Online research in philosophy

The most common philosophical view of existence is that existence amounts to existential quantification or is a second-order concept. A less common philosophical view is that existence is a first-order property distinguishing between nonexistent (past, possible, or merely intentional) objects and existing objects. An even less common philosophical view is that existence divides into different ‘modes of being’ for different kinds of entities. The aim of the present paper is to take a closer look at how the notion of existence is in fact expressed in natural language. In natural language, it appears, existence is not so much expressed by quantification, which can be shown to be neutral as regards any distinction between existent and nonexistent objects that one might draw. Rather existence is expressed by predicates, and that is, first-order predicates, such as in English 'exist', 'occur', and 'obtain'. The semantic behavior of such existence predicates reveals a notion of existence that divides into at least three different kinds of modes being, reflecting the distinction between endurance and perdurance, as well as their space-related analogues, but also the particular mode of being of such entities as states, facts, conditions, and laws.

The theory of special relativity (SR) is considered in the framework of classical first-order logic. The four axioms of SR are: The existence of a flat four-dimensional spacetime continuum, the existence of global inertial frames of reference and Einstein's two postulates. The propositions permitted are those involving the kinematics and dynamics of SR as formulated from an inertial frame; these give a complete description of the evolution of the universe. Assuming SR as consistent, there must exist a model M for SR in which F is an inertial frame in a non-trivial universe U(IBC) of material objects, with appropriate initial-boundary conditions IBC specified. Let P be defined as ``F is an inertial frame in U(IBC)''. Using Goedel's second incompleteness theorem, it is argued that P is undecidable in SR. Let Q be any proposition such that (Q --> P) is not a theorem of SR. If, in addition, (P --> Q) is a theorem of SR then Q must necessarily be true in M. It follows that there must exist a model N for SR in which Q is true and P is false, i.e., F is an accelerated frame in U(IBC). The philosophical and mathematical implications of this result for the consistency of SR are discussed.

The Successor Axiom asserts that every number has a successor, or in other words, that the number series goes on and on ad infinitum. The present work investigates a particular subsystem of Frege Arithmetic, called F, which turns out to be equivalent to second-order Peano Arithmetic minus the Successor Axiom, and shows how this system can develop arithmetic up through Gauss' Quadratic Reciprocity Law. It then goes on to represent questions of provability in F, and shows that F can prove its own consistency and indeed the consistency of stronger systems. So, arithmetic without the Successor Axiom has an exceptional combination of three chracteristics: it is natural, it is strong, and it proves its own, as well as stronger systems’, consistency.

In this paper we study the consistency strength of the theory $\mathbf\mathrm{ZFC} + (\exists\kappa \text{strong limit})(\forall\mu , and we prove the consistency of this theory relative to the consistency of the existence of a supercompact cardinal and an inaccessible above it.

Every recursively enumerable extension of arithmetic which obeys the disjunction property obeys the numerical existence property [Fr, 1]. The requirement of recursive enumerability is essential. For extensions of intuitionistic second order arithmetic by means of sentences (in its language) with no existential set quantifiers, the numerical existence property implies the set existence property. The restriction on existential set quantifiers is essential. The numerical existence property cannot be eliminated, but in the case of finite extensions of HAS, can be replaced by a weaker form of it. As a consequence, the set existence property for intuitionistic second order arithmetic can be proved within itself.

Using an axiomatization of second-order arithmetic (essentially second-order Peano Arithmetic without the Successor Axiom), arithmetic's basic operations are defined and its fundamental laws, up to unique prime factorization, are proven. Two manners of expressing a system's consistency are presented - the "Godel" consistency, where a wff is represented by a natural number, and the "real" consistency, where a wff is represented as a second-order sequence, which is a stronger notion. It is shown that the system can prove at least its Godel consistency and that closely allied systems can prove their real consistency.

The theory that ``consistency implies existence'' was put forward by Hilbert on various occasions around the start of the last century, and it was strongly and explicitly emphasized in his correspondence with Frege. Since (Gödel's) completeness theorem, abstractly speaking, forms the basis of this theory, it has become common practice to assume that Hilbert took for granted the semantic completeness of second order logic. In this paper I maintain that this widely held view is untrue to the facts, and that the clue to explain what Hilbert meant by linking together consistency and existence is to be found in the role played by the completeness axiom within both geometrical and arithmetical axiom systems.

As it is currently used, "foundations of arithmetic" can be a misleading expression. It is not always, as the name might indicate, being used as a plural term meaning X = {x : x is a foundation of arithmetic}. Instead it has come to stand for a philosophico-logical domain of knowledge, concerned with axiom systems, structures, and analyses of arithmetic concepts. It is a bit as if "rock" had come to mean "geology." The conflation of subject matter and its study is a serious one, because in the end, one can lose sight of what one should be doing in the first place. Perhaps it is taking matters too literally, but it seems that there is something to be said for taking the term to represent X. Doing so and accepting the term to have some kind of significance, it is then natural to focus on the question of what a foundation of arithmetic should be; and, if one exists, what one is. Whatever the case, that is what shall be done in this paper.

In "The Nature and Meaning of Numbers," Dedekind produces an original, quite remarkable proof for the holy grail in the foundations of elementary arithmetic, that there are an infinite number of things. It goes like this. [p, 64 in the Dover edition.] Consider the set S of things which can be objects of my thought. Define the function phi(s), which maps an element s of S to the thought that s can be an object of my thought. Then phi is evidently one-to-one, and the image of phi is contained in S. Indeed, it is properly contained in S, because I myself can be an object of my thoughts and so belong to S, but I myself am not a mere thought. Thus S is infinite.

I begin with a personal confession. Philosophical discussions of existence have always bored me. When they occur, my eyes glaze over and my attention falters. Basically ontological questions often seem best decided by banging on the table--rocks exist, fairies do not. Argument can appear long-winded and miss the point. Sometimes a quick distinction resolves any apparent difficulty. Does a falling tree in an earless forest make noise, ie does the noise exist? Well, if noise means that an ear must be there to hear it, then the answer to the question is evidently "no." But if noise means that, if there were (counterfactually) someone there, then he would hear it, then just as obviously, the answer becomes "yes.".