Under the assumption that δ is a Woodin cardinal and GCH holds, I show that if F is any class function from the regular cardinals to the cardinals such that (1) ${\kappa < {\rm cf}(F(\kappa))}$ , (2) ${\kappa < \lambda}$ implies ${F(\kappa) \leq F(\lambda)}$ , and (3) δ is closed under F, then there is a cofinality-preserving forcing extension in which 2 γ = F(γ) for each regular cardinal γ < δ, and in which δ remains Woodin. Unlike the analogous (...) results for supercompact cardinals [Menas in Trans Am Math Soc 223:61–91, (1976)] and strong cardinals [Friedman and Honzik in Ann Pure Appl Logic 154(3):191–208, (2008)], there is no requirement that the function F be locally definable. I deduce a global version of the above result: Assuming GCH, if F is a function satisfying (1) and (2) above, and C is a class of Woodin cardinals, each of which is closed under F, then there is a cofinality-preserving forcing extension in which 2 γ = F(γ) for all regular cardinals γ and each cardinal in C remains Woodin. (shrink)

An Easton function is a monotone function C from infinite regular cardinals to cardinals such that C has cofinality greater than α for each infinite regular cardinal α. Easton showed that assuming GCH, if C is a definable Easton function then in some cofinality-preserving extension, C=2α for all infinite regular cardinals α. Using “generic modification”, we show that over the ground model L, models witnessing Easton’s theorem can be obtained as inner models of L[0#], for Easton functions which (...) are L-definable with parameters at most . And using a gap 1 morass, we obtain an inner model of L[0#] with the same cofinalities as L in which is a strong limit cardinal and equals. (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)

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)

I propose to depict the relationship between Badiou’s philosophy and mathematics as a three-layered model. Philosophy as metaontology creates a metastructure, mathematics as ontology in the form of a condition of philosophy constitutes its situation, and mathematics as a multiple universe of all given axioms, theorems, techniques, interpretations, and systems (set theory, category theory, etc.) is an inconsistent multiplicity. So, we can interpret the relationship between philosophy and mathematics as the one between a metastructure and a situation. By using (...) class='Hi'>Easton’s theorem, we come to realise that philosophical concepts in the metastructure “quantitatively” exceed the elements that belong to mathematics as ontology. Therefore, philosophy as metaontology shows the limits of mathematics as ontology. (shrink)

We reprove Gitik's theorem that if the GCH holds and o(κ) = κ + 1 then there is a generic extension in which κ is still measurable and there is a closed unbounded subset C of κ such that every $\nu \in C$ is inaccessible in the ground model. Unlike the forcing used by Gitik. the iterated forcing $R_{\lambda +1}$ used in this paper has the property that if λ is a cardinal less then κ then $R_{\lambda + 1}$ (...) can be factored in V as $R_{\kappa + 1} = R_{\lambda + 1} \times R_{\lambda + 1, \kappa}$ where $\mid R_{\lambda +1}\mid \leq \lambda^+$ and $R_{\lambda + 1, \kappa}$ does not add any new subsets of λ. (shrink)

There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate this by examining Easton's strengthening (...) of Cohen's result:Theorem 1. There is a forcing extensionL[G] of L in which GCH fails at every regular cardinal.Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L[G] and V? In particular, are these models compatible, in the sense that they are inner models of a common third model? If not, then the failure of GCH at every regular cardinal is consistent only in a weak sense, as it can only hold in universes which are incompatible with the universe of all sets. Ideally, we would like L[G] to not only be compatible with V, but to be an inner model of V.We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals. (shrink)

An important technique in large cardinal set theory is that of extending an elementary embedding j: M → N between inner models to an elementary embedding j*: M[G] → N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the (...) generic G* is simply generated by the image of G. But in difficult cases, such as in Woodin's proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin's theorem as well as the new result that global domination at inaccessibles (the statement that d (κ) is less than 2κ for inaccessible κ, where d (κ) is the dominating number at κ) is internally consistent, given the existence of 0#. (shrink)

We argue that Löb's Theorem implies a limitation on mechanism. Specifically, we argue, via an application of a generalized version of Löb's Theorem, that any particular device known by an observer to be mechanical cannot be used as an epistemic authority (of a particular type) by that observer: either the belief-set of such an authority is not mechanizable or, if it is, there is no identifiable formal system of which the observer can know (or truly believe) it to (...) be the theorem-set. This gives, we believe, an important and hitherto unnoticed connection between mechanism and the use of authorities by human-like epistemic agents. (shrink)

Modality specificity in priming is taken as evidence for independent perceptual systems. However, Easton, Greene, and Srinivas (1997) showed that visual and haptic cross-modal priming is comparable in magnitude to within-modal priming. Where appropriate, perceptual systems might share like information. To test this, we assessed priming and recognition for visual and auditory events, within- and across- modalities. On the visual test, auditory study resulted in no priming. On the auditory priming test, visual study resulted in priming that was only marginally (...) less than within-modal priming. The priming results show that visual study facilitates identification on both visual and auditory tests, but auditory study only facilitates performance on the auditory test. For both recognition tests, within-modal recognition exceeded cross-modal recognition. The results have two novel implications for the understanding of perceptual priming: First, we introduce visual and auditory priming for spatio-temporal events as a new priming paradigm chosen for its ecological validity and potential for information exchange. Second, we propose that the asymmetry of the cross-modal priming observed here may reflect the capacity of these perceptual modalities to provide cross-modal constraints on ambiguity. We argue that visual perception might inform and constrain auditory processing, while auditory perception corresponds to too many potential visual events to usefully inform and constrain visual perception. (shrink)

Bell’s theorem admits several interpretations or ‘solutions’, the standard interpretation being ‘indeterminism’, a next one ‘nonlocality’. In this article two further solutions are investigated, termed here ‘superdeterminism’ and ‘supercorrelation’. The former is especially interesting for philosophical reasons, if only because it is always rejected on the basis of extra-physical arguments. The latter, supercorrelation, will be studied here by investigating model systems that can mimic it, namely spin lattices. It is shown that in these systems the Bell inequality can be (...) violated, even if they are local according to usual definitions. Violation of the Bell inequality is retraced to violation of ‘measurement independence’. These results emphasize the importance of studying the premises of the Bell inequality in realistic systems. (shrink)

Bell's theorem is expounded as an analysis in Bayesian inference. Assuming the result of a spin measurement on a particle is governed by a causal variable internal (hidden, “local”) to the particle, one learns about it by making a spin measurement; thence about the internal variable of a second particle correlated with the first; and from there predicts the probabilistic result of spin measurements on the second particle. Such predictions are violated by experiment: locality/causality fails. The statistical nature of (...) the observations rules out signalling; acausal, superluminal, or otherwise. Quantum mechanics is irrelevant to this reasoning, although its correct predictions of experiment imply that it has a nonlocal/acausal interpretation. Cramer's newtransactional interpretation, which incorporates this feature by adapting the Wheeler-Feynman idea of advanced and retarded processes to the quantum laws, is advocated. It leads to an invaluable way of envisaging quantum processes. The usual paradoxes melt before this, and one, the “delayed choice” experiment, is chosen for detailed inspection. Nonlocality implies practical difficulties in influencing hidden variables, which provides a very plausible explanation for why they have not yet been found; from this standpoint, Bell's theoremreinforces arguments in favor of hidden variables. (shrink)

The most popular English arithmetic of the 16th century was Robert Recorde’s Ground of Artes. Of the many surviving editions (15 before 1600), only two were published during Recorde’s lifetime, and a third was published shortly after his death. The first two differ materially from the latter, which was the basis of the long series of „augmented“ or „revised“ editions which appeared from 1561 to 1699. Because of their greater accessibility, discussions of 16th-century arithmetics usually cite these later editions, sometimes (...) tacitly ascribing material to the early editions which does not appear before 1558. (shrink)

We prove the following version of Hechler's classical theorem: For each partially ordered set (Q, ≤) with the property that every countable subset of Q has a strict upper bound in Q, there is a ccc forcing notion such that in the generic extension for each tall analytic P-ideal J (coded in the ground model) a cofinal subset of (J, ⊆*) is order isomorphic to (Q, ≤).

We prove that Voiculescu’s noncommutative version of the Weyl-von Neumann Theorem can be extended to all unital, separably representable $\mathrm {C}^\ast $ -algebras whose density character is strictly smaller than the cardinal invariant $\mathfrak {p}$. We show moreover that Voiculescu’s Theorem consistently fails for $\mathrm {C}^\ast $ -algebras of larger density character.

We all want to find a way to live fulfilling lives together in spite of our differences. What beliefs must be held in common if we are to do so? What beliefs must be excluded? And what are the implications for religion’s place in society? Philip Kitcher recognises that our answers to these questions have important implications for education, and he devotes a chapter of 'The Main Enterprise of the World' to the role of religion in education and wider society. (...) This paper is a critical response to that chapter. I argue that Kitcher is right to recognise that religious beliefs cannot be confined to the private sphere and thus they are a matter of educational concern. However, I question whether Kitcher makes an adequate case for the distinctive harmfulness of the beliefs that he wishes to exclude. I also question Kitcher’s suggestion that students be taught that all religions are ‘on a par’, epistemically speaking. I finish by shining a light on Kitcher’s proposal that education should aim for students to develop specific beliefs about religion, clarifying the proposal and highlighting some implications for education in practice. (shrink)

Bell's theorem is expounded as an analysis in Bayesian probabilistic inference. Assume that the result of a spin measurement on a spin-1/2 particle is governed by a variable internal to the particle (local, “hidden”), and examine pairs of particles having zero combined angular momentum so that their internal variables are correlated: knowing something about the internal variable of one tells us something about that of the other. By measuring the spin of one particle, we infer something about its internal (...) variable; through the correlation, about the internal variable of the second particle, which may be arbitrarily distant and is by hypothesis unchanged by this measurement (locality); and make (probabilistic) prediction of spin observations on the second particle. Each link in this chain has a counterpart in the Bayesian analysis of the situation. Irrespective of the details of the internal variable description, such prediction is violated by measurements on many particle pairs, so that locality—effectively the only physics invoked—fails. The time ordering of the two measurements is not Lorentz-invariant, implying acausality. Quantum mechanics is irrelevant to this reasoning, although its correct predictions of the statistics of the results imply it has a nonlocal—acausal interpretation; one such, the “transactional” interpretation, is presented to demonstrable advantage, and some misconceptions about quantum theory are pursued. The “unobservability” loophole in photonic Bell experiments is proven to be closed. It is shown that this mechanism cannot be used for signalling; signalling would become possible only if the hidden variables, which we insist must underlie the statistical character of the observations (the alternative is to give up), are uncovered in deviations from quantum predictions. Their reticence is understood as a consequence of their nonlocality: it is not easy to isolate and measure something nonlocal. Once the hidden variables are found, all the problems of quantum field theory and of quantum gravity might melt away. (shrink)

What should be the aim of LGBT-inclusive, state-mandated curricula in liberal, pluralist societies? In this article, I identify two distinct aims that such curricula might have. The first, LGBT Respect, aims to teach that LGBT individuals have equal political status and rights. The second, LGBT Approval, aims to teach a positive attitude towards LGBT relationships, including that there is nothing wrongful about these forms of relationship. I examine what arguments in favour of these different aims are available to the liberal (...) concerned with accommodating pluralism. To capture this concern, I adopt political liberalism's strict standard for legitimate political interventions that these must have justifications that all reasonable citizens can accept. This initially seems to recommend curricula that aim at LGBT Respect but stop short of LGBT Approval. Can the political liberal go any further? I propose and critically discuss the most promising arguments in favour of LGBT Approval, including the need to prevent harm to children, ensure political equality, and secure the social conditions needed for the development of the primary good of self-respect. I tentatively conclude that there exists a cumulative case for state-mandated curricula aiming at LGBT Approval, at least in the contingent, nonideal circumstances that currently obtain. (shrink)

This paper considers feministperspectives on the Human Rights Act. Itdiscusses the reasons why many feminists aresceptical regarding the impact the Act willhave on women''s lives, including theimplications for anti-discrimination law,problems with the framework of rights in theEuropean Convention and deeper difficulties facingfeminism in negotiating rights discourse. Whileacknowledging these problems, it is argued thatthere are grounds for a more positiveinterpretation of incorporation. Questions arethen raised about the nature and scope of rightsand the role of the state in challenging genderinequality.

I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument (...) by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. I also identify similar oversight in the GHZ variant of Bell's theorem, invalidating its claim of having found an inconsistency in the premisses of the argument by EPR for completing quantum mechanics. Conceptually, the oversight in both Bell's theorem and its GHZ variant traces back to the oversight in von Neumann's theorem against hidden variable theories identified by Grete Hermann in the 1930s. (shrink)

The period in the history of blood transfusion that I discuss is roughly 1628, the date of publication of Harvey’s work on blood circulation, De Motu Cordis, and 1668, the year of the first allegedly successful transfusion of blood into a human subject by a French physician Jean Denis, and the official order to prohibit the procedure. The subject of special interest in this history is Robert Desgabets, an early defender and teacher of the Cartesian philosophy at St. Maur, in (...) the region of Lorraine, France. Desgabets’ Discourse de la communication ou transfusion du sang Communication or Transfusion of Blood contains a defence and description of the procedure. This three page manuscript is a lecture delivered at one of the meetings held at M. de Montmor in July 1658. Letters and documents which describe the events of these conferences during Desgabets’ eight month stay in Paris in 1658, indicate that he had numerous discussions with the Cartesian physicist Jacques Rohault and also with another leading Cartesian scientist, Gerauld Cordemoy. In addition, Desgabets is said to have frequently attended the meetings where he showed his interest in many scientific questions. I provide evidence that Desgabets was not an arm-chair scientist: he performed experiments, and he designed and created an apparatus to carry out the procedure of blood transfusion. I show how Desgabets’s Cartesian conception of mechanism and body had theological as well as epistemological underpinnings and consequences. For Desgabets, experiments on blood transfusion were as much demonstrations of the truth of the Cartesian metaphysics as they were demonstrations of the possibility of transubstantiation. The role of experiment was to provide demonstrations of the first principles or truths of physics and theology by connecting them to the way God actually made the world. The upshot of Desgabets's treatment of expérience is that it is integral to the knowledge of essences, and hence to the discovery of the truth of things. In placing expérience at the very foundation of true knowledge, Desgabets reconstructs Cartesianism on an empiricist foundation. (shrink)

We rediscuss the Einstein-Podolsky-Rosen paradox in Bohm's spin version and oppose to it Bohr's controversial point of view. Then we explain Bell's theorem, Bell inequalities, and its consequences. We describe the experiment of Aspect, Dalibard, and Roger in detail. Finally we draw attention to the nonlocal structure of the underlying theory.

We show how to use diagrams in order to obtain straightforward completeness theorems for extensions of K4.3 and a very simple and constructive proof of Bull's theorem: every normal extension of S4.3 has the finite model property.

This paper proposes a solution to the problem of non-locality associated with Bell’s theorem, within the counterfactual approach to the problem. Our proposal is that a counterfactual definition of locality can be maintained, if a subsidiary hypothesis be rejected, “locality involving two counterfactuals”. This amounts to the acceptance of locality in the actual world, and a denial that locality is always valid in counterfactual worlds. This also introduces a metaphysical asymmetry between the factual and counterfactual worlds. This distinction is (...) analogous to what occurs in the derivations of Bell’s theorem which assume hidden-variables, where macroscopic locality can be maintained at the price of rejecting outcome independence. This can be interpreted as non-locality at the level of potentialities, which might be identified with the non-locality of counterfactual worlds. Our solution, presented for the CHSH inequality, is falsifiable, and we test it with two other setups, Bell’s original inequality and the EPR thought-experiment. (shrink)

We study the formal first order system TIND in the standard language of Gentzen's LK . TIND extends LK by the purely logical rule of term-induction, that is a restricted induction principle, deriving numerals instead of arbitrary terms. This rule may be conceived as the logical image of full induction.

G¨odel’s a theorem concerns an arithmetical statement and the truth of this statement does not depend on self-reference; nevertheless its interpretation is of tremendous interest. G¨odel’s theorem allows one to conclude that formal arithmetic is not axiomatizable. But there is another very interesting logico-philosophical result: the possibility of a statement to exist such that it is improvable in the object-theory and at the same time its truth is provable in the metatheory. It seems that in the real history (...) G¨odel’s theorem was absolutely unexpected, and even in retrospect, now it is difficult to point out logico-philosophical reasons for the above possibility. In my opinion though it was still conceivable at that time, but unfortunately the philosophers and logicians missed to notice it. In the terms of recursive arithmetic Hilbert’s programme can be formulated as follows: Build a decidable, complete, consistent logical system such that the class of derivable statements coincides with the class of the mathematical statements which are intuitively true; prove this equivalence by means of an arithmetization of the theory in the framework of the recursive arithmetic! Now, if the system is so strong as to admit the construction of the classical mathematics, then in particular it will contain the classical arithmetic. But Hilbert’s requirements reduce to a proof with the means of the recursive arithmetic, by arithmetization. So, on one hand the system will contain the arithmetic and, on the other hand it will be expressible in it, in order words the system will include its own syntax, as well as its own semantics. Now it is clear that Hilbert’s programme contains a possibility of a situation similar to some well known semantic antinomies, situations of self-reference. (shrink)

Recent re-evaluation of the question of the exact role of experience in the Cartesian philosophy has emerged from many quarters. The metaphysical issue of innate ideas has been raised by such scholars as McRae and Miles, and a close examination of the role of empirical enquiry and methodology in Cartesian science have been undertaken by Clarke, Garber, Buchdahl and Laudan, to mention only a few. These recent reappraisals of the role of experience in Descartes's philosophy have been cast mostly in (...) twentieth-century terms with a specific view to concerns of the present. ;My research centres on an examination of the philosophical works of three Cartesians, Antoine Le Grand, Robert Desgabets, and Pierre-Sylvain Regis. Though they are relatively minor figures in the history of ideas, their defence of Descartes's thesis concerning the free creation of the eternal truths, and their emphasis on the importance of experience, provide a unique opportunity to reevaluate the role of experience in the Cartesian philosophy in terms that are common in the period. ;There was a strongly empirical component in the philosophical doctrines and practices of these seventeenth-century Cartesians. Le Grand's view of truth as created and dependent on the will of God led him to assign experiment and experience the role of "making manifest" the operation of laws in the physical universe. Desgabets rejected innate ideas, and argued that all ideas depend on the operation of the senses. Regis emphasized the immutability and total dependence of truths and essences on the free will of God, and thereby undermined the rationalist foundation for the Cartesian doctrine of essences. ;My goal in this approach to the question of the nature and role of experience in the Cartesian philosophy is to explore the philosophical basis of the empiricist tendencies found in Descartes as developed by three of his disciples. My hope is that my study of Le Grand, Regis, and Desgabets, who are relatively unknown today, will not only serve to place them appropriately within the history of ideas, but will dislodge the common misconception of Cartesianism as having been an inherently rationalist philosophy. (shrink)

The paper considers the claim that quantum theories with a deterministic dynamics of objects in ordinary space-time, such as Bohmian mechanics, contradict the assumption that the measurement settings can be freely chosen in the EPR experiment. That assumption is one of the premises of Bell’s theorem. I first argue that only a premise to the effect that what determines the choice of the measurement settings is independent of what determines the past state of the measured system is needed for (...) the derivation of Bell’s theorem. Determinism as such does not undermine that independence . Only entanglement could do so. However, generic entanglement without collapse on the level of the universal wave-function can go together with effective wave-functions for subsystems of the universe, as in Bohmian mechanics. The paper argues that such effective wave-functions are sufficient for the mentioned independence premise to hold. (shrink)

From the foregoing discussion we can note that whilst Marx transcends the fact-value distinction he embraces neither a scientistic approach nor a moral theory. Rather he gives a sociological account of morality, illustrating that description and evaluation cannot be separated and that juridical conceptions need to be understood in relation to the mode of production in which they arise.30 In the absence of an absolute notion of justice it is mistaken to see Marx as offering a critique of capitalism based (...) on moral principles. Of course Marx had reasons for attacking capitalism but these are contained within his account of the capitalist mode of production. Yet whilst this theory is not a moral theory it cannot be described as a descriptive theory either. In Marx's work description and evaluation cannot be meaningfully separated.In portraying Communist society as the solution to problems generated within capitalist society Marx is not sketching out a picture of a morally superior society but rather considering the possibilities of an alternative way of life with its own moral and judicial standards. Indeed the fact that Marx did not provide blueprints for future Communist society is itself symptomatic of his awareness of the difficulties involved in the attempt to describe in detail a form of social life based upon principles different from our own. (shrink)

In this short survey article, I discuss Bell’s theorem and some strategies that attempt to avoid the conclusion of non-locality. I focus on two that intersect with the philosophy of probability: (1) quantum probabilities and (2) superdeterminism. The issues they raised not only apply to a wide class of no-go theorems about quantum mechanics but are also of general philosophical interest.