For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f,g\in\omega^\omega}$$\end{document} let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c^\forall_{f,g}}$$\end{document} be the minimal number of uniform g-splitting trees needed to cover the uniform f-splitting tree, i.e., for every branch ν of the f-tree, one of the g-trees contains ν. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c^\exists_{f,g}}$$\end{document} be the dual notion: For every branch ν, one of the g-trees guesses ν(m) infinitely often. We show that (...) it is consistent that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c^\exists_{f_\epsilon,g_\epsilon}{=}c^\forall_{f_\epsilon,g_\epsilon}{=}\kappa_\e psilon}$$\end{document} for continuum many pairwise different cardinals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa_\epsilon}$$\end{document} and suitable pairs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(f_\epsilon,g_\epsilon)}$$\end{document}. For the proof we introduce a new mixed-limit creature forcing construction. (shrink)

We show that the Abraham–Rubin–Shelah Open Coloring Axiom is consistent with a large continuum, in particular, consistent with [Formula: see text]. This answers one of the main open questions from [U. Abraham, M. Rubin and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of [Formula: see text]-dense real order types, Ann. Pure Appl. Logic 325(29) (1985) 123–206]. As in [U. Abraham, M. Rubin and S. Shelah, On the consistency of some partition (...) theorems for continuous colorings, and the structure of [Formula: see text]-dense real order types, Ann. Pure Appl. Logic 325(29) (1985) 123–206], we need to construct names for the so-called preassignments of colors in order to add the necessary homogeneous sets. However, the known constructions of preassignments (ours in particular) only work assuming the [Formula: see text]. In order to address this difficulty, we show how to construct such names with very strong symmetry conditions. This symmetry allows us to combine them in many different ways, using a new type of poset called a partition product. Partition products may be thought of as a restricted memory iteration with stringent isomorphism and coherent-overlap conditions on the memories. We finally construct, in [Formula: see text], the partition product which gives us a model of [Formula: see text] in which [Formula: see text]. (shrink)

For f, g $ \in \omega ^\omega $ let $c_{f,g}^\forall $ be the minimal number of uniform g-splitting trees (or: Slaloms) to cover the uniform f-splitting tree, i.e., for every branch v of the f-tree, one of the g-trees contains v. $c_{f,g}^\exists $ is the dual notion: For every branch v, one of the g-trees guesses v(m) infinitely often. It is consistent that $c_{f \in ,g \in }^\exists = c_{f \in ,g \in }^\forall = k_ \in $ for N₁ many (...) pairwise different cardinals $k_ \in $ and suitable pairs $(f_{ \in ,g \in } ).$ For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature forcing satisfies fusion and pure decision. We introduce decisiveness and use it to construct a variant of the countable support iteration of such forcings, which still satisfies fusion and pure decision. (shrink)

Let κ < λ be regular uncountable cardinals. Using a finite support iteration (in fact a matrix iteration) of ccc posets we obtain the consistency of b = a = κ < s = λ. If μ is a measurable cardinal and μ < κ < λ, then using similar techniques we obtain the consistency of b = κ < a = s = λ.

We show that is consistent with the existence of a -definable wellorder of the reals and a -definable ω-mad subfamily of [ω]ω.

We extend the work of Fischer et al. [6] by presenting a method for controlling cardinal characteristics in the presence of a projective wellorder and 2ℵ0>ℵ2. This also answers a question of Harrington [9] by showing that the existence of a Δ31 wellorder of the reals is consistent with Martinʼs axiom and 2ℵ0=ℵ3.

This is a survey paper which discusses the impact of large cardinals on provability of the Continuum Hypothesis. It was Gödel who first suggested that perhaps “strong axioms of infinity” could decide interesting set-theoretical statements independent over ZFC, such as CH. This hope proved largely unfounded for CH—one can show that virtually all large cardinals defined so far do not affect the status of CH. It seems to be an inherent feature of large cardinals that they (...) do not determine properties of sets low in the cumulative hierarchy if such properties can be forced to hold or fail by small forcings.The paper can also be used as an introductory text on large cardinals as it defines all relevant concepts. (shrink)

In this paper we consider whether L(R) has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(R) not to contain such a counterexample. Along the way we establish many results about nonstationary towers, non-reflecting stationary sets, generalizations of proper and semiproper forcing and Chang's conjecture.

This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly (...) pick out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum. (shrink)

We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$, as $\kappa (...) $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$ ). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert’s axiom of completeness. (shrink)

This paper examines the contribution of Gabrio Piola to continuum mechanics.Though he was undoubtably a skilled mathematician and a good mechanician, little is commonly known about his papers within the international scientific community, principally because a large part of the Italian school of mechanics was isolated in the first half of the XIXth century.We examine and comment on Piola’s most important papers, and compare them with those of his contemporaries Cauchy, Poisson and Kirchhoff.

There inevitably comes a time when even the best informed minds are tempted to yield to the lure of binarism: when we are no longer concerned with the details of a system, but rather with our vision of the relationship between man and the world. Without going any further it can already be said that the problem of existence is presented to us in terms of a man/world duality, as though we were unable to exceed our subjective vision of things (...) so as to assess their intrinsic reality.When linguists discovered that they could isolate discrete units called phonemes, they were tempted to oppose the discontinuity of these phonemes – reflecting the underlying discontinuity of the signifiers – to a continuum of prelinguistic experience, a continuum that is not arranged as discrete elements except with reference to any significant units of the language selected for the purpose of communicating this experience to others. At the outset we have nothing but a nebulous mass where only the application of a linguistic grid allows us to discern identifiable units. Even if, in the final analysis, it will prove necessary to doubt the universal validity of this vision of things, we must recognize its usefulness in concretizing a concept of linguistic facts – a concept that has for some time played a beneficent role in research. At least for some of us, this vision should reflect the manner in which the world is gradually organized in our minds each time a perceptible reality is isolated from its context at the instant it receives a name. Allow me to introduce an illustration from personal experience. It was the last day of July in 1914. In our little village in the Savoy the international tension – which would eventually end in general mobilization – was very perceptible; our classes were idle, and teachers and pupils had settled down in a field near the school. I was six. Sitting on the grass, my attention was attracted by a plant with large flat leaves. At that moment someone said the word plantain “plantain.” The plant had not been heretofore exactly unknown to me; it didn't strike me, at that moment, as a discovery; but it acquired a distinct existence only at that instant in the green carpet of the field when a signifier had definitively isolated it. It was most assuredly the gravity of the hour that engraved this incident in my memory. (shrink)

There are powerful fairness and efficiency arguments for making charitable donations to soup kitchens 100% deductible. These arguments have no purchase for donations to fund opulent church organs, yet these too are 100% deductible under the current tax code. This stark dichotomy is only the tip of the iceberg. Looking at a wider sampling of charitable gifts reveals a charitable continuum. Based on sliding scales for efficiency, multiple theories of fairness, pluralism, institutional competence and social welfare dictate that charitable (...) deductions should in most cases be fractions between zero and one. Moreover, the Central Limit Theorem strongly suggests that combining this welter of largely independent criteria with the wide variety of charitable gifts results in a classic bell-shaped normal curve of optimal deductions, with a peak at some central value and quickly decaying to zero at the extremes of 0% and 100%. Given that those are the only two options under the current tax code, the current charitable deduction regime inevitably makes large errors in most cases. Actually calculating a precise optimal percentage for each type of charitable donation is of course impractical. This Article suggests, however, that we can do much better than the systematically erroneous current charitable deduction. Granting a 100% deduction only for donations to the desperately poor, along with 50%, 25%, and 0% for gifts yielding progressively fewer efficiency, fairness, pluralism, and institutional competence benefits, promises to deliver a socially more desirable charitable deduction. (shrink)

The purpose of this research was to explore the concept of collaboration within a specific healthcare context and to include the perspectives of healthcare users, a position largely lacking in previous studies. In applying a critical theoretical approach, the focus was on, as an exemplar, mothers with newborn babies who had spent more than 48 hr in a special care nursery. Semistructured interviews were undertaken with child health nurses, midwives and mothers. The three key theoretical findings on collaboration generated in (...) the study point to layers of meanings around identity, knowledge and institutions of care. Findings from the interview data analysis were further examined through the lens of key policy documents. The research outcomes indicate that the concept of collaboration serves an important function in healthcare in obscuring the complexities and ambiguities that characterise the care continuum. The study concludes the need for a more critical approach to the assumptions that underlie the language of collaboration and the implications for practice in healthcare. (shrink)

The problem at the center of this essay is how one can reconcile the continuity of space with a monadological theory of matter, according to which matter is ultimately composed of simple elements, a problem that greatly exercised Leibniz, the Wolffians, and Kant. The underlying purpose of this essay is to illustrate my reading of Kant’s philosophical development, and of his relation to the Wolffians and Leibniz, according to which, (a), this development was fueled by ‘home-grown’ problems that arose within (...) the framework of the Wolffian philosophy from which Kant started out, and, (b), on his journey to critical idealism, Kant gradually moved away from Wolffianism, but closer to Leibniz, which, however, he came to realize only some years after the publication of the Critique of Pure Reason. This reading is illustrated by showing that the problem of how to reconcile the continuity of space with a monadological theory of matter is a problem that Kant inherits from Leibniz and the Wolffians, in whose thinking it already plays an important role, that Leibniz’s mature solution to the problem differs markedly from the Wolffian solution, and that Kant’s early, pre-critical solution is largely Wolffian, while his later critical solution is largely Leibnizian, as he himself notes with gleeful satisfaction. The discussion also reveals that this problem is one of the key problems that fueled Kant’s philosophical development and, eventually, led him to the discovery of transcendental idealism. (shrink)

We argue against Foreman’s proposal to settle the continuum hypothesis and other classical independent questions via the adoption of generic large cardinal axioms.

We explore the connection between combinatorial principles on uncountable cardinals, like stick and club, on the one hand, and the combinatorics of sets of reals and, in particular, cardinal invariants of the continuum, on the other hand. For example, we prove that additivity of measure implies that Martin’s axiom holds for any Cohen algebra. We construct a model in which club holds, yet the covering number of the null ideal is large. We show that for uncountable cardinals κ≤λ (...) and , if all subsets of λ either contain, or are disjoint from, a member of , then has size at least etc. As an application, we solve the Gross space problem under by showing that there is such a space over any countable field. In two appendices, we solve problems of Fuchino, Shelah and Soukup, and of Kraszewski, respectively. (shrink)

We argue for a perspective on bilingual heritage speakers as native speakers of both their languages and present results from a large-scale, cross-linguistic study that took such a perspective and approached bilinguals and monolinguals on equal grounds. We targeted comparable language use in bilingual and monolingual speakers, crucially covering broader repertoires than just formal language. A main database was the open-access RUEG corpus, which covers comparable informal vs. formal and spoken vs. written productions by adolescent and adult bilinguals with (...) heritage-Greek, -Russian, and -Turkish in Germany and the United States and with heritage-German in the United States, and matching data from monolinguals in Germany, the United States, Greece, Russia, and Turkey. Our main results lie in three areas. We found non-canonical patterns not only in bilingual, but also in monolingual speakers, including patterns that have so far been considered absent from native grammars, in domains of morphology, syntax, intonation, and pragmatics. We found a degree of lexical and morphosyntactic inter-speaker variability in monolinguals that was sometimes higher than that of bilinguals, further challenging the model of the streamlined native speaker. In majority language use, non-canonical patterns were dominant in spoken and/or informal registers, and this was true for monolinguals and bilinguals. In some cases, bilingual speakers were leading quantitatively. In heritage settings where the language was not part of formal schooling, we found tendencies of register leveling, presumably due to the fact that speakers had limited access to formal registers of the heritage language. Our findings thus indicate possible quantitative differences and different register distributions rather than distinct grammatical patterns in bilingual and monolingual speakers. This supports the integration of heritage speakers into the native-speaker continuum. Approaching heritage speakers from this perspective helps us to better understand the empirical data and can shed light on language variation and change in native grammars. Furthermore, our findings for monolinguals lead us to reconsider the state-of-the art on majority languages, given recurring evidence for non-canonical patterns that deviate from what has been assumed in the literature so far, and might have been attributed to bilingualism had we not included informal and spoken registers in monolinguals and bilinguals alike. (shrink)

We continue the investigation of Gregory trees and the Cantor Tree Property carried out by Hart and Kunen. We produce models of MA with the Continuum arbitrarily large in which there are Gregory trees, and in which there are no Gregory trees.

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle [math] introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from (...) above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle [math] for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak [math]-indescribability, can be canonically characterized through localized versions of the principle [math]. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak [math]-indescribability. (shrink)

Continuing, we study the Strong Downward Löwenheim–Skolem Theorems of the stationary logic and their variations. In Fuchino et al. it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters \. This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size \. We also consider a version of the stationary logic and show that the SDLS for this logic in internal interpretation (...) \\) for reflection down to \ is consistent under the assumption of the consistency of ZFC \ “the existence of a supercompact cardinal” and this SDLS implies that the continuum is weakly Mahlo. These three “axioms” in terms of SDLS are consequences of three instances of a strengthening of generic supercompactness which we call Laver-generic supercompactness. Existence of a Laver-generic supercompact cardinal in each of these three instances also fixes the cardinality of the continuum to be \ or \ or very large respectively. We also show that the existence of one of these generic large cardinals implies the “\” version of the corresponding forcing axiom. (shrink)

Einsteinian gravity, of which Newtonian gravity is a part, is fraught with the problem of singularity that has been established as a theorem by Hawking and Penrose. The _hypothesis_ that founds the basis of both Einsteinian and Newtonian theories of gravity is that bodies with unequal magnitudes of masses fall with the same acceleration under the gravity of a source object. Since, the Einstein’s equations is one of the assumptions that underlies the proof of the singularity theorem, therefore, the above (...) hypothesis is implicitly one of the founding pillars of the same. In this work, I demonstrate how one can possibly write a non-singular theory of gravity which manifests that the above mentioned hypothesis is only valid in an approximate sense in the “large distance” scenario. To mention a specific instance, under the gravity of the earth, a 5 kg and a 500 kg fall with accelerations which differ by approximately \(113.148\times 10^{-32}\) meter/sec \(^2\) and the more massive object falls with less acceleration. Further, I demonstrate why the concept of gravitational field is not definable in the “small distance” regime which automatically justifies why the Einstein’s and Newton’s theories fail to provide any “small distance” analysis. In course of writing down this theory, I demonstrate why the continuum hypothesis as spelled out by Goedel, is undecidable. The theory has several aspects which provide the following realizations: (i) Descartes’ self-skepticism concerning exact representation of numbers by drawing lines (ii) Born’s wish of taking into account “natural uncertainty in all observations” while describing “a physical situation” by means of “real numbers” (iii) Klein’s vision of having “a fusion of arithmetic and geometry” where “a point is replaced by a small spot” (iv) Goedel’s assertion about “non-standard analysis, in some version” being “the analysis of the future”. (shrink)

Many philosophers have claimed that there is a tension between the impenetrability of matter and the possibility of contact between continuous bodies. This tension has led some to claim that impenetrable continuous bodies could not ever be in contact, and it has led others to posit certain structural features to continuous bodies that they believe would resolve the tension. Unfortunately, such philosophical discussions rarely borrow much from the investigation of actual matter. This is probably largely because actual matter is not (...) continuous, and so it might seem as if discussion of the structure of continuous bodies is merely within the realm of philosophical thought experiments rather than actual scientific investigation. However, classical continuum mechanics models actual matter as if it were continuous, and it has implications about the structure of continuous bodies and about what contact and impenetrability are. This paper describes the relevant notions from classical continuum mechanics so as to resolve the alleged tension between contact and impenetrability. (shrink)

The continuum function αmaps to2α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β, image and α<β→F≤F. The classic example of an Easton function is the continuum function αmaps to2α on regular cardinals. If GCH holds then any Easton function is the continuum function on regular cardinals of some cofinality-preserving extension V[G]; we say that F is realised in V[G]. However if (...) we also wish to preserve measurable cardinals, new restrictions must be put on F. We say that κ is F-hypermeasurable iff there is an elementary embedding j:V→M with critical point κ such that H)Vsubset of or equal toM; j will be called a witnessing embedding. We will show that if GCH holds then for any Easton function F there is a cofinality-preserving generic extension V[G] such that if κ, closed under F, is F-hypermeasurable in V and there is a witnessing embedding j such that j≥F, then κ will remain measurable in V[G]. (shrink)

Periodic systems are considered whose increments in quantum energy grow with quantum number. In the limit of large quantum number, systems are found to give correspondence in form between classical and quantum frequency-energy dependences. Solely passing to large quantum numbers, however, does not guarantee the classical spectrum. For the examples cited, successive quantum frequencies remain separated by the incrementhI −1, whereI is independent of quantum number. Frequency correspondence follows in Planck's limit,h → 0. The first example is that (...) of a particle in a cubical box with impenetrable walls. The quantum emission spectrum is found to be uniformly discrete over the whole frequency range. This quality holds in the limitn → ∞. The discrete spectrum due to transitions in the high-quantum-number bound states of a particle in a box with penetrable walls is shown to grow uniformly discrete in the limit that the well becomes infinitely deep. For the infinitely deep spherical well, on the other hand, correspondence is found to be obeyed both in emission and configuration. In all cases studied the classical ensemble gives a continuum of frequencies. (shrink)

The equiconsistency of a measurable cardinal with Mitchell order o=κ++ with a measurable cardinal such that 2κ=κ++ follows from the results by W. Mitchell [13] and M. Gitik [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ .In Friedman and Honzik [5], we formulated and proved Eastonʼs theorem [4] in a large cardinal setting, using slightly stronger hypotheses than the lower bounds identified by Mitchell and Gitik , for a suitable μ, instead of the (...) cardinals with the appropriate Mitchell order).In this paper, we use a new idea which allows us to carry out the constructions in Friedman and Honzik [5] from the optimal hypotheses. It follows that the lower bounds identified by Mitchell and Gitik are optimal also with regard to the general behavior of the continuum function on regulars in the context of measurable cardinals. (shrink)

A question dating to Mardešić and Prasolov’s 1988 work [S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307(2) (1988) 725–744], and motivating a considerable amount of set theoretic work in the years since, is that of whether it is consistent with the ZFC axioms for the higher derived limits [Formula: see text] [Formula: see text] of a certain inverse system [Formula: see text] indexed by [Formula: see text] to simultaneously vanish. An equivalent formulation (...) of this question is that of whether it is consistent for all [Formula: see text]-coherent families of functions indexed by [Formula: see text] to be trivial. In this paper, we prove that, in any forcing extension given by adjoining [Formula: see text]-many Cohen reals, [Formula: see text] vanishes for all [Formula: see text]. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher-dimensional [Formula: see text]-system lemmas. This work removes all large cardinal hypotheses from the main result of [J. Bergfalk and C. Lambie-Hanson, Simultaneously vanishing higher derived limits, Forum Math. Pi 9 (2021) e4] and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of [Formula: see text] for all [Formula: see text]. (shrink)

In recent years, mtDNA and Y chromosome studies involving human populations from South Asia and the rest of the world have revealed new insights about the peopling of the world by anatomically modern humans during the late Pleistocene, some 40,000–60,000 years ago, over the southern coastal route from Africa. Molecular studies and archaeological record are both largely consistent with autochthonous differentiation of the genetic structure of the caste and tribal populations in South Asia. High level of endogamy created by numerous (...) social boundaries within and between castes and tribes, along with the influence of several evolutionary forces such as genetic drift, fragmentation and long‐term isolation, has kept the Indian populations diverse and distant from each other as well as from other continental populations. This review attempts to summarize recent genetic studies on Indian caste and tribal populations with the focus on the information embedded in the socially defined structure of Indian populations. BioEssays 29: 91–100, 2007. © 2006 Wiley Periodicals, Inc. (shrink)

In Plato, Politics and a Practical Utopia Kenneth Royce Moore offers a working model of Magnesia, the city of Plato's Laws. His method is to treat the “second-best city” “as if it were a real polis of the ancient world” (p. 82). Moore's conclusion is that Plato has created a “fairly large city”, with some unusual institutional features, but one that is “strangely practical” and firmly grounded in reality (p. ix). The Laws is often said to be a long (...) and rambling work showing “various signs of (…) - 12. Plato 12 (2012). (shrink)

We show the consistency of ${\frak o} <{\frak d}$ where ${\frak o}$ is the size of the smallest off-branch family, and ${\frak d}$ is as usual the dominating number. We also prove the consistency of ${\frak b} < {\frak a}$ with large continuum. Here, ${\frak b}$ is the unbounding number, and ${\frak a}$ is the almost disjointness number.

The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than $\aleph _{0}$ and smaller or equal than $\mathfrak {c}.$ Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of $\omega $ such that the intersection of any two of its elements is finite. A MAD family is a maximal (...) almost disjoint family. An ultrafilter $\mathcal {U}$ on $\omega $ is called a P-point if every countable $\mathcal {B\subseteq U}$ there is $X\in $ $\mathcal {U}$ such that $X\setminus B$ is finite for every $B\in \mathcal {B}.$ This kind of ultrafilters has been extensively studied, however there is still a large number of open questions about them.In the preliminaries we recall the principal properties of filters, ultrafilters, ideals, MAD families and cardinal invariants of the continuum. We present the construction of Shelah, Mildenberger, Raghavan, and Steprāns of a completely separable MAD family under $\mathfrak {s\leq a}.$ None of the results in this chapter are due to the author.The second chapter is dedicated to a principle of Sierpiński. The principle $\left $ of Sierpiński is the following statement: There is a family of functions $\left \{ \varphi _{n}:\omega _{1}\longrightarrow \omega _{1}\mid n\in \omega \right \} $ such that for every $I\in \left [ \omega _{1}\right ] ^{\omega _{1}}$ there is $n\in \omega $ for which $\varphi _{n}\left [ I\right ] =\omega _{1}.$ This principle was recently studied by Arnie Miller. He showed that this principle is equivalent to the following statement: There is a set $X=\left \{ f_{\alpha }\mid \alpha \alpha $ then $f_{\beta }\cap g$ is infinite. Miller showed that the principle of Sierpiński implies that non $\left =\omega _{1}.$ He asked if the converse was true, i.e., does non $\left =\omega _{1}$ imply the principle $\left $ of Sierpiński? We answer his question affirmatively. In other words, we show that non $\left =\omega _{1}$ is enough to construct an $\mathcal {IE}$ -Luzin set. It is not hard to see that the $\mathcal {IE}$ -Luzin set we constructed is meager. This is no coincidence, because with the aid of an inaccessible cardinal, we construct a model where non $\left =\omega _{1}$ and every $\mathcal {IE}$ -Luzin set is meager.The third chapter is dedicated to a conjecture of Hrušák. Michael Hrušák conjectured the following: Every Borel cardinal invariant is either at most non $\left $ or at least cov $\left $. Although the veracity of this conjecture is still an open problem, we were able to obtain some partial results: The conjecture is false for “Borel invariants of $\omega _{1}^{\omega }$ ” nevertheless, it is true for a large class of definable invariants. This is part of a joint work with Michael Hrušák and Jindřich Zapletal.In the fourth chapter we present a survey on destructibility of ideals and MAD families. We prove several classic theorems, but we also prove some new results. For example, we show that every almost disjoint family of size less than $\mathfrak {c}$ can be extended to a Cohen indestructible MAD family is equivalent to $\mathfrak {b=c}$. A MAD family $\mathcal {A}$ is Shelah–Steprāns if for every $X\subseteq \left [ \omega \right ] ^{<\omega }\setminus \left \{ \emptyset \right \} $ either there is $A\in \mathcal {I}\left $ such that $s\cap A\neq \emptyset $ for every $s\in X$ or there is $B\in \mathcal {I}\left $ that contains infinitely many elements of X $ denotes the ideal generated by $\mathcal {A}$ ). This concept was introduced by Raghavan which is connected to the notion of “strongly separable” introduced by Shelah and Steprāns. We prove that Shelah–Steprāns MAD families have very strong indestructibility properties: Shelah–Steprāns MAD families are indestructible for “many” definable forcings that does not add dominating reals. According to the author’s best knowledge, this is the strongest notion that has been considered in the literature so far. In spite of their strong indestructibility, Shelah–Steprāns MAD families can be destroyed by a ccc forcing that does not add unsplit or dominating reals. We also consider some strong combinatorial properties of MAD families and show the relationships between them.The fifth chapter is one of the most important chapters in the thesis. A MAD family $\mathcal {A}$ is called $+$ -Ramsey if every tree that branches into $\mathcal {I}\left $ -positive sets has an $\mathcal {I}\left $ -positive branch. Michael Hrušák’s first published question is the following: Is there a $+$ -Ramsey MAD family? It was previously known that such families can consistently exist. However, there was no construction of such families using only the axioms of ZFC. We solve this problem by constructing such a family without any extra assumptions.In the fourth and fifth chapters, we introduce several notions of MAD families, in the sixth chapter we prove several implications and non implications between them. We construct several MAD families with different properties.In the seventh chapter we build models without P -points. We show that there are no P -points after adding Silver reals either iteratively or by the side by side product. These results have some important consequences: The first one is that is its possible to get rid of P -points using only definable forcings. This answers a question of Michael Hrušák. We can also use our results to build models with no P -points and with arbitrarily large continuum, which was also an open question. These results were obtained with David Chodounský.prepared by Osvaldo Guzmán GonzálezE-mail : [email protected] : https://arxiv.org/abs/1810.09680. (shrink)

A large and growing body of literature has investigated the role of teachers’ agency in their career trajectories. However, far too little attention has been paid to English as a Foreign Language teachers’, especially female EFL teachers’, professional agency for their career development in the Chinese higher education setting. To address this gap, this study explores female EFL teachers’ professional agency from a self-discrepancy theory perspective, namely, how the participating teachers have perceived discrepancies in their professional development and how (...) they have enacted their professional agency to realize sustainable development. Based on a metaphor investigation of 167 teachers and interviews with nine of them, the current study found that there are certain discrepancies between female EFL teachers’ self-guides and actual selves concerning their professional identity construction; female EFL teachers’ professional agency is manifested in the continuum of iteration, practical evaluation, and projectivity processes, as well as in the entity of personal and environmental factors; and 3) female EFL teachers’ professional agency and gender identity are closely intertwined with each other. This study can offer implications for teacher agency research and female teachers’ sustainable development at large. (shrink)

It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is (...) countable and that the continuum is a proper class. We examine several principles based on maximality considerations in this framework, and show how some (namely Ordinal Inner Model Hypotheses) enable us to incorporate standard set theories (including ZFC with large cardinals added). We conclude that the systems considered raise questions concerning the foundational purposes of set theory. (shrink)

We show that the existence of a ${\Pi^1_1}$ -definable mad family is consistent with the existence of a ${\Delta^{1}_{3}}$ -definable well-order of the reals and ${\mathfrak{b}=\mathfrak{c}=\aleph_3}$.

We determine the large cardinal consistency strength of the existence of a λ-supercompact cardinal κ such that equation image fails at λ. Indeed, we show that the existence of a λ-supercompact cardinal κ such that 2λ ≥ θ is equiconsistent with the existence of a λ-supercompact cardinal that is also θ-tall. We also prove some basic facts about the large cardinal notion of tallness with closure.

Ethical challenges that arise within healthcare delivery institutions are currently categorized as either clinical or organizational, based on the type of issue. Despite this common binary issue-based methodology, empirical study and increasing academic dialogue indicate that a clear line cannot easily be drawn between organizational and clinical ethics. Disagreement around end-of-life treatments, for example, often spawn value differences amongst parties at both organizational and clinical levels and requires a resolution to address both the case at hand and large-scale underlying (...) system-level confounders. I refer to issues that contain elements of both clinical and organizational issues as hybrids and propose a new taxonomy to characterize hybrid cases. I contend that salient contextual features of an ethical issue, such as where it is identified, who it impacts and where it is ideally resolved in relation to its scope of impact, should inform procedure. Implementation of a Hybrid taxonomy viewing ethical issues as existing on a continuum furthers that end. The primary goals are to 1) systematize thinking about ethical issues that arise within healthcare delivery institutions and 2) allow the content of the ethical challenge to drive the process, rather than continuing to rely on the traditional binary issue-based choice. Failure to capture the complexity of hybrid situations perpetuates incomplete information and ultimately an inchoate resolution that creates more questions than answers. (shrink)

Contemporary political philosophers disagree about whether theories of justice should be utopian or realistic. Contributors to this volume largely deny that the choice between realism and idealism is binary. Their contributions represent a continuum between realism and idealism that best represents the contemporary state of the debate.

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μ-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

Background Respect for autonomy is a key principle in bioethics. However, respecting autonomy in practice is complex because most people define themselves and make decisions influenced by a complex network of social relationships. The extent to which individual autonomy operates for each partner within the context of decision-making within marital or similar relationships is largely unexplored. This paper explores issues related to decision-making by couples for health care and the circumstances under which such a practice should be respected as compatible (...) with autonomous decision-making. Discussion We discuss the concept of autonomy as it applies to persons and to actions, human interdependency and gender roles in decision-making, the dynamics and outcomes of couples’ joint decision-making, and the ethics of couples’ joint decision-making. We believe that the extent to which couples’ joint decision-making might be deemed ethically acceptable will vary depending on the context. Given that in many traditional marriages the woman is the less dominant partner, we consider a spectrum of scenarios of couples’ joint decision-making about a woman’s own health care that move from those that are acceptably autonomous to those that are not consistent with respecting the woman’s autonomous decision-making. To the extent that there is evidence that both members of a couple understand a decision, intend it, and that neither completely controls the other, couples’ joint decision-making should be viewed as consistent with the principle of respect for the woman’s autonomy. At the other end of the spectrum are decisions made by the man without the woman’s input, representing domination of one partner by the other. Conclusions We recommend viewing the dynamics of couples’ joint decision-making as existing on a continuum of degrees of autonomy. This continuum-based perspective implies that couples’ joint decision-making should not be taken at face value but should be assessed against the specific cultural, ethnic, and religious backgrounds and personal circumstances of the individuals in question. (shrink)

Determinism, Freedom, and Moral Responsibility brings together nine substantial essays on determinism, freedom, and moral responsibility in antiquity by Susanne Bobzien. The essays present the main ancient theories on these subjects, ranging historically from Aristotle followed by the Epicureans, the early Stoics, several later Stoics, and up to Alexander of Aphrodisias in the third century CE. -/- The author discusses questions about rational and autonomous human agency and their compatibility with a large range of important philosophical issues, including their (...) compatibility with divine predetermination and other theological questions; with atomism and continuum theory and with the physical sciences more generally; with the determination of character and its development from childhood through nature and nurture; with epistemic features such as ignorance of circumstances; with theories of necessity, possibility and contingency; with external or internal preceding causes and impediments; and with folk theories of fatalism. Room is also given to the questions of how human autonomous agency is related to moral development, to virtue and wisdom, and to blame and praise. -/- Historically unified, philosophically profound, and methodologically rigorous, Bobzien’s essays show that in Classical and Hellenistic philosophy these topics were all debated without reference to freedom to do otherwise or to a free will, and that the latter two notions were fully developed only later. The volume will be of interest both to philosophers and to historians of philosophy, with more than half of the essays accessible to advanced undergraduates. (shrink)

The maximum entropy principle is widely used to determine non-committal probabilities on a finite domain, subject to a set of constraints, but its application to continuous domains is notoriously problematic. This paper concerns an intermediate case, where the domain is a first-order predicate language. Two strategies have been put forward for applying the maximum entropy principle on such a domain: applying it to finite sublanguages and taking the pointwise limit of the resulting probabilities as the size n of the sublanguage (...) increases; selecting a probability function on the language as a whole whose entropy on finite sublanguages of size n is not dominated by that of any other probability function for sufficiently large n. The entropy-limit conjecture says that, where these two approaches yield determinate probabilities, the two methods yield the same probabilities. If this conjecture is found to be true, it would provide a boost to the project of seeking a single canonical inductive logic—a project which faltered when Carnap's attempts in this direction succeeded only in determining a continuum of inductive methods. The truth of the conjecture would also boost the project of providing a canonical characterisation of normal or default models of first-order theories. Hitherto, the entropy-limit conjecture has been verified for languages which contain only unary predicate symbols and also for the case in which the constraints can be captured by a categorical statement of quantifier complexity. This paper shows that the entropy-limit conjecture also holds for categorical statements of complexity, for various non-categorical constraints, and in certain other general situations. (shrink)

We show that assuming modest large cardinals, there is a definable class of ordinals, closed and unbounded beneath every uncountable cardinal, so that for any closed and unbounded subclasses $P, Q, {\langle L[P],\in,P \rangle }$ and ${\langle L[Q],\in,Q \rangle }$ possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. Examples of such P are Card, the class of uncountable cardinals, I the uniform indiscernibles, or for any n the class $C^{n}{=_{{\operatorname {df}}}}\{ \lambda \, (...) | \, V_{\lambda } \prec _{{\Sigma }_{n}}V\}$ ; moreover the theory of such models is invariant under ZFC-preserving extensions. They also all have a rich structure satisfying many of the usual combinatorial principles and a definable wellorder of the reals. The inner model constructed using definability in the language augmented by the Härtig quantifier is thus also characterized. (shrink)

The current experiments examine mental health clinicians’ beliefs about biological, psychological, and environmental bases of the DSM‐IV‐TR mental disorders and the consequences of those causal beliefs for judging treatment effectiveness. Study 1 found a large negative correlation between clinicians’ beliefs about biological bases and environmental/psychological bases, suggesting that clinicians conceptualize mental disorders along a single continuum spanning from highly biological disorders (e.g., autistic disorder) to highly nonbiological disorders (e.g., adjustment disorders). Study 2 replicated this finding by having clinicians (...) list what they thought were the specific causes of nine familiar mental disorders and rate their bio–psycho–environmental bases. Study 3 further found that clinicians believe medication to be more effective for biologically based mental disorders and psychotherapy to be more effective for psychosocially based mental disorders. These results demonstrate that even expert mental health clinicians make strong distinctions between psychological and biological phenomena. (shrink)

Music impacting on speech processing is vividly evidenced in most reports involving professional musicians, while the question of whether the facilitative effects of music are limited to experts or may extend to amateurs remains to be resolved. Previous research has suggested that analogous to language experience, musicianship also modulates lexical tone perception but the influence of amateur musical experience in adulthood is poorly understood. Furthermore, little is known about how acoustic information and phonological information of lexical tones are processed by (...) amateur musicians. This study aimed to provide neural evidence of cortical plasticity by examining categorical perception of lexical tones in Chinese adults with amateur musical experience relative to the non-musician counterparts. Fifteen adult Chinese amateur musicians and an equal number of non-musicians participated in an event-related potential (ERP) experiment. Their mismatch negativities (MMNs) to lexical tones from Mandarin Tone 2–Tone 4 continuum and non-speech tone analogs were measured. It was hypothesized that amateur musicians would exhibit different MMNs to their non-musician counterparts in processing two aspects of information in lexical tones. Results showed that the MMN mean amplitude evoked by within-category deviants was significantly larger for amateur musicians than non-musicians regardless of speech or non-speech condition. This implies the strengthened processing of acoustic information by adult amateur musicians without the need of focused attention, as the detection of subtle acoustic nuances of pitch was measurably improved. In addition, the MMN peak latency elicited by across-category deviants was significantly shorter than that by within-category deviants for both groups, indicative of the earlier processing of phonological information than acoustic information of lexical tones at the pre-attentive stage. The results mentioned above suggest that cortical plasticity can still be induced in adulthood, hence non-musicians should be defined more strictly than before. Besides, the current study enlarges the population demonstrating the beneficial effects of musical experience on perceptual and cognitive functions, namely, the effects of enhanced speech processing from music are not confined to a small group of experts but extend to a large population of amateurs. (shrink)

Open-source communities that focus on content rely squarely on the contributions of invisible strangers in cyberspace. How do such communities handle the problem of trusting that strangers have good intentions and adequate competence? This question is explored in relation to communities in which such trust is a vital issue: peer production of software (FreeBSD and Mozilla in particular) and encyclopaedia entries (Wikipedia in particular). In the context of open-source software, it is argued that trust was inferred from an underlying ‘hacker (...) ethic’, which already existed. The Wikipedian project, by contrast, had to create an appropriate ethic along the way. In the interim, the assumption simply had to be that potential contributors were trustworthy; they were granted ‘substantial trust’. Subsequently, projects from both communities introduced rules and regulations which partly substituted for the need to perceive contributors as trustworthy. They faced a design choice in the continuum between a high-discretion design (granting a large amount of trust to contributors) and a low-discretion design (leaving only a small amount of trust to contributors). It is found that open-source designs for software and encyclopaedias are likely to converge in the future towards a mid-level of discretion. In such a design the anonymous user is no longer invested with unquestioning trust. (shrink)

We address three fundamental questions: What does it mean for an entity to be living? What is the role of inter-organismic collaboration in evolution? What is a biological individual? Our central argument is that life arises when lineage-forming entities collaborate in metabolism. By conceiving of metabolism as a collaborative process performed by functional wholes, which are associations of a variety of lineage-forming entities, we avoid the standard tension between reproduction and metabolism in discussions of life – a tension particularly evident (...) in discussions of whether viruses are alive. Our perspective assumes no sharp distinction between life and non-life, and does not equate life exclusively with cellular or organismal status. We reach this conclusion through an analysis of the capabilities of a spectrum of biological entities, in which we include the pivotal case of viruses as well as prions, plasmids, organelles, intracellular and extracellular symbionts, unicellular and multicellular life-forms. The usual criterion for classifying many of the entities of our continuum as non-living is autonomy. This emphasis on autonomy is problematic, however, because even paradigmatic biological individuals, such as large animals, are dependent on symbiotic associations with many other organisms. These composite individuals constitute the metabolic wholes on which selection acts. Finally, our account treats cooperation and competition not as polar opposites but as points on a continuum of collaboration. We suggest that competitive relations are a transitional state, with multi-lineage metabolic wholes eventually outcompeting selfish competitors, and that this process sometimes leads to the emergence of new types or levels of wholes. Our view of life as a continuum of variably structured collaborative systems leaves open the possibility that a variety of forms of organized matter – from chemical systems to ecosystems – might be usefully understood as living entities. (shrink)

Respect for autonomy is a key principle in bioethics. However, respecting autonomy in practice is complex because most people define themselves and make decisions influenced by a complex network of social relationships. The extent to which individual autonomy operates for each partner within the context of decision-making within marital or similar relationships is largely unexplored. This paper explores issues related to decision-making by couples for health care and the circumstances under which such a practice should be respected as compatible with (...) autonomous decision-making. We discuss the concept of autonomy as it applies to persons and to actions, human interdependency and gender roles in decision-making, the dynamics and outcomes of couples’ joint decision-making, and the ethics of couples’ joint decision-making. We believe that the extent to which couples’ joint decision-making might be deemed ethically acceptable will vary depending on the context. Given that in many traditional marriages the woman is the less dominant partner, we consider a spectrum of scenarios of couples’ joint decision-making about a woman’s own health care that move from those that are acceptably autonomous to those that are not consistent with respecting the woman’s autonomous decision-making. To the extent that there is evidence that both members of a couple understand a decision, intend it, and that neither completely controls the other, couples’ joint decision-making should be viewed as consistent with the principle of respect for the woman’s autonomy. At the other end of the spectrum are decisions made by the man without the woman’s input, representing domination of one partner by the other. We recommend viewing the dynamics of couples’ joint decision-making as existing on a continuum of degrees of autonomy. This continuum-based perspective implies that couples’ joint decision-making should not be taken at face value but should be assessed against the specific cultural, ethnic, and religious backgrounds and personal circumstances of the individuals in question. (shrink)

In the quest and search for a physical theory of everything from the macroscopic large body matter to the microscopic elementary particles, with strange and weird concepts springing from quantum physics discovery, irreconcilable positions and inconvenient facts complicated physics – from Newtonian physics to quantum science, the question is- how do we close the gap? Indeed, there is a scientific and mathematical fireworks when the issue of quantum uncertainties and entanglements cannot be explained with classical physics. The Copenhagen interpretation (...) is an expression of few wise men on quantum physics that was largely formulated from 1925 to 1927 namely by Niels Bohr and Werner Heisenberg. From this point on, there is a divergence of quantum science into the realms of indeterminacy, complementarity and entanglement which are principles expounded in Yijing, an ancient Chinese knowledge constructed on symbols, with a vintage of at least 3 millennia, with broken and unbroken lines to form stacked 6-line structure called the hexagram. It is premised on probability development of the hexagram in a space-time continuum. The discovery of the quantization of action meant that quantum physics could not convincingly explain the principles of classical physics. This paper will draw the great departure from classical physics into the realm of probabilistic realities. The probabilistic nature and reality interpretation had a significant influence on Bohr’s line of thought. Apparently, Bohr realized that speaking of disturbance seemed to indicate that atomic objects were classical particles with definite inherent kinematic and dynamic properties (Hanson, 1959). Disturbances, energy excitation and entanglements are processual evolutionary phases in Yijing. This paper will explore the similarities in quantum physics and the methodological ways where Yijing is pivoted to interpret observable realities involving interactions which are uncontrollable and probabilistic and forms an inseparable unity due to the entanglement, superposition Transgressing disciplinary boundaries in the discussion of Yijing, originally from the Western Zhou period (1000-750 BC), over a period of warring states and the early imperial period (500-200 BC) which was compiled, transcribed and transformed into a cosmological texts with philosophical commentaries known as the “Ten Wings” and closely associated with Confucius (551- 479 BC) with the Copenhagen Interpretation (1925-1927) by the few wise men including Niel Bohr and Werner Heisensberg would seem like a subversive undertaking. Subversive as the interpretations from Yijing is based on wisdom derived from thousands of years from ancient China to recently discovered quantum concepts. The subversive undertaking does seem to violate the sanctuaries of accepted ways in looking at Yijing principles, classical physics and quantum science because of the fortified boundaries that have been erected between Yijing and the sciences. Subversive as this paper may be, it is an attempt to re-cast an ancient framework where indeterminism, complementarity, non-linearity entanglement, superposition and probability interpretation is seen in today quantum’s realities. (shrink)