Adding higher types to set theory differs from adding inaccessible cardinals, in that higher type arguments apply to all sets rather than just ordinary ones. Levy's reflection axiom is justified, by considering the principle that we can pretend that the universe is a set, together with methods of Gaifman [8]. We reprove some results of Gaifman, and some facts about Levy's reflection axiom, including the fact that adding higher types yields no new theorems about sets. Some remarks on (...) standard models are made. An obvious strengthening of Levy's axiom to higher types is considered, which implies the existence of indescribable cardinals. Other remarks about larger cardinals are made; some questions of Gloede [9] are settled. Finally we argue that the evidence for V = L is strong, and that CH is certainly true. MSC: 03E30, 03E55. (shrink)

Introduction -- The end of history and the Kantian moment -- The progressive mind and the Islamist challenge -- The incoherence of the philosophers -- The language of progress and the closure of the European mind -- The networked global order -- All roads lead to China -- Maxims or axioms? -- The revenge of politics and the search for order -- Conclusion -- Afterword : the study of international relations and the erosion of acdemic integrity.

This chapter contains sections titled: Introduction: Questions of Meaning Theories of Meaning Language, Mind and Metaphysics: Questions of Priority Semantic Theories: Davidson's Programme Analysing the Concept of Meaning: Grice's Programme Pragmatics: Conversational Implicature and Relevance Theory.

There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou’s probabilistic bisimilarity, whereas the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator that there is such a process with an uncountable Zhou ordinal.

This chapter contains section titled: Discussion.

We give a new characterization of the class of completely representable cylindric algebras of dimension 2 #lt; n ≤ w via special neat embeddings. We prove an independence result connecting cylindric algebra to Martin's axiom. Finally we apply our results to finite-variable first order logic showing that Henkin and Orey's omitting types theorem fails for Ln, the first order logic restricted to the first n variables when 2 #lt; n#lt;w. Ln has been recently (and quite extensively) studied as (...) a many-dimensional modal logic. (shrink)

The main result of this paper is a formal argument for the principle of maximizing expected utility that does not rely on the law of large numbers. Unlike the well-known arguments by Savage and von Neumann & Morgenstern, this argument does not presuppose the sure-thing principle or the independence axiom. The principal idea is to use the concept of transformative decision rules for decomposing the principle of maximizing expected utility into a sequence of normatively reasonable subrules. It is shown (...) that this procedure provides a resolution of Allais's paradox that cannot be obtained by Savage-style or von Neumann & Morgenstern-style arguments. (shrink)

We give a new characterization of the class of completely representable cylindric algebras of dimension 2 #lt; n w via special neat embeddings. We prove an independence result connecting cylindric algebra to Martin''s axiom. Finally we apply our results to finite-variable first order logic showing that Henkin and Orey''s omitting types theorem fails for L n, the first order logic restricted to the first n variables when 2 #lt; n#lt;w. L n has been recently (and quite extensively) studied as (...) a many-dimensional modal logic. (shrink)

This paper presents an axiomatization of the principle of maximizing expected utility that does not rely on the independence axiom or sure-thing principle. Perhaps more importantly the new axiomatization is based on an ex ante approach, instead of the standard ex post approach. An ex post approach utilizes the decision maker's preferences among risky acts for generating a utility and a probability function, whereas in the ex ante approach a set of preferences among potential outcomes are on the input (...) side of the theory and the decision maker's preferences among risky acts on the output side. (shrink)

We show that it is consistent that Martin's axiom holds, the continuum is large, and yet the dual distributivity number ℌ is κ1. This answers a question of Halbeisen.

We deal with the consistency strength of ZFC + variants of MA + suitable sets of reals are measurable (and/or Baire, and/or Ramsey). We improve the theorem of Harrington and Shelah [2] repairing the asymmetry between measure and category, obtaining also the same result for Ramsey. We then prove parallel theorems with weaker versions of Martin's axiom (MA(σ-centered), (MA(σ-linked)), MA(Γ + ℵ 0 ), MA(K)), getting Mahlo, inaccessible and weakly compact cardinals respectively. We prove that if there exists (...) r ∈ R such that ω L[ r] 1 = ω 1 and MA holds, then there exists a ▵ 1 3 -selective filter on ω, and from the consistency of ZFC we build a model for ZFC + MA(I) + every ▵ 1 3 -set of reals is Lebesgue measurable, has the property of Baire and is Ramsey. (shrink)

Kurt Gödel is almost as famous—one might say “notorious”—for his extreme platonist views as he is famous for his mathematical theorems. Moreover his platonism is not a myth; it is well-documented in his writings. Here are two platonist declarations about set theory, the first from his paper about Bertrand Russell and the second from the revised version of his paper on the Continuum Hypotheses.Classes and concepts may, however, also be conceived as real objects, namely classes as “pluralities of things” or (...) as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence.But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.The first statement is a platonist declaration of a fairly standard sort concerning set theory. What is unusual in it is the inclusion of concepts among the objects of mathematics. This I will explain below. The second statement expresses what looks like a rather wild thesis. (shrink)

A physician and spokesman for the French Ideologues, Pierre-JeanGeorges Cabanis (1757-1808) stands at the crossroads of several influential developments in modern culture--Enlightenment optimism about human perfectibility, the clinical method in medicine, and the formation and adaptation of liberal social ideals in the French Revolution. This first major study of Cabanis in English traces the influences of these developments on his thought and career. Originally published in 1980. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously (...) out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

These new essays on J. L. Austin's philosophy constitute the first major study of his thought in decades.

Assuming an inaccessible cardinal $\kappa$ , there is a generic extension in which $MA + 2^{\aleph_0} = \kappa$ holds and the reals have a $\Delta^2_1$ well-ordering.

It is shown that every locally countable upper semi-lattice of cardinality the continuum can be embedded into the Turing degrees, assuming Martin’s Axiom.

We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ℵ1 dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ℵ1 dense sets related to the rectangle refining property, and prove that they are really weaker fragments.

Why was it that Francis Bacon, trained for high political office, devoted himself to proposing a celebrated and sweeping reform of the natural sciences? Julian Martin's investigative study looks at Bacon's family context, his employment in Queen Elizabeth's security service and his radical critique of the relationship between the Common Law and the Monarchy, to find the key to this important question. Deeply conservative and elitist in his political views, Bacon adapted Tudor strategies of State management and bureaucracy, the (...) social anxieties and prejudices of the late-Elizabethan governing elite, and a principal intellectual resource of the English governing classes - the Common Law - into a novel vision and method for the sciences. Bacon's axiom that 'Knowledge is Power' takes on far-reaching implications in Martin's challenging argument that the reform of natural philosophy was a central part of an audacious plan to strengthen the powers of the Crown in the State. (shrink)

Martin's axiom is equivalent to the statement that the universe is absolute under ccc forcing extensions for Σ 1 sentences with a subset of $\kappa, \kappa , as a parameter.

This book is a collection of papers presented at an interdisciplinary workshop at the Calgary Institute for the Humanities in May 1980. The three broad issues covered are: the physician-patient relationship, the allocation of responsibility among doctors and nurses, and the political and social framework of the health care system. The first set of essays is concerned with the moral and legal aspects of the physician-patient relationship. The link between knowledge and power is examined as well as the moral dilemmas (...) posed by medical technology. These initial essays would alone justify this publ. (shrink)

We strengthen a result of Harrington and Shelah by showing that, unless ω1 is an inaccessible cardinal in L, a relatively weak fragment of Martin's axiom implies that there exists a δ13 set of reals without the property of Baire.

We present a number of results involving almost disjoint sets and Martin's axiom. Included is an example, due to K. Kunen, of a c.c.c. partial order without property K whose product with every c.c.c. partial order is c.c.c.

We show that either PFA + or Martin's maximum implies Fleissner's Axiom R, a reflection principle for stationary subsets of P ℵ 1 (λ). In fact, the "plus version" (for one term denoting a stationary set) of Martin's axiom for countably closed partial orders implies Axiom R.