Usually the Lorentz transformations are derived from the conservation of the spacetime interval. We propose here a way of obtaining spacetime transformations between two inertial frames directly from symmetry, the isotropy of the space and principle of relativity. The transformation is uniquely defined except for a constant e, that depends only on the process of synchronization of clocks inside each system. Relativistic velocity addition is obtained, and it is shown that the set of velocities is a bounded symmetric domain. If (...) e=0, Galilean transformations are obtained. If e>0, the speed 1/√e and a spacetime interval are conserved. By assuming constancy of the speed of light, we get e=1/c 2 and the transformation between the frames becomes the Lorentz transformation. If e<0, a proper speed and a Hilbertian norm are conserved. (shrink)

We give a geometric realization of space-time spinors and associated representations, using the Jordan triple structure associated with the Cartan factors of type 4, the so-called spin factors. We construct certain representations of the Lorentz group, which at the same time realize bosonic spin-1 and fermionic spin- $${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0em}\!\lower0.7ex\hbox{$2$}}$$ wave equations of relativistic field theory, showing some unexpected relations between various low-dimensional Lorentz representations. We include a geometrically and physically motivated introduction to Jordan triples and spin factors.

This volume brings Cassirer s work into the arena of contemporary debates both within and outside of philosophy. All articles offer a fresh and contemporary look at one of the most prolific and important philosophers of the 20th century. The papers are authored by a wide array of scholars working in different areas, such as epistemology, philosophy of culture, sociology, psychopathology, philosophy of science and aesthetics.".

We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. (...) G. Kanovei, On descriptive forms of the countable axiom of choice, Investigations on nonclassical logics and set theory, Work Collect., Moscow, 3-136 ]. (shrink)

The Tulis thesis becomes even more powerful when the constitutional revolution he describes is put in its Progressive‐Era context. The public had long demanded social reforms designed to curb or replace laissez‐faire capitalism, which was seen as antithetical to the interests of ordinary working people. But popular demands for social reform went largely unmet until the 1910s. Democratizing political reforms, such as the rhetorical presidency, were designed to facilitate “change” by finally giving the public the power to enact social reforms. (...) The resulting political order has created systemic pressure for policy demagoguery in place of rational deliberation. Mass political mobilization seems to be better achieved by contests of grand principle that pit the well‐meaning supporters of obviously needed reforms against “villains and conspirators,” than by technical discussions of the possibly counterproductive effects of those reforms. (shrink)

Assuming the existence of a weakly compact hypermeasurable cardinal we prove that in some forcing extension א ω is a strong limit cardinal and א ω+2 has the tree property. This improves a result of Matthew Foreman (see [2]).

It is widely held that individuals who are unable to provide informed consent should be enrolled in clinical research only when the risks are low, or the research offers them the prospect of direct benefit. There is now a rich literature on when the risks of clinical research are low enough to enroll individuals who cannot consent. Much less attention has focused on which benefits of research participation count as ‘direct’, and the few existing accounts disagree over how this crucial (...) concept should be defined. This disagreement raises concern over whether those who cannot consent, including children and adults with severe dementia, are being adequately protected. The present paper attempts to address this concern by considering first what additional protections are needed for these vulnerable individuals. This analysis suggests that the extant definitions of direct benefits either provide insufficient protection for research subjects or pose excessive obstacles to appropriate research. This analysis also points to a modified definition of direct benefits with the potential to avoid these two extremes, protecting individuals who cannot consent without blocking appropriate research. (shrink)

ABSTRACTHow can political actors identify which putative expert is truly expert, given that any putative expert may be wrong about a given policy question; given that experts may therefore disagree with one another; and given that other members of the polity, being non-expert, can neither reliably adjudicate inter-expert disagreement nor detect when a consensus of experts is misguided? This would not be an important question if the problems dealt with by politics were usually simple ones, in the sense that the (...) answer to them is self-evident. But to the extent that political problems are complex, expertise is required to answer them—although if such expertise exists, we are unlikely to know who has it. This conundrum is illustrated by the financial crisis. The consensus views of financial regulators prior to the crisis appear to have been mistaken, in that the regulators not only failed to anticipate a crisis of the type that occurred, but adopted regulations that may have encouraged the concentration of mortgage risk in financial institutions. Similarly, once the crisis broke, academic financial experts, financial regulators, and journalists—who communicate information from one branch of the division of epistemic labor to another—converged on a narrative of the crisis that, once again, may plausibly be described as inaccurate, but that nonetheless has come to shape the understanding of the crisis shared by the rest of the polity. (shrink)

Immanuel Kant's Metaphysical Foundations of Natural Science (1786) provides metaphysical foundations for the application of mathematics to empirically given nature. The application that Kant primarily has in mind is that achieved in Isaac Newton's Principia (1687). Thus, Kant's first chapter, the Phoronomy, concerns the mathematization of speed or velocity, and his fourth chapter, the Phenomenology, concerns the empirical application of the Newtonian notions of true or absolute space, time, and motion. This paper concentrates on Kant's second and third chapters—the Dynamics (...) and the Mechanics, respectively—and argues that they are best read as providing a transcendental explanation of the conditions for the possibility of applying the (mathematical) concept of quantity of matter to experience. Kant again has in mind the empirical measures of this quantity that Newton fashions in the Principia, and he aims to make clear, in particular, how Newton achieves a universal measure for all bodies whatsoever by projecting the static quantity of terrestrial weight into the heavens by means of the theory of universal gravitation. Kant is not attempting to prove a priori what Newton has established empirically but, rather, to clarify the character of Newton's mathematization by building Newton's empirical measures into the very concept of matter that is articulated in the Metaphysical Foundations. (shrink)

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)

In thinking and talking about rights, including property rights, it seems natural to put the argument in either moral or legal terms. From the former viewpoint, rights are part of a description of what actions are right or wrong. The fact that I have a right to do something is an argument, although not necessarily a sufficient argument, that someone who prevents me from doing it is acting wrongly.

The target article proposes that people perceive social robots as depictions rather than as genuine social agents. We suggest that people might instead view social robots as social agents, albeit agents with more restricted capacities and moral rights than humans. We discuss why social robots, unlike other kinds of depictions, present a special challenge for testing the depiction hypothesis.

Let k ≥ 2 and f:Nk Æ [1,k] and n ≥ 1 be such that there is no x1 < ... < xk+1 £ n such that f(x1,...,xk) = f(x1,...,xk+1). Then we want to find g:Nk+1 Æ [1,3] such that there is no x1 < ... < xk+2 £ n such that g(x1,...,xk+1) = g(x2,...,xk+2). This reducees adjacent Ramsey in k dimensions with k colors to adjacent Ramsey in k+1 dimensions with 3 colors.

The main result in this paper is a method for obtaining derivation trees from sentences of certain formal grammars. No parsing algorithm was previously known to exist for these grammars.Applied to Montague's PTQ the method produces all parses that could correspond to different meanings. The technique directly addresses scope and reference and provides a framework for examining these phenomena. The solution for PTQ is implemented in an efficient and useful computer program.

We show that, assuming the existence of the canonical inner model with one Woodin cardinal $M_1 $, there is a model of $ZFC$ in which the nonstationary ideal on $\omega _1 $ is $\aleph _2 $-saturated and whose reals admit a ${\rm{\Sigma }}_4^1 $-wellorder.

Taylor, Sandel, Walzer, and MacIntyre waver between granting the community authority over the individual and limiting this authority so severely that communitarianism becomes a dead letter. The reason for this vacillation can be found in the aspiration of each theorist to base liberal values‐equality and liberty—on particularism. Communitarians compound liberal formalism by adding to the liberal goal, individual autonomy, the equally abstract aim of grounding autonomy in a communally shared identity. Far from returning political theory to substantive considerations of the (...) good, communitarianism legitimizes really existing liberal politics—the politics of the nation‐state. (shrink)

We now fix A ⊆ Q. We study a fundamental class of digraphs associated with A, which we call the A-digraphs. An A,kdigraph is a digraph (Ak,E), where E is an order invariant subset of A2k in the following sense. For all x,y ∈ A2k, if x,y have the same order type then x ∈ E ↔ y ∈ E.

Loose, unconnected, free-floating, mobile: this is the modern individual, at least in comparison with the immediate past.

We force a property of cardinals first proved relatively consistent by Sargsyan, that of being supercompact but not equation image-supercompact, starting from a model of set theory which does not satisfy equation image and that contains supercompact cardinals.

Vopenka [2] proved long ago that every set of ordinals is set-generic over HOD, Gödel's inner model of hereditarily ordinal-definable sets. Here we show that the entire universe V is class-generic over, and indeed over the even smaller inner model $\mathbb{S}=$, where S is the Stability predicate. We refer to the inner model $\mathbb{S}$ as the Stable Core of V. The predicate S has a simple definition which is more absolute than any definition of HOD; in particular, it is possible (...) to add reals which are not set-generic but preserve the Stable Core. (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)

E. T. Jaynes' resolution of Bertrand's paradox in terms of invariance principles is criticized. An experimental setup is considered which generates general solutions to Bertrand's problem by rotating a line around a point a distancer+d from a circle of radiusr. The general solution obtained is neither translationally nor scale invariant, but depends on the value ofr/d. Only in the limitr/d » 0, when the line is just translating across the circle, is the distribution translationally invariant and scale invariant. In this (...) limiting case the distribution found is just that obtained by Jaynes. (shrink)

The sensible response to conflicts of interest is impaired by misconceptions and sloppy usage of terminology. Apparent and potential are widely misused modifiers for conflicts. Excessive legislative focus on financial interests limits understanding of the scope and significance of researchers' conflicts of interest. There is no moral or ethical failing in having a conflict of interest; the problem occurs when conflicts are not disclosed appropriately and when conflicts are allowed to bias research, teaching, or practice. Avoidance and prevention should be (...) applied to bias, not conflicts. (shrink)

The Complete Theory of Everything (CTE) is based on certain axioms of indiscernibility. Such axioms of indiscernibility have been given a philosophical justification by Kit Fine. I want to report on an attempt to give strong indiscernibility axioms which might also be subject to such philosophical analysis, and which prove the consistency of set theory; i.e., ZFC or more. In this way, we might obtain a (new kind of) philosophical consistency proof for mathematics.

An unprovable theorem is a mathematical result that can-not be proved using the com-monly accepted axioms for mathematics (Zermelo-Frankel plus the axiom of choice), but can be proved by using the higher infinities known as large cardinals. Large car-dinal axioms have been the main proposal for new axioms originating with Gödel.

We present a way out of Russell’s paradox for sets in the form of a direct weakening of the usual inconsistent full comprehension axiom scheme, which, with no additional axioms, interprets ZFC. In fact, the resulting axiomatic theory 1) is a subsystem of ZFC + “there exists arbitrarily large subtle cardinals”, and 2) is mutually interpretable with ZFC + the scheme of subtlety.

We discuss several axiomatizations of set theory in first order predicate calculus with epsilon and a constant symbol W, starting with the simple system K(W) which has a strong equivalence with ZF without Foundation. The other systems correspond to various extensions of ZF by certain large cardinal hypotheses. These axiomatizations are unusually simple and uncluttered, and are highly suggestive of underlying philosophical principles that generate higher set theory.

In mathematics, we back up our discoveries with rigorous deductive proofs. Mathematicians develop a keen instinctive sense of what makes a proof rigorous. In logic, we strive for a *theory* of rigorous proofs.

We give a complete proof that all 3 quantifier sentences in the primitive notation of set theory (Œ,=), are decided in ZFC, and in fact in a weak fragment of ZF without the power set axiom. We obtain information concerning witnesses of 2 quantifier formulas with one free variable. There is a 5 quantifier sentence that is not decided in ZFC (see [Fr02]).

S. Adams, W. Ambrose, A. Andretta, H. Becker, R. Camerlo, C. Champetier, J.P.R. Christensen, D.E. Cohen, A. Connes. C. Dellacherie, R. Dougherty, R.H. Farrell, F. Feldman, A. Furman, D. Gaboriau, S. Gao, V. Ya. Golodets, P. Hahn, P. de la Harpe, G. Hjorth, S. Jackson, S. Kahane, A.S. Kechris, A. Louveau,, R. Lyons, P.-A. Meyer, C.C. Moore, M.G. Nadkarni, C. Nebbia, A.L.T. Patterson, U. Krengel, A.J. Kuntz, J.-P. Serre, S.D. Sinel'shchikov, T. Slaman, Solecki, R. Spatzier, J. Steel, D. Sullivan, S. (...) Thomas, A. Valette, V.S. Varadarajan, B. Velickovic, B. Weiss, J.D.M. Wright, R.J. Zimmer. (shrink)

Provide a formal system that is a conservative extension of PA for Π02 sentences, and even a conservative extension of HA, that supports the worry free smooth development of constructive analysis in the style of Errett Bishop.

I don’t remember if I got as high as 2-390, but I distinctly remember taking my first logic course - as a Freshman - with Hartley Rogers, in Fall 1964 - here in 2-190. Or was it in 2-290?