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We introduce a general theory of functions called Flow. We prove ZF, non-well founded ZF and ZFC can be immersed within Flow as a natural consequence from our framework. The existence of strongly inaccessible cardinals is entailed from our axioms. And our first important application is the introduction of a model of Zermelo-Fraenkel set theory where the Partition Principle (PP) holds but not the Axiom of Choice (AC). So, Flow allows us to answer to the oldest open problem in set theory: if PP entails AC., Comment: 37 pages, 4 Figures

Quasi-set theory $\cal Q$ allows us to cope with certain collections of objects where the usual notion of identity is not applicable, in the sense that $x = x$ is not a formula, if $x$ is an arbitrary term. $\cal Q$ was partially motivated by the problem of non-individuality in quantum mechanics. In this paper I discuss the range of explanatory power of $\cal Q$ for quantum phenomena which demand some notion of indistinguishability among quantum objects. My main focus is on the double-slit experiment, a major physical phenomenon which was never modeled from a quasi-set-theoretic point of view. The double-slit experiment strongly motivates the concept of degrees of indistinguishability within a field-theoretic approach, and that notion is simply missing in $\cal Q$. Nevertheless, other physical situations may eventually demand a revision on quasi-set theory axioms, if someone intends to use it in the quantum realm for the purpose of a clear discussion about non-individuality. I use this opportunity to suggest another way to cope with identity in quantum theories., Comment: 23 pages, no figures

In the early 1990's A. Elitzur and L. Vaidman proposed an interaction free measurement (IFM) that allows researchers to find infinitely fragile objects without destroying them. But Elitzur-Vaidman IFM has been used only to determine the position of opaque objects. In this paper, we propose an extension of such a technique that allows measurement of classical electric and magnetic fields. Our main goal is to offer a framework for future investigations about the role of the measurement processes, expanding the physical properties that are measurable by means of IFM., Comment: 10 pages, 3 figures

Elementary particles in quantum mechanics (QM) are indistinguishable when sharing the same intrinsic properties and the same quantum state. So, we can consider quantum particles as non-individuals, although non-individuality is usually considered as a consequence of the formalism of QM, since the entanglement of states forbids any labelling process. We show how to consider non-individuality as one of the basic principles of QM, instead of a logical consequence. The advantages of our framework are discussed as well. We also show that even in classical particle mechanics it is possible to consider the existence of non-individual particles. One of our main contributions is to show how to derive the apparent individuality of classical particles from the assumption that all physical objects are non-individuals., Comment: 23 pages, no figures

It is well known that in Zermelo-Fraenkel (ZF) set theory any finite set is decidable. In this paper we discuss an extension of ZF where this result is no longer valid. Such an extension is quasi-set theory and it has its origin on problems motivated by quantum mechanics., Comment: 9 pages, no figures

Quasi-set theory is a first order theory without identity, which allows us to cope with non-individuals in a sense. A weaker equivalence relation called ``indistinguishability'' is an extension of identity in the sense that if $x$ is identical to $y$ then $x$ and $y$ are indistinguishable, although the reciprocal is not always valid. The interesting point is that quasi-set theory provides us a useful mathematical background for dealing with collections of indistinguishable elementary quantum particles. In the present paper, however, we show that even in quasi-set theory it is possible to label objects that are considered as non-individuals. We intend to prove that individuality has nothing to do with any labelling process at all, as suggested by some authors. We discuss the physical interpretation of our results., Comment: 11 pages, no figures

In a recent paper Andrei N. Soklakov explained the foundations of the Lagrangian formulation of classical particle mechanics by means of Kolmogorov complexity. In the present paper we use some of Soklakov ideas in order to derive the second law of thermodynamics. Our main result is that the complexity of a thermal system corresponds to its entropy., Comment: 4 pages, no figures

According to Cantor, a set is a collection into a whole of defined and separate (we shall say distinct) objects. So, a natural question is ``How to treat as `sets' collections of indistinguishable objects?". This is the aim of quasi-set theory, and this problem was posed as the first of present day mathematics, in the list resulting from the Congress on the Hilbert Problems in 1974. Despite this pure mathematical motivation, quasi-sets have also a strong commitment to the way quantum physics copes with elementary particles. In this paper, we discuss the axiomatics of quasi-set theory and sketch some of its applications in physics. We also show that quasi-set theory allows us a better and deeper understanding of the role of the concept of equality in mathematics., Comment: LaTeX, 32 pages, 3 figures

We use Padoa's principle of independence of primitive symbols in axiomatic systems in order to show that time is dispensable in continuum thermodynamics, according to the axiomatic formulation of Gurtin and Williams. We also show how to define time by means of the remaining primitive concepts of Gurtin and Williams system. Finally, we introduce thermodynamics without any explicit reference to time and briefly discuss some physical and philosophical consequences of our main results., Comment: LaTeX, 11 pages, no figures. This is a preliminary version before we submit it for publication in a journal

We use Padoa's principle of independence of primitive symbols in axiomatic systems in order to discuss the mathematical role of time and space-time in some classical physical theories. We show that time is eliminable in Newtonian mechanics and that space-time is also dispensable in Hamiltonian mechanics, Maxwell's electromagnetic theory, the Dirac electron, classical gauge fields, and general relativity., Comment: LaTeX, 12 pages, no figures. This is a new version

We present an axiomatic framework for what we call Mach's mechanics, inspired on the ideas by A. K. T. Assis and P. Graneau about relational mechanics. We show that contrarily to what is suggested by these authors, Mach's principle does not need to be committed with any Weber's like gravitational force. Actually it is possible to settle an axiomatic framework for non-relativistic classical particle mechanics which incorporates Mach's principle and is consistent with any force that obeys the newtonian action-reaction principle. We make further criticisms concerning relational mechanics., Comment: 16 pages, plain LaTeX, no figures

Quasi-set theory allows us a non trivial relation between indistinguishability and nonlocality into the context of Einstein- Podolsky-Rosen experiment. Quasi-set theory is a set theory which provides a manner for dealing with collections of indistinguishable but not identical elementary particles., Comment: RevTeX, 4 pages, 1 figure

We introduce the concept of indexed identity, where the usual notion of identity is a particular case. Our mathematical framework allows us a generalized method for `indexing' predicates, which corresponds to `fuzzification' of properties, in an intuitive sense. It can be established a relationship between this indexed mathematics and fuzzy mathematics. We show some few examples that illustrate our ideas., Comment: LaTeX, 9 pages, no figures

Quasi-set theory provides us a mathematical background for dealing with collections of indistinguishable elementary particles. In this paper, we show how to obtain the usual statistics (Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac) into the scope of quasi-set theory. We also show that, in order to derive Maxwell-Boltzmann statistics, it is not necessary to assume that the particles are distinguishable. In other words, Maxwell-Boltzmann statistics is possible even in an ensamble of indistinguishable particles, at least from the theoretical point of view. The main goal of this paper is to provide the mathematical grounds of a quasi-set-theoretical framework for statistical mechanics., Comment: LaTeX, 19 pages, 3 figures

We discuss the babylonian method of extracting the root square of a number, from the point of view of modern mathematics. We also speculate that the babylonian mathematics was rich enough for a generalization of this method, despite the lack of general statements and justified procedures in their mathematics., Comment: LaTeX, 4 pages, no figures

K-theory allows us to define an analytical condition for the existence of `false' gauge field copies through the use of the Atiyah-Singer index theorem. After establishing that result we discuss a possible extension of the same result without the help of the index theorem and suggest possible related lines of work., Comment: LaTeX, 7 pages, no figures

A condition for the existence of false gauge field copies in terms of the Lefschetz number of a certain differential operator is presented., Comment: LaTeX, 4 pages, no figures

We present an axiomatic framework for non-relativistic classical particle mechanics, inspired on Tati's ideas about a non-space-time description for physics. The main advantage of our picture is that it allows us to describe causality without any reference to elapsed time intervals., Comment: 9 pages, no figures, LaTeX. This paper will appear in Philos. Natur. 2 (1999)

We propose that the quantum vacuum may be considered as a gas of virtual photons which carry a non-vanishing linear momentum as well as a non-vanishing energy. We study, in particular, the Casimir effect in order to show that these virtual photons should satisfy a Fermi-Dirac statistics, which implies a non-zero temperature of the vacuum state., Comment: 3 pages, revtex, no figures, 9 references. This work is a new version of a paper submitted for publication in a major journal of physics

We propose a realistic and nonlocal interpretation for quantum mechanics, which requires new mathematical, physical and philosophical foundations for space-time. Our theory violates Bell's inequality. We also discuss the cat paradox., Comment: 11 pages, no figures, 19 references, LaTeX

Quasi-set theory provides a mathematical background for dealing with collections of indistinguishable elementary particles. In this paper, we show how to obtain the quantum statistics into the scope of quasi-set theory and discuss the Helium atom, which represents the simplest example where indistinguishability plays an important role. A brief discussion about indistinguishability and interference is also presented as well as other related lines of work. One of the advantages of our approach is that one of the most basic principles of quantum theory, namely, the Indistinguishability Postulate, does not need to be assumed even implicetely in the axiomatic basis of quantum mechanics., Comment: 20 pages, no figures, one table

We have recently showed that it is possible to deal with collections of indistinguishable elementary particles (in the context of quantum mechanics) in a set-theoretical framework by using hidden variables, in a sense. In the present paper we use such a formalism as a model for quasi-set theory. Quasi-set theory, based on Zermelo-Fraenkel set theory, was developed for dealing with collections of indistinguishable but, in a sense, not identical objects., Comment: LaTeX, 14 pages, no figures, one table

An axiomatics for indistinguishability of elementary particles in terms of hidden variables is presented in a manner which depart from the standard approaches usually given to hidden variables. Quantum distribution functions are also discussed and some related lines of work are suggested., Comment: 17 pages, LaTeX format

We use a local theory of photons purely as particles to model the single-photon experiment proposed by Tan, Walls, and Collett. Like Tan et al. we are able to derive a violation of Bell's inequalities for photon counts coincidence measurements. Our local probabilistic theory does not use any specific quantum mechanical calculations., Comment: LaTeX, 11 pages, one figure (in LaTeX), submitted to Foundations of Physics Letters

We show one can use classical fields to modify a quantum optics experiment so that Bell's inequalities will be violated. This happens with continuous random variables that are local, but we need to use the correlation matrix to prove there can be no joint probability distribution of the observables., Comment: LaTeX, 10 pages, one figure (in LaTeX)

In previous works Suppes and de Barros used a pure particle model to derive interference effects, where individual photons have well-defined trajectories, and hence no wave properties. In the present paper we extend that description to account for the Casimir effect. We consider that the linear momentum $\sum\frac{1}{2}\hbar {\bf k}$ of the vacuum state in quantum electrodynamics corresponds to the linear momentum of virtual photons. The Casimir effect, in the cases of two parallel plates and the solid ball, is explained in terms of the pressure caused by the photons. Contrary to quantum electrodynamics, we assume a finite number of virtual photons., Comment: LaTeX file, 11 pages, no figures. Key words: photon, trajectories, QED, Casimir effect, quantum vacuum