Non-standard models were introduced by Skolem, first for set theory, then for Peano arithmetic. In the former, Skolem found support for an anti-realist view of absolutely uncountable sets. But in the latter he saw evidence for the impossibility of capturing the intended interpretation by purely deductive methods. In the history of mathematics the concept of a nonstandard model is new. An analysis of some major innovations–the discovery of irrationals, the use of negative and complex numbers, the modern concept (...) of function, and non-Euclidean geometry–reveals them as essentially different from the introduction of non-standard models. Yet, non-Euclidean geometry, which is discussed at some length, is relevant to the present concern; for it raises the issue of intended interpretation. The standard model of natural numbers is the best candidate for an intended interpretation that cannot be captured by a deductive system. Next, I suggest, is the concept of a wellordered set, and then, perhaps, the concept of a constructible set. One may have doubts about a realistic conception of the standard natural numbers, but such doubts cannot gain support from non-standard models. Attempts to utilize non-standard models for an anti-realist position in mathematics, which appeal to meaning-as-use, or to arguments of the kind proposed by Putnam, fail through irrelevance, or lead to incoherence. Robinson’s skepticism, on the other hand, is a coherent position, though one that gives up on providing a detailed philosophical account. The last section enumerates various uses of non-standard models. (shrink)

I review some theoretical ideas in cosmology different from the standard “Big Bang”: the quasi-steady state model, the plasma cosmology model, non-cosmological redshifts, alternatives to non-baryonic dark matter and/or dark energy, and others. Cosmologists do not usually work within the framework of alternative cosmologies because they feel that these are not at present as competitive as the standard model. Certainly, they are not so developed, and they are not so developed because cosmologists do not work on them. It (...) is a vicious circle. The fact that most cosmologists do not pay them any attention and only dedicate their research time to the standard model is to a great extent due to a sociological phenomenon . We might well wonder whether cosmology, our knowledge of the Universe as a whole, is a science like other fields of physics or a predominant ideology. (shrink)

In his doctor's thesis [1], Henkin has shown that if a formal logic is consistent, and sufficiently complex, then it must admit a non-standard model. In particular, he showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of ≦.For (...) the purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either: The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model. That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation ≦. That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation ≦. (shrink)

In the context of life sciences, we are constantly confronted with information that possesses precise semantic values and appears essentially immersed in a specific evolutionary trend. In such a framework, Nature appears, in Monod’s words, as a tinkerer characterized by the presence of precise principles of self-organization. However, while Monod was obliged to incorporate his brilliant intuitions into the framework of first-order cybernetics and a theory of information with an exclusively syntactic character such as that defined by Shannon, research advances (...) in recent decades have led not only to the definition of a second-order cybernetics but also to an exploration of the boundaries of semantic information. As H. Atlan states, on a biological level "the function self-organizes together with its meaning". Hence the need to refer to a conceptual theory of complexity and to a theory of self-organization characterized in an intentional sense. There is also a need to introduce, at the genetic level, a distinction between coder and ruler as well as the opportunity to define a real software space for natural evolution. The recourse to non-standard model theory, the opening to a new general semantics, and the innovative definition of the relationship between coder and ruler can be considered, today, among the most powerful theoretical tools at our disposal in order to correctly define the contours of that new conceptual revolution increasingly referred to as metabiology. This book focuses on identifying and investigating the role played by these particular theoretical tools in the development of this new scientific paradigm. Nature "speaks" by means of mathematical forms: we can observe these forms, but they are, at the same time, inside us as they populate our organs of cognition. In this context, the volume highlights how metabiology appears primarily to refer to the growth itself of our instruments of participatory knowledge of the world. (shrink)

In response to Bhupinder Singh Anand''s article CAN WE REALLY FALSIFY TRUTH BY DICTAT? in THE REASONER II, 1, January 2008,that denies the existence of nonstandard models of Peano Arithmetic, we prove from Compactness the existence of such models.

A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a “geometric technique” used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman’s theorem on the existence of rank-initial embeddings between countable non- (...) class='Hi'>standard models of the fragment \ + \-Separation of \; and Gaifman’s technique of iterated ultrapowers is employed to show that any countable model of \ can be elementarily rank-end-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of “strong rank-cut” is characterized in terms of the theory \, and in terms of fixed-point sets of self-embeddings. (shrink)

The existence of non-standard numbers in first-order arithmetics is a semantic obstacle for modelling our arithmetical skills. This article argues that so far there is no adequate approach to overcome such a semantic obstacle, because we can also find out, and deal with, non-standard elements in Turing machines.

We give an elementary introduction to non-standard analysis and its applications to the theory of stochastic processes. This is based on a joint book with J. E. Fenstad, R. Høegh-Krohn and T. Lindstrøm. In particular we give a discussion of an hyperfinite theory of Dirichlet forms with applications to the study of the Hamiltonian for a quantum mechanical particle in the potential created by a polymer. We also discuss new results on the existence of attractive polymer measures in dimension (...) d ≤ 5, with applications to the (φ 1 2 φ 2 2 )d-mode1 of interacting quantum fields. (shrink)

This paper provides an explicit description of a model for intuitionistic non-standard arithmetic, which can be formalized in a constructive metatheory without the axiom of choice.

This thesis studies collider phenomenology of physics beyond the Standard Model at the Large Hadron Collider (LHC). It also explores in detail advanced topics related to Higgs boson and supersymmetry - one of the most exciting and well-motivated streams in particle physics. In particular, it finds a very large enhancement of multiple Higgs boson production in vector-boson scattering when Higgs couplings to gauge bosons differ from those predicted by the Standard Model. The thesis demonstrates that due to the (...) loss of unitarity, the very large enhancement for triple Higgs boson production takes place. This is a truly novel finding. The thesis also studies the effects of supersymmetric partners of top and bottom quarks on the Higgs production and decay at the LHC, pointing for the first time to non-universal alterations for two main production processes of the Higgs boson at the LHC-vector boson fusion and gluon-gluon fusion. Continuing the exploration of Higgs boson and supersymmetry at the LHC, the thesis extends existing experimental analysis and shows that for a single decay channel the mass of the top quark superpartner below 175 GeV can be completely excluded, which in turn excludes electroweak baryogenesis in the Minimal Supersymmetric Model. This is a major new finding for the HEP community. This thesis is very clearly written and the introduction and conclusions are accessible to a wide spectrum of readers. (shrink)

We propose a set theory, called NRFST, in which the Cantorian axiom of infinity is negated, and a new notion of infinity is introduced via non standard methods, i. e. via adequate notions of standard and internal, two unary predicates added to the language of ZF. After some initial results on NRFST, we investigate its relative consistency with respect to ZF and Kawai's WNST.

In this paper, we show within RCA 0 that weak Konig's lemma is necessary and sufficient to prove that any (separable) compact group has a Haar measure. Within WKL 0 , a Haar measure is constructed by a non-standard method based on a fact that every countable non-standard model of WKL 0 has a proper initial part isomorphic to itself [10].

This new edition of work that has evolved over the past seven years completes the derivation of the form of The Standard Model from quantum theory and the extension of the Theory of Relativity to superluminal transformations. The much derided form of The Standard Model is established from a consideration of Lorentz and superluminal relativistic space-time transformations. So much so that other approaches to elementary particle theory pale in comparison. In previous work color SU(3) was derived from space-time (...) considerations. This book shows that the SU(2) U(1) Weak Interaction sector follows directly from the extension of Lorentz transformations to superluminal (faster-than-light) transformations. In fact, ElectroWeak symmetry is shown to have an SU(2) U(1) U(1) symmetry that naturally includes WIMPs (Weakly Interacting Massive Particles), a candidate for Dark Matter. Thus the form of the Standard Model is dictated by space-time considerations fulfilling a dream of Einstein. Using a new matrix formulation of Logic - Operator Logic - we show that Dirac-like equations can be constructed. This edition also derives broken SU(4) four fermion generations as well as a non-Higgsian (Dimensional Mass Generation) derivation of fermion and ElectroWeak gauge boson masses from GL(4). The program of the book is completed by reprinting the book Quantum Theory of the Third Kind, which describes Two-Tier quantum field theory. This theory yields finite results (No Feynman diagram calculations diverge!) in The Standard Model and Quantum Gravity calculations to all orders in perturbation theory. Lastly, the book also contains a description of Operator Logic, Probabilistic Operator Logic, Quantum Operator Logic (for q-number statements). A comprehensive derivation of the form of The Standard Model from geometric considerations. (shrink)

It is usually accepted in the literature that negation is a contradictory-forming operator and that two statements are contradictories if and only if it is logically impossible for both to be true and logically impossible for both to be false. These two premises have been used by Hartley Slater [Slater, 1995] to argue that paraconsistent negation is not a “real” negation because a sentence and its paraconsistent negation can be true together. In this paper we claim that a counterpart of (...) Slater´s argument can be directed against the negation operator of classical logic. Carnap’s discovery that there are models of classical propositional logic with non-standard or non-normal interpretations of the connectives will be used to build such an argument. One such non-normal valuation which can be added to the set of classically admissible valuations without altering the set of theorems or the set of valid consequences assigns true to every well-formed formula and, therefore, assigns a designated value to every formula and its negation. We ponder the consequences of these arguments for the claims that paraconsistent negations are not genuine negations and that the negation of classical logic is a contradictory-forming operator. (shrink)

In this paper, I argue that the distinction between standard and non-standard pragmatic implications, originally used to differentiate among types of conversational implicatures, applies to the family of contents—traditionally referred to as ‘presuppositions’—that exhibit projective behaviour. Following the scholars working within the Question Under Discussion model of communication, I distinguish between two types of projective implications: suppositions and presuppositions narrowly construed. Next, I identify two rules of appropriateness that govern the use of, respectively, supposition-triggering and presupposition-triggering expressions. Finally, (...) I argue that the ostentatious violation of the rules in question gives rise to non-standard projective implications, whereas their observance results in standard suppositions and presuppositions; I also use the idea of discourse coherence to develop a sketchy account of the mechanisms underlying the functioning of non-standard projective implications. (shrink)

We describe an explicit construction of algebraic models for theories with non-standard elements either with classical or constructive logic. The corresponding truthvalue algebra in our construction is a complete algebra of subsets of some concrete decidable set. This way we get a quite finitistic notion of true which reflects a notion of the deducibility of a given theory. It enables us to useconstructive, proof-theoretical methods for theories with non-standard elements. It is especially useful in the case of (...) theories with constructive logic where algorithmic properties are essential. (shrink)

This paper presents an analysis of the sorites paradox for collective nouns and gradable adjectives within the framework of classical logic. The paradox is explained by distinguishing between qualitative and quantitative representations. This distinction is formally represented by the use of a different mathematical model for each type of representation. Quantitative representations induce Archimedean models, but qualitative representations induce non-Archimedean models. By using a non-standard model of \ called \, which contains infinite and infinitesimal numbers, the two (...) paradoxes are shown to have distinct structures. The sorites paradox for collective nouns arises from the use of infinite numbers, whereas the sorites paradox for gradable adjectives arises from the use of infinitesimal numbers. Each paradox can be traced to a different source of vagueness. The sorites paradox for collective nouns is caused by \, and the sorites paradox for gradable adjectives is caused by \ \. If correct, this analysis implies that infinite and infinitesimal numbers are cognitively real, and that they play a role in the semantic interpretation of natural language. (shrink)

In ${\mathbf{H}}$ , a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of ${\mathbf{H}}$ , and we prove an analogy of Hájek’s theorem with a very simple procedure.

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...) teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

We prove that every countable non-standard model of WKL0 has a proper initial part isomorphic to itself. This theorem enables us to carry out non-standard arguments over WKL0.