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Bhupinder Singh Anand

Publications and source records attributed to Bhupinder Singh Anand.

At least 37 records · Page 2Linked to original sources
arXiv2005

Is the Halting problem effectively solvable non-algorithmically, and is the Goedel sentence in NP, but not in P?

We consider the thesis that an arithmetical relation, which holds for any, given, assignment of natural numbers to its free variables, is Turing-decidable if, and only if, it is the standard representation of a PA-provable formula. We show that, classically, such a thesis is, both, unverifiable and irrefutable, and, that it implies the Turing Thesis is false; that Goedel's arithmetical predicate R(x), treated as a Boolean function, is in the complexity class NP, but not in P; and that the Halting problem is effectively solvable, albeit not algorithmically.

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arXiv2005

An arguable inconsistency in ZF

Classical theory proves that every primitive recursive function is strongly representable in PA; that formal Peano Arithmetic, PA, and formal primitive recursive arithmetic, PRA, can both be interpreted in Zermelo-Fraenkel Set Theory, ZF; and that if ZF is consistent, then PA+PRA is consistent. We show that PA+PRA is inconsistent; it follows that ZF, too, is inconsistent.

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arXiv2004

Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?

Classical interpretations of Goedel's formal reasoning imply that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is essentially unverifiable. However, a language of general, scientific, discourse cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic is verifiably complete. We show how some paradoxical concepts of Quantum mechanics can be expressed, and interpreted, naturally under a constructive definition of mathematical truth.

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arXiv2003

Can Laplace's formula model a deterministic universe that is irreducibly probabilistic?

If we assume the Thesis that any classical Turing machine T, which halts on every n-ary sequence of natural numbers as input in a determinate time t(n), determines a PA-provable formula, whose standard interpretation is an n-ary arithmetical relation f(x1, ..., xn) that holds if, and only if, T halts, then we can define Laplace's formula recursively such that it can model the state of a deterministic quantum universe that is irreducibly probabilistic.

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arXiv2003

How definitive is the standard interpretation of Goedel's Incompleteness Theorem?

Standard interpretations of Goedel's "undecidable" proposition, [(Ax)R(x)], argue that, although [~(Ax)R(x)] is PA-provable if [(Ax)R(x)] is PA-provable, we may not conclude from this that [~(Ax)R(x)] is PA-provable. We show that such interpretations are inconsistent with a standard Deduction Theorem of first order theories.

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arXiv2003

Is the Halting probability a Dedekind real number?

In a recent historical overview, Cristian S. Calude, Elena Calude, and Solomon Marcus identify eight stages in the development of the concept of a mathematical proof in support of an ambitious conjecture: we can express classical mathematical concepts adequately only in a mathematical language in which both truth and provability are essentially unverifiable. In this paper we show, firstly, that the concepts underlying their thesis can, however, be interpreted constructively; and, secondly, that an implicit thesis in the authors' arguments implies that the probability of a given Turing machine halting on a given input cannot be expressed as a Dedekind real number.

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arXiv2003

Why we must heed Wittgenstein's "notorious paragraph"

We argue that, although Wittgenstein's reservations on Goedel's interpretation of his own formal reasoning are, indeed, of historical importance, the uneasiness that academicians and philosophers continue to sense, and express, over standard interpretations of Goedel's formal reasoning - even seventy years after the publication of his seminal 1931 paper - is of much greater significance, and relevance, to us today.

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arXiv2003

Are there parts of our arithmetical competence that no sound formal system can duplicate?

In 1995, David Chalmers opined as implausible that there may be parts of our arithmetical competence that no sound formal system could ever duplicate. We prove that the recursive number-theoretic relation x=Sb(y 19|Z(y)) - which is algorithmically verifiable since Goedel's recursive function Sb(y 19|Z(y)) is Turing-computable - cannot be "duplicated" in any consistent formal system of Arithmetic.

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arXiv2003

Is there a duality in the classical acceptance of non-constructive, foundational, concepts as axiomatic?

We consider a philosophical question that is implicit in Selmer Bringsjord's paper, "The narrational case against Church's Thesis": If, as Mendelson argues, the classically accepted definitions of foundational concepts such as "partial recursive function", "function", "(Tarskian) truth", "set" etc. are vague and imprecise - hence possibly non-constructive and intuitionistically objectionable - then replacing one non-constructive concept by another may be psychologically unappealing, but it should be meta-mathematically valid and acceptable.

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arXiv2003

Is there a "loophole" in Goedel's interpretation of his formal reasoning and its consequences?

We formally define a "mathematical object" and "set". We then argue that expressions such as "(Ax)F(x)", and "(Ex)F(x)", in an interpretation M of a formal theory P, may be taken to mean "F(x) is true for all x in M", and "F(x) is true for some x in M", respectively, if, and only if, the predicate letter "F" is a mathematical object in P. In the absence of a proof, the expressions "(Ax)F(x)", and "(Ex)F(x)", can only be taken to mean that "F(x) is true for any given x in M", and "It is not true that F(x) is false for any given x in M", respectively, indicating that the predicate "F(x)" is well-defined, and effectively decidable individually, for any given value of x, but that there may be no uniform effective method (algorithm) for such decidability. We show how some paradoxical concepts of Quantum Mechanics can then be expressed in a constructive interpretation of standard Peano's Arithmetic.

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arXiv2003

Three beliefs that lend illusory legitimacy to Cantor's diagonal argument

Whatever other beliefs there may remain for considering Cantor's diagonal argument as mathematically legitimate, there are three that, prima facie, lend it an illusory legitimacy; they need to be explicitly discounted appropriately. The first, Cantor's diagonal argument defines a non-countable Dedekind real number; the second, Goedel uses the argument to define a formally undecidable, but interpretively true, proposition; and the third, Turing uses the argument to define an uncomputable Dedekind real number.

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arXiv2003

Can Turing machines capture everything we can compute?

If we define classical foundational concepts constructively, and introduce non-algorithmic effective methods into classical mathematics, then we can bridge the chasm between truth and provability, and define computational methods that are not Turing-computable.

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arXiv2003

The formal roots of Platonism

We present some arguments for the thesis that a set-theoretic inspired faith, in the ability of intuitive truth to faithfully reflect relationships between elements of a Platonic universe, may be as misplaced as an assumption that such truth cannot be expressed in a constructive, and effectively verifiable, manner.

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arXiv2003

Can we express every transfinite concept constructively?

In a forthcoming book, professional computer scientist and physicist Paul Budnik presents an exposition of classical mathematical theory as the backdrop to an elegant thesis: we can interpret any model of a formal system of Peano Arithmetic in an appropriate, digital, computational language. In this paper we attempt - without addressing the question of whether or not Budnik succeeds in establishing his thesis convincingly - to identify dogmas of standard interpretations of classical mathematical theory that appear to be implicit in Budnik's exposition, and to correspond to them dogmas of a constructive interpretation of classical theory.

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arXiv2003

Reviewing Goedel's and Rosser's meta-reasoning of undecidability

I review the classical conclusions drawn from Goedel's meta-reasoning establishing an undecidable proposition GUS in standard PA. I argue that, for any given set of numerical values of its free variables, every recursive arithmetical relation can be expressed in PA by different, but formally equivalent, propositions. This asymmetry yields alternative Representation and Self-reference meta-Lemmas. I argue that Goedel's meta-reasoning can thus be expressed avoiding any appeal to the truth of propositions in the standard interpretation IA of PA. This now establishes GUS as decidable, and PA as omega-inconsistent. I argue further that Rosser's extension of Goedel's meta-reasoning involves an invalid deduction.

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arXiv2003

Omega-inconsistency in Goedel's formal system: a constructive proof of the Entscheidungsproblem

If we apply an extension of the Deduction meta-Theorem to Goedel's meta-reasoning of "undecidability", we can conclude that Goedel's formal system of Arithmetic is not omega-consistent. If we then take the standard interpretation "(Ax)(F(x)" of the PA-formula [(Ax)F(x)] to mean "There is a general, x-independent, routine to establish that F(x) holds for all x", instead of "F(x) holds for all x", it follows that a constructively interpreted omega-inconsistent system proves Hilbert's Entscheidungsproblem negatively.

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arXiv2003

Goedel's Incompleteness Theorems hold vacuously

In an earlier paper, "Omega-inconsistency in Goedel's formal system: a constructive proof of the Entscheidungsproblem" (math/0206302), I argued that a constructive interpretation of Goedel's reasoning establishes any formal system of Arithmetic as omega-inconsistent. It follows from this that Goedel's Theorem VI holds vacuously. In this paper I show that Goedel's Theorem XI essentially states that, if we assume there is a P-formula [Con(P)] whose standard interpretation is equivalent to the assertion "P is consistent", then [Con(P)] is not P-provable. I argue that there is no such formula.

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