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Large-Scale Mathematical Formalization Projects: Coq, Implementation of Mathematics in Set Theory, Metamath, Mizar System, Nicolas Bourbaki, Principia

Large-Scale Mathematical Formalization Projects: Coq, Implementation of Mathematics in Set Theory, Metamath, Mizar System, Nicolas Bourbaki, Principia


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About the Book

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 24. Chapters: Coq, Implementation of mathematics in set theory, Metamath, Mizar system, Nicolas Bourbaki, Principia Mathematica. Excerpt: This article examines the implementation of mathematical concepts in set theory. The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC (the dominant set theory) and in NFU, the version of Quine's New Foundations shown to be consistent by R. B. Jensen in 1969 (here understood to include at least axioms of Infinity and Choice). What is said here applies also to two families of set theories: on the one hand, a range of theories including Zermelo set theory near the lower end of the scale and going up to ZFC extended with large cardinal hypotheses such as "there is a measurable cardinal"; and on the other hand a hierarchy of extensions of NFU which is surveyed in the New Foundations article. These correspond to different general views of what the set-theoretical universe is like, and it is the approaches to implementation of mathematical concepts under these two general views that are being compared and contrasted. It is not the primary aim of this article to say anything about the relative merits of these theories as foundations for mathematics. The reason for the use of two different set theories is to illustrate that multiple approaches to the implementation of mathematics are feasible. Precisely because of this approach, this article is not a source of "official" definitions for any mathematical concept. The following sections carry out certain constructions in the two theories ZFC and NFU and compare the resulting implementations of certain mathematical structures (such as the natural numbers). Mathematical theories prove theorems (and nothing else). So saying that a theory allows the construction of a certain object means that it is a theorem of that theory that that object exists. This is a statement about a definition of the form "the x such that exists," where is a formula of our language: the theory proves the existence of "the x such that " just in case it is a theorem that "there is one and only one x such that ." (S


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Product Details
  • ISBN-13: 9781156350027
  • Publisher: Books LLC, Wiki Series
  • Publisher Imprint: Books LLC, Wiki Series
  • Height: 246 mm
  • No of Pages: 26
  • Spine Width: 1 mm
  • Weight: 68 gr
  • ISBN-10: 1156350026
  • Publisher Date: 23 Apr 2013
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Sub Title: Coq, Implementation of Mathematics in Set Theory, Metamath, Mizar System, Nicolas Bourbaki, Principia
  • Width: 189 mm


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