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Randomizing a Model
"... A randomization of a first order structure M is a new structure with certain closure properties whose universe is a set K of "random elements" of M. Randomizations assign probabilities to sentences of the language of M with new constants from K. Our main theorem shows that all randomizations of M ..."
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A randomization of a first order structure M is a new structure with certain closure properties whose universe is a set K of "random elements" of M. Randomizations assign probabilities to sentences of the language of M with new constants from K. Our main theorem shows that all randomizations of M are models of the same first order theory T , which has a nice set of axioms and admits elimination of quantifiers. Moreover, the class of substructures of models of T is characterized by a natural set V of universal axioms of T , so that T is the model completion of V . 1 Introduction A common theme in mathematics is to start with a first order structure M, and introduce a new structure K which has a set K of "random elements" of M as a universe and which assigns probabilities to sentences of the language of M with new constants from K. There are several ways to do this; three wellknown examples will be given in this introduction, and many others will be given in Section 4. The aim of thi...
Tennenbaum’s Theorem for Models of Arithmetic
, 2006
"... This paper discusses Tennenbaum’s Theorem in its original context of models of arithmetic, which states that there are no recursive nonstandard models of Peano Arithmetic. We focus on three separate areas: the historical background to the theorem; an understanding of the theorem and its relationship ..."
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This paper discusses Tennenbaum’s Theorem in its original context of models of arithmetic, which states that there are no recursive nonstandard models of Peano Arithmetic. We focus on three separate areas: the historical background to the theorem; an understanding of the theorem and its relationship with the Gödel–Rosser Theorem; and extensions of Tennenbaum’s theorem to diophantine problems in models of arthmetic, especially problems concerning which diophantine equations have roots in some model of a given theory of arithmetic.
Generating Counterexamples for Structural Inductions by Exploiting Nonstandard Models
"... Abstract. Induction proofs often fail because the stated theorem is noninductive, in which case the user must strengthen the theorem or prove auxiliary properties before performing the induction step. (Counter)model finders are useful for detecting nontheorems, but they will not find any counterexa ..."
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Abstract. Induction proofs often fail because the stated theorem is noninductive, in which case the user must strengthen the theorem or prove auxiliary properties before performing the induction step. (Counter)model finders are useful for detecting nontheorems, but they will not find any counterexamples for noninductive theorems. We explain how to apply a wellknown concept from firstorder logic, nonstandard models, to the detection of noninductive invariants. Our work was done in the context of the proof assistant Isabelle/HOL and the counterexample generator Nitpick. 1
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."
Skolem and Gödel
, 1996
"... ... and Gödel (1906–1978) are the two greatest logicians of the century. Yet their styles, philosophies, and careers are strikingly different. Gödel had already published some of his great works and had become world renowned by the time he was 25 years of age. Skolem began to publish his important p ..."
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... and Gödel (1906–1978) are the two greatest logicians of the century. Yet their styles, philosophies, and careers are strikingly different. Gödel had already published some of his great works and had become world renowned by the time he was 25 years of age. Skolem began to publish his important papers only after he was 30, and his impact grew slowly over the years. Gödel was meticulous in writing for publication and published little after he reached 45. Skolem wrote informally, often even casually, continuing to publish into the last days of his life. Gödel was a wellknown absolutist and Platonist who had devoted much effort to studying and writing philosophy. Skolem was inclined to finitism and relativism, and rarely attempted to offer an articulate presentation of his coherent and fruitful philosophical viewpoint about the nature of mathematics and mathematical activity. Apart from mathematical logic, Gödel made contributions to the philosophy of mathematics and to fundamental physics. Skolem divided his work almost equally between logic and other parts of discrete mathematics, particularly algebra and number theory. For many years I have been deeply involved with Gödel’s work and his life. Even though I was for a long time intensely interested in Skolem’s work in logic and made a careful study of it in the sixties, since then I have not followed carefully the important applications and developments of Skolem’s ideas by many logicians. I know very little about his life and his work in fields other than logic. In my opinion, there is much room for interesting and instructive studies of Skolem’s work and his life. One attraction for me in coming here to give the
Realizability with a Local Operator of A.M. Pitts
, 2013
"... We study a notion of realizability with a local operator J which was first considered by A.M. Pitts in his thesis [7]. Using the SuslinKleene theorem, we show that the representable functions for this realizability are exactly the hyperarithmetical ( ∆ 1 1) functions. We show that there is a realiz ..."
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We study a notion of realizability with a local operator J which was first considered by A.M. Pitts in his thesis [7]. Using the SuslinKleene theorem, we show that the representable functions for this realizability are exactly the hyperarithmetical ( ∆ 1 1) functions. We show that there is a realizability interpretation of nonstandard arithmetic, which, despite its classical character, lives in a very nonclassical universe, where the Uniformity Principle holds and König’s Lemma fails. We conjecture that the local operator gives a useful indexing of the hyperarithmetical functions.
When is.999... less than 1?
"... We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is “an infinite number of 9s ” merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone’s “semicolon ” notation? Is it p ..."
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We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is “an infinite number of 9s ” merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone’s “semicolon ” notation? Is it possible to choose a canonical alternative interpretation? Should unital evaluation of the symbol.999... be inculcated in a prelimit teaching environment? The problem of the unital evaluation is hereby examined from the preR, prelim viewpoint of the student. 1.