The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form Vκ where κ is a strongly inaccessible cardinal and Vκ denotes the κth level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depend on the context in (...) which they are embedded. The context here will be the theory CZF− of constructive Zermelo–Fraenkel set theory but without -Induction . Let INAC be the statement that for every set there is an inaccessible set containing it. CZF−+INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF−+INAC also has a realizability interpretation in type theory which gives its theorems a direct computational meaning. The main result presented here is that the proof theoretic ordinal of CZF−+INAC is a small ordinal known as the Feferman–Schütte ordinal Γ0. (shrink)

This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory and Martin-Löf's intuitionistic theory of types. This paper treats Mahlo's π-numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non-datur, is equivalent to ZF plus (...) the assertion that there are inaccessibles of all transfinite orders. Finally, the theorems of that extension of CZF are interpreted in an extension of Martin-Löf's intuitionistic theory of types by a universe. (shrink)

We construct models for the level-by-level equivalence between strong compactness and supercompactness containing failures of the Generalized Continuum Hypothesis at inaccessible cardinals. In one of these models, no cardinal is supercompact up to an inaccessible cardinal, and for every inaccessible cardinal $\delta $, $2^{\delta }\gt \delta ^{++}$. In another of these models, no cardinal is supercompact up to an inaccessible cardinal, and the only inaccessible cardinals at which GCH holds are also measurable. These results extend and generalize earlier work of (...) the author. (shrink)

In liberal democracies it is now a commonplace that public debates in the institutionalized political sphere should involve only arguments and reasons that are in principle intelligible, accessible and acceptable to all citizens. Many political theorists take the view that religious arguments and reasons do not meet these requirements. My article interrogates this widely held position, considering each of the three requirements in turn. Motivating my discussion is the view that religious beliefs and practices should not be regarded as essentially (...) private matters, with discussion of their validity confined to some antecedently demarcated sphere. Rather, claims made for the validity of religious beliefs and practices should be thematized and evaluated in public processes of deliberation, opening them to possible challenges from other citizens, irrespective of whether these other citizens are religious believers. My article offers a freedom-based argument for this position. (shrink)

The religious inquirer is, however, in a tough spot, for she is subject to norms that it appears she cannot jointly satisfy. On the one hand, there are norms for the conduct of one's doxastic life, which do not emanate from any particular religious tradition, that enjoin us to be conscientious in our believings. In her view, conforming to these norms does not license having religious beliefs: there are simply too many evidential impediments to having such beliefs, ranging from deep (...) theoretical challenges to the fundamental religious claims to questions about the reliability of the fundamental religious texts and traditions. Call these norms, which the religious inquirer holds that religious belief does not satisfy, external epistemic norms. On the other hand, there are norms that emanate from within the religious traditions themselves that speak against engaging in a religious way of life in order to enjoy its considerable goods by doing such things as playing the role of a believer or going through the ritualistic motions. Instead, these norms seem to call for genuine faith, which is understood to be something approximating what the true believer has. Call these norms, which place restrictions on how one could legitimately engage in a religious way of life, the internal religious norms. (shrink)

Throughout history philosophers have relentlessly pursued what may be called "inaccessible domains." This book explores how the traditions of existential phenomenology relate to Freudian psychoanalysis. A clear, succinct, and systematic account of the philosophical presuppositions of psychoanalytic theory and practice, this work offers a deeper and richer understanding and appreciation of Freudian thought, as well as its antecedents and influences. With its unique perspective on Freud's work, Apprehending the Inaccessible puts readers in a better position to appreciate his contributions and (...) evaluate the relationship between his and other philosophical world views. The authors, both of whom have extensive backgrounds in philosophy and psychology, present balanced critical analyses of crucial developments in, for example, the evolution of the Freudian notion of the unconscious, and the engagement of existential phenomenology with Freudian psychoanalysis. Askay and Farquhar then consider-often for the first time-individual thinkers' reflections on and interpretations of Freud, ranging from the primary figures in existential phenomenology to the most prominent figures in the existential psychoanalytic movement. Even as their work offers a new approach to Freudian thought, it reasserts the importance of alternative views found in existential phenomenology as those views pertain to psychoanalysis and the question of apprehending the inaccessible. (shrink)

Suppose L = { <,...} is any countable first order language in which < is interpreted as a linear order. Let T be any complete first order theory in the language L such that T has a κ-like model where κ is an inaccessible cardinal. Such T proves the Inaccessibility Scheme. In this paper we study elementary end extensions of models of the inaccessibility scheme.

In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of 2κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^\kappa $$\end{document}, κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} inaccessible, and study its associated ideal of null sets and notion of measurability. This issue was addressed by Shelah ), arXiv:0904.0817, Problem 0.5) and concerns the definition of a forcing which is κκ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa ^\kappa $$\end{document}-bounding, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,>\,$$\end{document}cov_λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\_\lambda $$\end{document}), arXiv:0904.0817, Problem 0.5), and in this paper we independently reprove this result by using a different type of construction. This also contributes to a line of research adressed in the survey paper :439–456, 2016). (shrink)

Argues, contra Dummett, that the platonist need not be any more committed than the intuitionist to the notion that there are arithmetical truths in principle inaccessible to any finite intelligence.

The claim that language is in principle inaccessible to consciousness may look counterintuitive but is not as challenging as finding an answer to the subsequent question of why that must be the case -- if language is a function that is in the service of consciousness and we cannot imagine why language would have existed at all without the existence of consciousness. On the one hand, language is the cognitive capacity that seems best fit to support consciousness in its monitoring (...) and control functions; on the other hand, language learning (learning the rules of one's own language), language structure and language processing turn out upon closer scrutiny to be in principle inaccessible to consciousness. I present a set of arguments in favour of the thesis that language is in principle inaccessible to consciousness on the basis of a set of asymmetries between sentence structure and the structure of consciousness. If the thesis in question is on the right track, we have to face two basic problems. The first deals with linguistics method(s), namely how can we study a very complex mental phenomenon like language if it is not available to introspection? The second problem is related to the question put in the title of this article. The suggested answer is along the lines that inaccessibility of language to consciousness enables a cognitive architecture that can run a Cartesian theatre. (shrink)

Abstract:In this article, Kundakunda's theory of omniscience is defended using formal principles derived from set theory. More precisely, analogous features in the work of Jain mystic Kundakunda and the German mathematician Georg Cantor are described, demonstrating that both thinkers demanded an independently existent, transcendental Absolute to render consistent their own systems of thought. Both of their projects entailed resolving the formal quandary of inaccessibility, or the inability for any sequential, determinate objectifications to ever mereologically sum up to a genuine (...) 'Absolute'—that is, a 'gestalt' that is 'more than the sum of its parts'. (shrink)

We prove optimal lower bounds on the computation time for several well-known test problems on a quite realistic computational model: the random access machine. These lower bound arguments may be of special interest for logicians because they rely on finitary analogues of two important concepts from mathematical logic: inaccessible numbers and order indiscernibles.

Dans cet article, on utilise la théorie des dilatateurs pour décrire les premiers récursivement inaccessible et Mahlo.

The usual problems of measurement and its meaning are complicated and magnified when the object of study is in principle and in fact inaccessible. When a phenomenon occurs in a place where our instruments cannot reach, what can the relation between the instrument, its reading, and the phenomenon be? This essay asks how researchers have addressed questions about inaccessible processes of Earth's magnetic field on the surface, at the edge of space and under its surface. This case takes us beyond (...) simple contrasts of inductive and deductive methods, or field sciences versus basic sciences, to a messier, more nuanced view of inherently complex sciences. The article focuses on investigations of Earth's magnetism and electricity, but its perspectives are applicable to many areas of recent science. (shrink)

This paper offers a critical phenomenological view of the concept of access intimacy, a term coined by disability justice advocate Mia Mingus. Access intimacy refers to a mode of relation between disabled people or between disabled and non-disabled people that can be born of concerted cultivation or instantly intimated and centrally concerns the feeling of someone genuinely understanding and anticipating another’s access needs. Putting in conversation this notion of intimacy with Kym Maclaren’s critical phenomenological account of intimacy, I show how (...) accessibility is not about what one person or institution can do for another but involves an ongoing, interpersonal process of relating and taking responsibility for our inevitable encroachment on one another in ways that enhance one another’s freedom. (shrink)

We show that a coding principle introduced by J. Moore with respect to all ladder systems is equiconsistent with the existence of a strongly inaccessible cardinal. We also show that a coding principle introduced by S. Todorcevic has consistency strength at least of a strongly inaccessible cardinal.

The distinction between uninformed learning (type-1) and learning based on recoding using prior information (type-2) helps to clarify some long-standing psychological problems, including misunderstanding of mathematics by children, the need for active construction of concepts in cognitive development, and the difficulty of configural learning tasks. However, an alternative to recoding some type-2 tasks is to represent the input as separate dimensions, which are processed jointly. This preserves the original structure, but is subject to processing capacity limitations.

In this paper we propose that cosmological fine-tuning arguments, when levied in support of the existence of Intelligent Designers or Multiverses, are much less interesting than they are thought to be. Our skepticism results from tracking the distinction between merely epistemic or logical possibilities on one hand and nonepistemic possibilities, such as either nomological or metaphysical possibilities, on the other. We find that fine-tuning arguments readily conflate epistemic or logical possibilities with nonepistemic possibilities and we think that this leads to (...) treating the search for an explanation of fine-tuning as analogous to standard empirical theorizing about first-order nomological matters, when in fact the two investigational enterprises are profoundly different. Similar conflation occurs when fine-tuning arguments do not carefully distinguish between different interpretations of probabilities within the arguments. Finally, these arguments often rely on spatial analogies, which are often misleading precisely in that they encourage the conflation of epistemic and nonepistemic possibility. When we pay attention to the distinctions between merely epistemic versus nonepistemic modalities and probabilities, the extant arguments in favor of intelligent designers or multiverses, or even for the nonepistemic improbability of fine-tuning, consist of empirically unconstrained speculation concerning relevant nonepistemic modal facts. (shrink)

We show the following results on Wainer's notation for a minimal subrecursive inaccessible ordinal τ: First, we give a constructive proof of the collapsing theorem. Secondly, we prove that the slow-growing hierarchy and the fast-growing hierarchy up to τ have elementary properties on increase and domination, which completes Wainer's proof that τ is a minimal subrecursive inaccessible. Our results are obtained by showing a strong normalization theorem for the term structure of the notation. MSC: 03D20, 03F15.

Let T be the standard Veblen 1908 ordinal notation system for Γ0 as defined, for example, in Schütte's 1977 textbook [13] on Proof Theory. We define a slight modification of the standard assignment of fundamental sequences for the limit ordinals in T and prove that Γ0 is subrecursively inaccessible for this assignment, i.e. the induced slow and fast growing hierarchy match up at Γ0 for the first time.The results of this paper also indicate that φε00 may be considered as a (...) new slow growing ordinal of PA in the sense that the induced slow growing hierarchy up to φε00 classifies the PA-provablyrecursive functions.We show how the results of this paper can be used to bui d a subrecursive hierarchyover the predicative ordinals in the spirit of Wainer's 1970 abstract [16 ]. (shrink)

In this paper first order theories for nonmonotone inductive definitions are introduced, and a proof-theoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given.

In this paper it is shown that the meet-inaccessible degrees are dense in R. The construction uses an 0''-priority argument. As a consequence, the meet-inaccessible degrees and the meet-accessible degrees give a partition of R into two sets, either of which is a nontrivial dense subset of R and generates R - {0} under joins (thus an automorphism base of R).

We investigate the cofinality of the strong measure zero ideal for κ inaccessible and show that it is independent of the size of 2κ.

This article engages Agamben’s view that philosophy and poetry need to remarry, to heal a fracture that springs from the origin of Western culture between knowing and having the (inaccessible) object. While Agamben would like philosophy to wax more poetic (to have the object) and poetry to show more awareness of its philosophical implications (to know the object), he also encourages direct interventions between these two arenas. This essay thus stages an interpenetration of poetic writing and philosophy (Agamben with a (...) little Derrida). James’s embodiment of Agamben’s theory of melancholia in The Aspern Papers set alongside Brodsky’s reflection of Agamben’s notions of nudity, the specter, and the messianic in Watermark—in relation to depictions of Venice—leads to the realization that Agamben’s messianic renders melancholia inoperative. Agamben’s messianic vision that finds salvation in loss cancels the desirability of clinging to the lost object by showing the redemption of its passage. (shrink)

In this paper I prove the following theorems which are the converses of some results of Judah and Laver (1983) and of Judah and Marshall (1993).-IfKM+ATW is not an extension by definition ofKM (and the model involved is well founded), then the existence of two inaccessible cardinals is consistent with ZF.-IfKM+ATW is not a conservative extension ofKM (and the model involved is well founded), then the existence of an inaccessible number of inaccessible cardinals is consistent with ZF.whereKM is Kelley Morse (...) theory andKM+ATW isKM with types of well-orders. (shrink)