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36
Naming and Diagonalization, from Cantor to Gödel to Kleene
 in Logic Journal of the IGPL, 22 pages, and on Gaifman’s website
, 2006
"... Gödel’s incompleteness results apply to formal theories for which syntactic constructs can be given names, in the same language, so that some basic syntactic operations are representable in the theory. It is now customary to derive these results from the fixed point theorem (also known as the reflec ..."
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Gödel’s incompleteness results apply to formal theories for which syntactic constructs can be given names, in the same language, so that some basic syntactic operations are representable in the theory. It is now customary to derive these results from the fixed point theorem (also known as the reflection theorem), which asserts the existence of sentences that “speak about themselves”. Let T be the theory and, for each wff φ, letpφqbe the term that serves as its name. Then the theorem says that, for any wff α(v) (with one free variable), there exists a sentence β for which: T ` β ↔ α(pβq) β is sometimes called the fixed point of α(v). All that is needed for the fixed point theorem is that the diagonal function, the one that maps each φ(v) toφ(p(φ(v)q)), be representable in T. The construction of β is more transparent if we assume that the functions is represented by a term of the language, diag(x). This means that the following holds for each φ(v): T ` diag(pφ(v)q) =pφ(pφ(v)q)q (Here ‘= ’ is the equality sign of the formal language; we use it also to denote equality in our metalanguage.) In other words, we can prove in T, for each particular argument, what the
BEYOND UNDECIDABLE
, 2006
"... Abstract. The predicate complementary to the wellknown Gödel’s provability predicate is defined. From its recursiveness new consequences concerning the incompleteness argumentation are drawn and extended to new results of consistency, completeness and decidability with regard to Peano Arithmetic an ..."
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Abstract. The predicate complementary to the wellknown Gödel’s provability predicate is defined. From its recursiveness new consequences concerning the incompleteness argumentation are drawn and extended to new results of consistency, completeness and decidability with regard to Peano Arithmetic and the first order predicate calculus.
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, that the only numeric functions that can be calculated by effective means are the recursive ones (which are the same, extensionally, as the Turingcomputable numeric functions). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
Historical Projects in Discrete Mathematics and Computer Science
"... A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itse ..."
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A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development
Length Of Polynomial Ascending Chains And Primitive Recursiveness
, 1992
"... In a polynomial ring K[X 1 ; : : : ; Xn ] over a field, let I 0 ae I 1 ae \Delta \Delta \Delta ae I s be a strictly ascending chain of ideals, with the condition that every I i can be generated by elements of degree not greater than f(i). A. Seidenberg showed that there is a bound on the length s ..."
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In a polynomial ring K[X 1 ; : : : ; Xn ] over a field, let I 0 ae I 1 ae \Delta \Delta \Delta ae I s be a strictly ascending chain of ideals, with the condition that every I i can be generated by elements of degree not greater than f(i). A. Seidenberg showed that there is a bound on the length s of such a chain depending only on n and f , which is recursive in f for every n and primitive recursive in f for n = 2. In this paper we give a better bound, expressed in a rather simple way in terms of f , which is attained when f is an increasing function. We prove that it is primitive recursive in f for all n. We also show that, on the contrary, there is no bound which is primitive recursive in n in general.
Church Without Dogma: Axioms for computability
"... Abstract. Church’s and Turing’s theses assert dogmatically that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computabilty. I present analyses of calculability that are embedded in a rich historical and philosophical context, lead to pre ..."
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Abstract. Church’s and Turing’s theses assert dogmatically that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computabilty. I present analyses of calculability that are embedded in a rich historical and philosophical context, lead to precise concepts, and dispense with theses. To investigate effective calculability is to analyze processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work of Gandy, I formulate boundedness and locality conditions for two types of calculators, namely, human computing agents and mechanical computing devices (or discrete machines). The distinctive feature of the latter is that they can carry out parallel computations. Representing human and machine computations by discrete dynamical systems, the boundedness and locality conditions can be captured through axioms for Turing computors and Gandy machines; models of
Extension of embeddings in the recursively enumerable degrees
"... The extension of embeddings problem for the recursively enumerable degrees R = (R;!; 0; 0 0) asks for given finite partially ordered sets P ` Q with least and greatest elements, whether every embedding of P into R can be extended to an embedding of Q into R. Many of the landmark theorems giving an a ..."
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The extension of embeddings problem for the recursively enumerable degrees R = (R;!; 0; 0 0) asks for given finite partially ordered sets P ` Q with least and greatest elements, whether every embedding of P into R can be extended to an embedding of Q into R. Many of the landmark theorems giving an algebraic insight into R assert either extension or nonextension of embeddings. We extend, strengthen, and unify these results and their proofs to produce complete and complementary criteria and techniques to analyze instances of extension and nonextension. We conclude that the full extension of embeddings problem is decidable.
The ChurchTuring Thesis as an Immature Form of the ZuseFredkin Thesis (More Arguments in Support of the “Universe as a Cellular Automaton” Idea)
"... In [1] we have shown a strong argument in support of the "Universe as a computer " idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only "some kind of computer", but also a concrete computational model known as a ..."
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In [1] we have shown a strong argument in support of the "Universe as a computer " idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only "some kind of computer", but also a concrete computational model known as a "cellular automaton".
Computation vs. Information Processing: Why Their Difference Matters to Cognitive Science
"... Since the cognitive revolution, it’s become commonplace that cognition involves both computation and information processing. Is this one claim or two? Is computation the same as information processing? The two terms are often used interchangeably, but this usage masks important differences. In this ..."
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Since the cognitive revolution, it’s become commonplace that cognition involves both computation and information processing. Is this one claim or two? Is computation the same as information processing? The two terms are often used interchangeably, but this usage masks important differences. In this paper, we distinguish information processing from computation and examine some of their mutual relations, shedding light on the role each can play in a theory of cognition. We recommend that theorists of cognition be explicit and careful in choosing 1 notions of computation and information and connecting them together. Much confusion can be avoided by doing so.
DISCONTINUOUS PHENOMENA
"... ABSTRACT. We discuss the relationship between discontinuity and definability in the Turing degrees, with particular reference to degree invariant solutions to Post’s Problem. Proofs of new results concerning definability in lower cones are outlined. 1. ..."
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ABSTRACT. We discuss the relationship between discontinuity and definability in the Turing degrees, with particular reference to degree invariant solutions to Post’s Problem. Proofs of new results concerning definability in lower cones are outlined. 1.