We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length ω to computations on tapes of arbitrary ordinal length. We show that a set of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of Gödel's constructible universe L. This characterization can be used to prove the generalized continuum hypothesis in L.

We generalize ordinary register machines on natural numbers to machines whose registers contain arbitrary ordinals. Ordinal register machines are able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the sets of ordinals in a model of the Zermelo-Fraenkel axioms ZFC. This allows the following characterization of computable sets: a set of ordinals is ordinal register (...) computable if and only if it is an element of Gödel’s constructible universe L. (shrink)

We generalize standard Turing machines, which work in time ω on a tape of length ω, to α-machines with time α and tape length α, for α some limit ordinal. We show that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of α-recursive or α-recursively enumerable are equivalent to being computable or computably enumerable by an α-machine, respectively. We emphasize the (...) algorithmic approach to admissible recursion theory by indicating how the proof of the Sacks–Simpson theorem, i.e., the solution of Post’s problem in α-recursion theory, could be based on α-machines, without involving constructibility theory. (shrink)

Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by $\mathscr{P}(\omega)/\mathrm{finite}$. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are \emph{no} intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality (...) of the associated measures. (shrink)

By using the notion of a simplified (κ,1)-morass, we construct κ-thin-tall, κ-thin-thick and, in a forcing extension, κ-very thin-thick superatomic Boolean algebras for every infinite regular cardinal κ.

We survey the definition and fundamental properties of the family of short core models, which extend the core model K of Dodd and Jensen to include α-sequences of measurable cardinals . The theory is applied to various combinatorial principles to get lower bounds for their consistency strengths in terms of the existence of sequences of measurable cardinals. We consider instances of Chang's conjecture, ‘accessible’ Jónsson cardinals, the free subset property for small cardinals, a canonization property of ω ω , and (...) a non-closure property of elementary embeddings of the universe. In some cases, equiconsistencies are obtained. (shrink)

Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times register contents are defined by appropriate limit operations. In this paper, we examine the ITRMs introduced by the third and fourth author (Koepke and Miller in Logic and Theory of Algorithms LNCS, pp. 306–315, 2008), where a register content at a limit time is (...) set to the lim inf of previous register contents if that limit is finite; otherwise the register is reset to 0. The theory of these machines has several similarities to the infinite time Turing machines (ITTMs) of Hamkins and Lewis. The machines can decide all ${\Pi^1_1}$ sets, yet are strictly weaker than ITTMs. As in the ITTM situation, we introduce a notion of ITRM-clockable ordinals corresponding to the running times of computations. These form a transitive initial segment of the ordinals. Furthermore we prove a Lost Melody theorem: there is a real r such that there is a program P that halts on the empty input for all oracle contents and outputs 1 iff the oracle number is r, but no program can decide for every natural number n whether or not ${n \in r}$ with the empty oracle. In an earlier paper, the third author considered another type of machines where registers were not reset at infinite lim inf’s and he called them infinite time register machines. Because the resetting machines correspond much better to ITTMs we hold that in future the resetting register machines should be called ITRMs. (shrink)

We give a proof ofTheorem 1. Let κ be the smallest cardinal such that the free subset property Fr ω (κ,ω 1)holds. Assume κ is singular. Then there is an inner model with ω1 measurable cardinals.

We propose a simple notion of "extender" for coding large elementary embeddings of models of set theory. As an application we present a self-contained proof of the theorem by D. Martin and J. Steel that infinitely many Woodin cardinals imply the determinacy of every projective set.

We give the proof of a theorem of Jensen and Zeman on the existence of a global □ sequence in the Core Model below a measurable cardinal κ of Mitchell order ) equal to κ++, and use it to prove the following theorem on mutual stationarity at n.Let ω1 denote the first uncountable cardinal of V and set to be the class of ordinals of cofinality ω1.TheoremIf every sequence n m. In particular, there is such a model in which for (...) all sufficiently large m<ω, the class of measurables λ with oM≥ωm is, in V, stationary below m+2. (shrink)

We prove that every homogeneously Souslin set is coanalytic provided that either (a) 0long does not exist, or else (b) V = K, where K is the core model below a μ-measurable cardinal.

Using the Friedman-Koepke Hyperfine Structure Theory of [2], we provide a short construction of a gap 1 morass in the constructible universe.

We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ${\neg {\rm AC}_\omega}$ in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiconsistency of the following theories: “ZFC + There is a proper class of regular almost (...) Ramsey cardinals”, and “ZF + DC + All infinite cardinals except possibly successors of singular cardinals are almost Ramsey”. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

We show that for all natural numbers n, the theory “ZF + DC $_{\aleph_n}$ + $\aleph_{\omega}$ is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + $\aleph_{\omega_1}$ is an ω 2-Rowbottom cardinal carrying an ω 2-Rowbottom filter and ω 1 is regular” has the same consistency strength as the theory “ZFC + There exist ω 1 measurable cardinals”. We (...) also discuss some generalizations of these results. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of $\aleph _{\omega}$ . Using symmetric collapses to $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ , (...) or $\aleph _{\omega _{2}}$ , we show that injective failures at $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ , or $\aleph _{\omega _{2}}$ can have relatively mild consistency strengths in terms of Mitchell orders of measurable cardinals. Injective failures of both the aforementioned theorem of Shelah and Silver's theorem that GCH cannot first fail at a singular strong limit cardinal of uncountable cofinality are also obtained. Lower bounds are shown by core model techniques and methods due to Gitik and Mitchell. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

Using the analysis developed in our earlier paper, we show that every uncountable cardinal in Gitik's model of in which all uncountable cardinals are singular is almost Ramsey and is also a Rowbottom cardinal carrying a Rowbottom filter. We assume that the model of is constructed from a proper class of strongly compact cardinals, each of which is a limit of measurable cardinals. Our work consequently reduces the best previously known upper bound in consistency strength for the theory math formula (...) + “All uncountable cardinals are singular” + “Every uncountable cardinal is both almost Ramsey and a Rowbottom cardinal carrying a Rowbottom filter”. (shrink)

As I write this, in November 1971, people are dying in East Bengal from lack of food, shelter, and medical caxc. The suffering and death that are occurring there now axe not inevitable, 1101; unavoidable in any fatalistic sense of the term. Constant poverty, a cyclone, and a civil war have turned at least nine million people into destitute refugees; nevertheless, it is not beyond Lhe capacity of the richer nations to give enough assistance to reduce any further suffering to (...) very small proportions. The decisions arid actions of human beings can prevent this kind of suffering. Unfortunately, human beings have not made the necessary decisions. At the individual level, people have, with very few exceptions, not responded to the situation in any signihczmt way. Generally speaking, people have not given large sums to relief funds; they have not written t0 their parliamentaxy representatives demanding increased government assistance; they have not demonstrated in the streets, held symbolic fasts, or done anything else directed toward providing thc refugees with the means to satisfy their essential needs. At thc government level, no govcrmmcnt has given the sort of massive aid that would enable the refugees to survive fm: more than a few days. Britain, for instance, has given rather more than most countries. It has, to date, given £14,750,000. For comparative purposes, B1~itain’s share of thc nonreccverabla development costs of the Anglo-French Concorde project is already in excess of £275,000,000, and on present estimates will reach £440,000,¤00. The implication is that the British government values a supersonic transport more than thirty times as.. (shrink)

This paper argues that a set of questioning attitudes are among the foundations of human and animal minds. While both verbal questioning and states of curiosity are generally explained in terms of metacognitive desires for knowledge or true belief, I argue that each is better explained by a prelinguistic sui generis type of mental attitude of questioning. I review a range of considerations in support of such a proposal and improve on previous characterizations of the nature of these attitudes. I (...) then broaden their explanatory scope to include a number of forms of exploratory search. The paper has three main goals: to characterize the nature of the questioning attitudes, outlining their causal role and type of content; to argue that they are fundamental components of the mind, being widespread among animals and not constructed during ontogeny out of other attitudes; and to suggest that they explain a great deal more behavior than one might think. (shrink)

Since its original publication in 1975, this groundbreaking work has awakened millions of people to the existence of "speciesism"—our systematic disregard of nonhuman animals—inspiring a worldwide movement to transform our attitudes to animals and eliminate the cruelty we inflict on them. In Animal Liberation, author Peter Singer exposes the chilling realities of today’s "factory farms" and product-testing procedures—destroying the spurious justifications behind them, and offering alternatives to what has become a profound environmental and social as well as moral issue. (...) An important and persuasive appeal to conscience, fairness, decency, and justice, it is essential reading for the supporter and the skeptic alike. (shrink)

We propose a solution to the problem of logical omniscience in what we take to be its fundamental version: as concerning arbitrary agents and the knowledge attitude per se. Our logic of knowledge is a spin-off from a general theory of thick content, whereby the content of a sentence has two components: an intension, taking care of truth conditions; and a topic, taking care of subject matter. We present a list of plausible logical validities and invalidities for the logic of (...) knowledge per se for arbitrary agents, and isolate three explanatory factors for them: the topic-sensitivity of content; the fragmentation of knowledge states; the defeasibility of knowledge acquisition. We then present a novel dynamic epistemic logic that yields precisely the desired validities and invalidities, for which we provide expressivity and completeness results. We contrast this with related systems and address possible objections. (shrink)

Single-premise epistemic closure is the principle that: if one is in an evidential position to know that P where P entails Q, then one is in an evidential position to know that Q. In this paper, I defend the viability of opposition to closure. A key task for such an opponent is to precisely formulate a restricted closure principle that remains true to the motivations for abandoning unrestricted closure but does not endorse particularly egregious instances of closure violation. I focus (...) on two brands of epistemic theory that naturally incorporate closure restrictions. The first type holds that the truth value of a knowledge ascription is relative to a relevant question. The second holds that the truth value of a knowledge ascription is relative to a relevant topic. For each approach, I offer a formalization of a leading theory from the literature and use this formalization to evaluate the theory’s adequacy in terms of a precise set of desiderata. I conclude that neither theory succeeds in meeting these desiderata, casting doubt on the viability of the underlying approaches. Finally, I offer a novel variant of the topic-sensitive approach that fares better. (shrink)

Introduction These essays have been written over a period of about ten years and have already been published separately in various places. ...

I lay out the framework for my theory of sensory imagination in “Imagining as a guide to possibility.” Sensory imagining involves mental imagery , and crucially, in describing the content of imagining, I distinguish between qualitative content and assigned content. Qualitative content derives from the mental image itself; for visual imaginings, it is what is “pictured.” For example, visually imagine the Philadelphia Eagles defeating the Pittsburgh Steelers to win their first Super Bowl. You picture the greenness of the field and (...) the football’s brown oblong shape. Some of what you imagine isn’t explicitly pictured, however. That it is Sunday, that it is the Super Bowl: these facts are assigned. (shrink)

Political decisions have the potential to greatly impact our lives. Think of decisions in relation to abortion or climate change, for example. This makes political legitimacy an important normative concern. But what makes political decisions legitimate? Are they legitimate in virtue of having support from the citizens? Democratic conceptions of political legitimacy answer in the affirmative. Such conceptions righly highlight that legitimate political decision-making must be sensitive to disagreements among the citizens. But what if democratic decisions fail to track what (...) there is most reason to do? What if a democratically elected government fails to take measures necessary to protect its population from threats related to climate change? Peter argues that the legitimacy of political decisions doesn't just depend on respect for the citizens' will; and defends a novel hybrid conception of political legitimacy, called the Epistemic Accountability conception. According to this conception, political legitimacy also depends on how political decision-making responds to evidence for what there is most reason to do. Grounds for Political Legitmacy starts with an overview of the main ways in which philosophers have thought about political legitimacy, and identifies the epistemic accountability conception as an overlooked alternative. It then develops the epistemic accountability conception of political legitimacy and discusses its implications for legitimate political decision-making. Considering the norms that should govern political debate, it examines the role of experts in politics, and probes the responsibilities of democratically elected political leaders and as well as of citizens. (shrink)

Author: Koepke Ireneusz Title: ANDRZEJ WALICKI AND “WARSAW SCHOOL OF THE HISTORY OF IDEAS” (Andrzej Walicki i „warszawska szkoła historyków (historii) idei”) Source: Filo-Sofija year: 2005, vol:.5, number: 2005/1, pages: 259-276 Keywords: ANDRZEJ WALICKI, WARSAW SCHOOL OF THE HISTORY OF IDEAS Discipline: PHILOSOPHY Language: POLISH Document type: ARTICLE Publication order reference (Primary author’s office address): E-mail: www:This article is an attempt of presenting Andrzej Walicki connections with the so called Warsaw School of the History of Ideas and his research (...) on basic methodological issues created by this “school”, which are linked with the history of ideas and the history of philosophy. The most important are research rules which has been used by A. Walicki, and their consequences on his intellectual work as well as some critical remarks. (shrink)