxiv.org/pdf/2209.05659.pdf. (shrink)