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Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
Contents
, 2008
"... We present the Curry–Howard correspondence for constructive logic via natural deduction, typed λcalculus and cartesian closed categories. We then examine how the correspondence may be extended to classical logic and nonconstructive proofs, and discuss some of the problems and questions ..."
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We present the Curry–Howard correspondence for constructive logic via natural deduction, typed λcalculus and cartesian closed categories. We then examine how the correspondence may be extended to classical logic and nonconstructive proofs, and discuss some of the problems and questions
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.