In this paper, we introduce a system of nonstandard second-order arithmetic $\mathsf{ns}$-$\mathsf{WWKL_0}$ which consists of $\mathsf{ns}$-$\mathsf{BASIC}$ plus Loeb measure property. Then we show that $\mathsf{ns}$-$\mathsf{WWKL_0}$ is a conservative extension of $\mathsf{WWKL_0}$ and we do Reverse Mathematics for this system.

This work contributes to the program of studying effective versions of “almost-everywhere” theorems in analysis and ergodic theory via algorithmic randomness. Consider the setting of Cantor space {0,1}N with the uniform measure and the usual shift. We determine the level of randomness needed for a point so that multiple recurrence in the sense of Furstenberg into effectively closed sets P of positive measure holds for iterations starting at the point. This means that for each k∈N there is an (...) n such that n,2n,…,kn shifts of the point all end up in P. We consider multiple recurrence into closed sets that possess various degrees of effectiveness: clopen, Π10 with computable measure, and Π10. The notions of Kurtz, Schnorr, and Martin-Löf randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the k commuting shift operators on {0,1}Nk. (shrink)

Numerous learning tasks can be described as the process of extrapolating patterns from observed data. One of the driving intuitions behind the theory of algorithmic randomness is that randomness amounts to the absence of any effectively detectable patterns: it is thus natural to regard randomness as antithetical to inductive learning. Osherson and Weinstein [11] draw upon the identification of randomness with unlearnability to introduce a learning-theoretic framework (in the spirit of formal learning theory) for modelling algorithmic (...) randomness. They define two success criteria—specifying under what conditions a pattern may be said to have been detected by a computable learning function—and prove that the collections of data sequences on which these criteria cannot be satisfied respectively correspond to the set of weak 1-randoms and the set of weak 2-randoms. This learning-theoretic approach affords an intuitive perspective on algorithmic randomness, and it invites the question of whether restricting attention to learning-theoretic success criteria comes at an expressivity cost. In other words, is the framework expressive enough to capture most core algorithmic randomness notions and, in particular, Martin-Löf randomness—arguably, the most prominent algorithmic randomness notion in the literature? In this paper, we answer the latter question in the affirmative by providing a learning-theoretic characterisation of Martin-Löf randomness. We then show that Schnorr randomness, another central algorithmic randomness notion, also admits a learning-theoretic characterisation in this setting. (shrink)

Some random processes, like a series of coin flips, can produce outcomes that seem particularly remarkable or striking. This paper explores an epistemic puzzle that arises when thinking about these outcomes and asking what, if anything, we can justifiably believe about them. The puzzle has no obvious solution, and any theory of epistemic justification will need to contend with it sooner or later. The puzzle proves especially useful for bringing out the differences between three prominent theories; the probabilist theory, the (...) normic theory and a theory recently defended by Goodman and Salow. (shrink)

Developing an analogue of Solovay reducibility in the higher recursion setting, we show that results from the classical computably enumerable case can be extended to the new context.

This article discusses some ethical principles for distributing pandemic influenza vaccine and other indivisible goods. I argue that a number of principles for distributing pandemic influenza vaccine recently adopted by several national governments are morally unacceptable because they put too much emphasis on utilitarian considerations, such as the ability of the individual to contribute to society. Instead, it would be better to distribute vaccine by setting up a lottery. The argument for this view is based on a purely consequentialist account (...) of morality; i.e. an action is right if and only if its outcome is optimal. However, unlike utilitarians I do not believe that alternatives should be ranked strictly according to the amount of happiness or preference satisfaction they bring about. Even a mere chance to get some vaccine matters morally, even if it is never realized. (shrink)

Festschrift for Michiel van Lambalgen on the occasion of his retirement as professor of logic and cognitive science at the University of Amsterdam.

One can consider μ‐Martin‐Löf randomness for a probability measure μ on 2ω, such as the Bernoulli measure given. We study Bernoulli randomness of sequences in with parameters, and we reintroduce Bernoulli normality, where the uniform distribution of digits is replaced with a Bernoulli distribution. We prove the equivalence of three characterizations of Bernoulli normality. We show that every Bernoulli random real is Bernoulli normal, and this has the corollary that the set of Bernoulli normal reals has full (...) Bernoulli measure in. We give an algorithm for computing Bernoulli normal sequences from normal sequences so that we can give explicit examples of Bernoulli normal reals. We investigate an application of randomness to iterated function systems. Finally, we list a few further questions relating to Bernoulli randomness and Bernoulli normality. (shrink)

A set $B\subseteq\mathbb{N}$ is called low for Martin-Löf random if every Martin-Löf random set is also Martin-Löf random relative to B . We show that a $\Delta^0_2$ set B is low for Martin-Löf random if and only if the class of oracles which compress less efficiently than B , namely, the class $\mathcal{C}^B=\{A\ |\ \forall n\ K^B(n)\leq^+ K^A(n)\}$ is countable (where K denotes the prefix-free complexity and $\leq^+$ denotes inequality modulo a constant. It follows that $\Delta^0_2$ (...) is the largest arithmetical class with this property and if $\mathcal{C}^B$ is uncountable, it contains a perfect $\Pi^0_1$ set of reals. The proof introduces a new method for constructing nontrivial reals below a $\Delta^0_2$ set which is not low for Martin-Löf random. (shrink)

It is known that the box dimension of any Martin-Löf random closed set of ${\{0,1\}^\mathbb{N}}$ is ${\log_2(\frac{4}{3})}$ . Barmpalias et al. [J Logic Comput 17(6):1041–1062, 2007] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for computing the Hausdorff dimension of almost every random closed set of ${\{0,1\}^\mathbb{N}}$ , and propose a general method for random closed (...) sets in other spaces. We further find both the appropriate dimensional Hausdorff measure and the exact Hausdorff dimension for such random closed sets. (shrink)

It is time for a new paper about open questions in the currently very active area of randomness and computability. Ambos-Spies and Kučera presented such a paper in 1999 [1]. All the question in it have been solved, except for one: is KL-randomness different from Martin-Löf randomness? This question is discussed in Section 6.Not all the questions are necessarily hard—some simply have not been tried seriously. When we think a question is a major one, and therefore (...) likely to be hard, we indicate this by the symbol ▶, the criterion being that it is of considerable interest and has been tried by a number of researchers. Some questions are close contenders here; these are marked by ▷. With few exceptions, the questions are precise. They mostly have a yes/no answer. However, there are often more general questions of an intuitive or even philosophical nature behind. We give an outline, indicating the more general questions.All sets will be sets of natural numbers, unless otherwise stated. These sets are identified with infinite strings over {0, 1}. Other terms used in the literature are sequence and real. (shrink)

A semimeasure is a generalization of a probability measure obtained by relaxing the additivity requirement to superadditivity. We introduce and study several randomness notions for left-c.e. semimeasures, a natural class of effectively approximable semimeasures induced by Turing functionals. Among the randomness notions we consider, the generalization of weak 2-randomness to left-c.e. semimeasures is the most compelling, as it best reflects Martin-Löf randomness with respect to a computable measure. Additionally, we analyze a question of Shen, a (...) positive answer to which would also have yielded a reasonable randomness notion for left-c.e. semimeasures. Unfortunately, though, we find a negative answer, except for some special cases. (shrink)

Dalam buku ini Kardinal Jorge Mario Bergoglio—saat itu masih Uskup Agung Buenos Aires dan sejak 13 Maret 2013 menjadi Paus Fransiskus—dan Rabi Abraham Skorka berdialog tentang sejumlah masalah agama, kehidupan, keluarga, politik, dan masyarakat yang mereka lihat sebagai tantangan besar pada abad ke-21 ini. Dialog itu mulai dan berakhir dengan penukaran pandangan tentang topik dialog sendiri sebagaimana mereka usahakan. Latarnya adalah Argentina yang karena sejarahnya telah lupa akan seni untuk saling mendengarkan dan berbicara dengan satu sama lain. Di tengah kebuntuan (...) itu kedua tokoh agama ini mulai berjumpa dalam suatu seri percakapan yang terus berkembang. Atas usul Skorka, sejumlah dari percakapan mereka akhirnya dipublikasikan. Buku dialog ini merupakan ilustrasi menarik tentang apa yang kemudian akan ditulis oleh Paus Fransiskus dalam Seruan Apostolik Evangelii Gaudium (Sukacita Injil, 2013) tentang dialog sosial (no. 238-258). Sejauh menyangkut dialog antaragama dalam buku ini segera tampak suatu perbedaan dengan pendahulunya, Paus Benediktus XVI, yang selalu menghindari ibadat bersama dalam perjumpaan antar agama. Sebagai bagian dialog, Bergoglio tidak segan memenuhi undangan untuk berpartisipasi, berbicara, berdoa, dan minta didoakan dalam ibadat sinagoga. Juga di Buenos Aires saat itu tidak semua orang dapat menerima hal itu dengan mudah.................. Dialog dua warga Buenos Aires ini berakar dalam konteks masyarakat Argentina, dengan sejarahnya tersendiri seperti conquista, peronisme, rezim militer, dan seterusnya. Setelah Uskup Buenos Aires menjadi Uskup Roma, dialog itu sudah diterjemahkan dalam banyak bahasa. Saya belum menemukan terjemahannya dalam bahasa Indonesia. Karena menyangkut masalah mendasar pluralisme agama, kehidupan, dan masyarakat, dialog-dialog ini inspiratif di manapun. Tentu dialog sudah mulai berlangsung juga dalam masyarakat kita, tetapi buahnya dalam terbitan seperti ini masih kurang kelihatan. Koleksi makalah-makalah hasil aneka diskusi forum kita belum memperlihatkan intensitas perjumpaan seperti yang terlihat dalam buku ini. (Martin Harun, Guru Besar Ilmu Teologi Emeritus, Sekolah Tinggi Filsafat Driyarkara, Jakarta). (shrink)

Unlike Martin‐Löf randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and further, provides a general method for abstracting “bit‐wise” definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previously (...) known results. The paper contains a new type of randomness—endomorphism randomness—which the author hopes will shed light on the open question of whether Kolmogorov‐Loveland randomness is equivalent to Martin‐Löf randomness. The last section contains ideas for future research. (shrink)

We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0'. This contrasts with a result of Kučera and Terwijn [5] on sets that are low for the class of Martin-Löf random reals.

The study of Martin‐Löf randomness on a computable metric space with a computable measure has seen much progress recently. In this paper we study Martin‐Löf randomness on a more general space, that is, a computable topological space with a computable measure. On such a space, Martin‐Löf randomness may not be a natural notion because there is no universal test, and Martin‐Löf randomness and complexity randomness (defined in this paper) do not coincide (...) in general. We show that SCT3 is a sufficient condition for the existence and coincidence, and study how much we can weaken this condition. (shrink)

We investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$. We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$ weakly 2-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $RC{A_0}$ the existence of (...) computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Löf randoms, and we describe a sense in which this result is nearly optimal. (shrink)

Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but (...) not Martin-Löf random, and provide a new characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of "Schnorr reducibility" which allows us to calibrate the Schnorr complexity of reals. We define the class of "Schnorr trivial" reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members. (shrink)

We show that for each computable ordinal $\alpha > 0$ it is possible to find in each Martin-Löf random ${\rm{\Delta }}_2^0 $ degree a sequence R of Cantor-Bendixson rank α, while ensuring that the sequences that inductively witness R’s rank are all Martin-Löf random with respect to a single countably supported and computable measure. This is a strengthening for random degrees of a recent result of Downey, Wu, and Yang, and can be understood as a randomized version of (...) it. (shrink)

Using possibly infinite computations on universal monotone Turing machines, we prove Martin-Löf randomness in ∅' of the probability that the output be in some set.

The concept of randomness has been unjustly neglected in recent philosophical literature, and when philosophers have thought about it, they have usually acquiesced in views about the concept that are fundamentally flawed. After indicating the ways in which these accounts are flawed, I propose that randomness is to be understood as a special case of the epistemic concept of the unpredictability of a process. This proposal arguably captures the intuitive desiderata for the concept of randomness; at least (...) it should suggest that the commonly accepted accounts cannot be the whole story and more philosophical attention needs to be paid. 1. Randomness in science1.1Random systems1.2Random behaviour1.3Random sampling1.4Caprice, arbitrariness and noise2. Concepts of randomness2.1Von Mises/church/martin-löf randomness2.2KCS-randomness3. Randomness is unpredictability: preliminaries3.1Process and product randomness3.2Randomness is indeterminism?4. Predictability4.1Epistemic constraints on prediction4.2Computational constraints on prediction4.3Pragmatic constraints on prediction4.4Prediction defined5. Unpredictability6. Randomness is unpredictability6.1Clarification of the definition of randomness6.2Randomness and probability6.3Subjectivity and context sensitivity of randomness7. Evaluating the analysis[R]andomness … is going to be a concept which is relative to our body of knowledge, which will somehow reflect what we know and what we don't know. Henry E. Kyburg, Jr ([1974], p. 217)Phenomena that we cannot predict must be judged random. Patrick Suppes ([1984], p. 32). (shrink)

We study randomness beyond${\rm{\Pi }}_1^1$-randomness and its Martin-Löf type variant, which was introduced in [16] and further studied in [3]. Here we focus on a class strictly between${\rm{\Pi }}_1^1$and${\rm{\Sigma }}_2^1$that is given by the infinite time Turing machines introduced by Hamkins and Kidder. The main results show that the randomness notions associated with this class have several desirable properties, which resemble those of classical random notions such as Martin-Löf randomness and randomness notions defined (...) via effective descriptive set theory such as${\rm{\Pi }}_1^1$-randomness. For instance, mutual randoms do not share information and a version of van Lambalgen’s theorem holds.Towards these results, we prove the following analogue to a theorem of Sacks. If a real is infinite time Turing computable relative to all reals in some given set of reals with positive Lebesgue measure, then it is already infinite time Turing computable. As a technical tool towards this result, we prove facts of independent interest about random forcing over increasing unions of admissible sets, which allow efficient proofs of some classical results about hyperarithmetic sets. (shrink)

We study the sets that are computable from both halves of some (Martin–Löf) random sequence, which we call 1/2-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the K-trivial sets. We characterize 1/2-bases as the sets computable from both halves of Chaitin’s Ω, and as the sets that obey the cost function c(x,s)=Ωs−Ωx−−−−−−−√. Generalizing these results yields a dense hierarchy (...) of subideals in the K-trivial degrees: For k<n, let Bk/n be the collection of sets that are below any k out of n columns of some random sequence. As before, this is an ideal generated by its c.e. elements and the random sequence in the definition can always be taken to be Ω. Furthermore, the corresponding cost function characterization reveals that Bk/n is independent of the particular representation of the rational k/n, and that Bp is properly contained in Bq for rational numbers p<q. These results are proved using a generalization of the Loomis–Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyze arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the K-trivial sets; we can calculate from the family which Bp it characterizes. We finish by studying the union of Bp for p<1; we prove that this ideal consists of the sets that are robustly computable from some random sequence. This class was previously studied by Hirschfeldt [D. R. Hirschfeldt, C. G. Jockusch, R. Kuyper and P. E. Schupp, Coarse reducibility and algorithmic randomness, J. Symbolic Logic81(3) (2016) 1028–1046], who showed that it is a proper subclass of the K-trivial sets. We prove that all such sets are robustly computable from Ω, and that they form a proper subideal of the sets computable from every (weakly) LR-hard random sequence. We also show that the ideal cannot be characterized by a cost function, giving the first such example of a Σ03 subideal of the K-trivial sets. (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)

We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}$ of continuous functions on ${2^{\mathbb N}}$ . A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random ${\Delta^0_2}$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any ${y \in (...) 2^{\mathbb N}}$ , there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. (shrink)

Background: Moral distress has been described as a major problem for the nursing profession, and in recent years, a considerable amount of research has been undertaken to examine its causes and effects. However, few research projects have been performed that examined the moral distress of an entire nation’s nurses, as this particular study does. Aim/objective: The purpose of this study was to determine the frequency and intensity of moral distress experienced by registered nurses in New Zealand. Research design: The research (...) involved the use of a mainly quantitative approach supported by a slightly modified version of a survey based on the Moral Distress Scale–Revised. Participants and research context: In total, 1500 questionnaires were sent out at random to nurses working in general areas around New Zealand and 412 were returned, giving an adequate response rate of 27%. Ethical considerations: The project was evaluated and judged to be low risk and recorded as such on 22 February 2011 via the auspices of the Massey University Human Ethics Committee. Findings: Results indicate that the most frequent situations to cause nursing distress were (a) having to provide less than optimal care due to management decisions, (b) seeing patient care suffer due to lack of provider continuity and (c) working with others who are less than competent. The most distressing experiences resulted from (a) working with others who are unsafe or incompetent, (b) witnessing diminished care due to poor communication and (c) watching patients suffer due to a lack of provider continuity. Of the respondents, 48% reported having considered leaving their position due to the moral distress. Conclusion: The results imply that moral distress in nursing remains a highly significant and pertinent issue that requires greater consideration by health service managers, policymakers and nurse educators. (shrink)

We prove that degrees that are low for Kurtz randomness cannot be diagonally non-recursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmune-free non-DNR degrees, which are also exactly the degrees that are low for weak 1-genericity. We also consider Low(M, Kurtz), the class of degrees a such that every element of M is a-Kurtz random. These are characterised when M is the class of Martin-Löf random, computably random, or Schnorr (...) random reals. We show that Low(ML, Kurtz) coincides with the non-DNR degrees, while both Low(CR, Kurtz) and Low(Schnorr, Kurtz) are exactly the non-high, non-DNR degrees. (shrink)

Let f be a computable function from finite sequences of 0ʼs and 1ʼs to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that (...) f-randomness relative to a PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree. (shrink)

Demuth tests generalize Martin-Löf tests in that one can exchange the m-th component a computably bounded number of times. A set fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that GmGm+1 for each m, we have weak Demuth randomness.We show that a weakly Demuth random set can be high and , yet not superhigh. Next, any c.e. set Turing below a Demuth random set is (...) strongly jump-traceable.We also prove a basis theorem for non-empty classes P. It extends the Jockusch–Soare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2-random set does not compute a 2-fixed point free function. (shrink)

We investigate the role of continuous reductions and continuous relativization in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van Lambalgen’s theorem and the Miller–Yu/Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterizations of lowness for Martin-Löf randomness. We also characterize computing higher [Formula: (...) see text]-trivial sets by higher random sequences. We give a separation between higher notions of randomness, in particular between higher weak 2-randomness and [Formula: see text]-randomness. To do so we investigate classes of functions computable from Kleene’s [Formula: see text] based on strong forms of the higher limit lemma. (shrink)

We use Demuth randomness to study strong lowness properties of computably enumerable sets, and sometimes of Δ20 sets. A set A⊆N is called a base for Demuth randomness if some set Y Turing above A is Demuth random relative to A. We show that there is an incomputable, computably enumerable base for Demuth randomness, and that each base for Demuth randomness is strongly jump-traceable. We obtain new proofs that each computably enumerable set below all superlow (...) class='Hi'>Martin-Löf random sets is strongly jump traceable, using Demuth tests. The sets Turing below each ω2-computably approximable Martin-Löf random set form a proper subclass of the bases for Demuth randomness, and hence of the strongly jump traceable sets. The c.e. sets Turing below each ω2-computably approximable Martin-Löf random set satisfy a new, very strong combinatorial lowness property called ω-traceability. (shrink)

In algorithmic randomness, when one wants to define a randomness notion with respect to some non-computable measure λ, a choice needs to be made. One approach is to allow randomness tests to access the measure λ as an oracle . The other approach is the opposite one, where the randomness tests are completely effective and do not have access to the information contained in λ . While the Hippocratic approach is in general much more restrictive, there (...) are cases where the two coincide. The first author showed in 2010 that in the particular case where the notion of randomness considered is Martin-Löf randomness and the measure λ is a Bernoulli measure, classical randomness and Hippocratic randomness coincide. In this paper, we prove that this result no longer holds for other notions of randomness, namely computable randomness and stochasticity. (shrink)

We prove a number of results in effective randomness, using methods in which Π⁰₁ classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.

Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin–Löf. We extend this result and demonstrate that QM is algorithmic $$\omega$$ -random and generic, precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo–Fraenkel Solovay (...) random on infinite-dimensional Hilbert spaces. Moreover, it is more likely that there exists a standard transitive ZFC model M, where QM is expressed in reality, than in the universe V of sets. Then every generic quantum measurement adds to M the infinite sequence, i.e. random real $$r\in 2^{\omega }$$, and the model undergoes random forcing extensions M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit, therefore showing the structural resemblance of both in this limit. We discuss several questions regarding measurability and possible practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC; however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself, the issues considered here remain hidden to a great extent. (shrink)

What should morally conscientious agents do if they must choose among options that are somewhat right and somewhat wrong? Should one select an option that is right to the highest degree, or would it perhaps be more rational to choose randomly among all somewhat right options? And how should lawmakers and courts address behaviour that is neither entirely right nor entirely wrong? In this first book-length discussion of the 'gray area' in ethics, Martin Peterson challenges the assumption that rightness (...) and wrongness are binary properties and explores acts which are neither entirely right nor entirely wrong, but rather a bit of both. Including discussions of white lies and the permissibility of abortion, Peterson's book presents a gradualist theory of right and wrong designed to answer these and other practical questions about the gray area in ethics. (shrink)

When the probability of causes, and the probability of effects, given causes, are each randomly assigned, entropy ‘usually’ increases.

“Freedom” is a phenomenon in the natural world. This phenomenon—and indirectly the question of free will—is explored using a variety of systems-theoretic ideas. It is argued that freedom can emerge only in systems that are partially determined and partially random, and that freedom is a matter of degree. The paper considers types of freedom and their conditions of possibility in simple living systems and in complex living systems that have modeling subsystems. In simple living systems, types of freedom include independence (...) from fixed materiality, internal rather than external determination, activeness that is unblocked and holistic, and the capacity to choose or alter environmental constraint. In complex living systems, there is freedom in satisfaction of lower level needs that allows higher potentials to be realized. Several types of freedom also manifest in the modeling subsystems of these complex systems: in the transcending of automatism in subjective experience, in reason as instrument for passion yet also in reason ruling over passion, in independence from informational colonization by the environment, and in mobility of attention. Considering the wide range of freedoms in simple and complex living systems allows a panoramic view of this diverse and important natural phenomenon. (shrink)

In Knowledge and Lotteries, John Hawthorne offers a diagnosis of our unwillingness to believe, of a given lottery ticket, that it will lose a fair lottery – no matter how many tickets are involved. According to Hawthorne, it is natural to employ parity reasoning when thinking about lottery outcomes: Put roughly, to believe that a given ticket will lose, no matter how likely that is, is to make an arbitrary choice between alternatives that are perfectly balanced given one’s evidence. It’s (...) natural to think that parity reasoning is only applicable to situations involving lotteries dice, spinners etc. – in short, situations in which we are reasoning about the outcomes of a putatively random process. As I shall argue in this paper, however, there are reasons for thinking that parity reasoning can be applied to any proposition that is less than certain given one’s evidence. To see this, we need only remind ourselves of a kind of argument employed by John Pollock and Keith Lehrer in the 1980s. If this argument works, then believing any uncertain proposition, no matter how likely it is, involves a (covert) arbitrary or capricious choice – an idea that contains an obvious sceptical threat. (shrink)

We formalise the notion of those infinite binary sequences z that admit a single program P which expresses the entire algorithmical structure of z. Such a program P minimizes the information which must be used in a relative computation for z. We propose two concepts with different strength for this notion, the learnable and the super-learnable sequences. We establish three different equivalent characterizations of learnable (super-learnable, resp.) sequences. In particular, we prove that a sequences z is learnable (super-learnable, resp.) if (...) and only if there is a computable probability measure p such that p is Schnorr (Martin-Lof, resp.) p-random. There is a recursively enumerable sequence which is not learnable. The learnable sequences are invariant with respect to all total and effective transformations of infinite binary sequences. (shrink)