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* Dauben, Joseph W. ‘The Trigonometric Background to Georg Cantor’s Theory of Sets’. Archive for History of Exact Sciences 7, no. 3 (1 January 1971): 181–216. doi:10.1007/BF00357216.
 
* Dauben, Joseph W. ‘The Trigonometric Background to Georg Cantor’s Theory of Sets’. Archive for History of Exact Sciences 7, no. 3 (1 January 1971): 181–216. doi:10.1007/BF00357216.
 
* Doyle, Peter G., and Cecil Qiu. ‘Division by Four’. arXiv:1504.01402 [math], 6 April 2015. http://arxiv.org/abs/1504.01402.
 
* Doyle, Peter G., and Cecil Qiu. ‘Division by Four’. arXiv:1504.01402 [math], 6 April 2015. http://arxiv.org/abs/1504.01402.
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== 노트 ==
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# Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.<ref name="ref_969a">[https://en.wikipedia.org/wiki/Set_theory Set theory]</ref>
 +
# The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s.<ref name="ref_969a" />
 +
# The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment.<ref name="ref_969a" />
 +
# Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams.<ref name="ref_969a" />
 +
# But not only was the basis of set theory shaken by Rusell's antinomy; logic itself was endangered.<ref name="ref_4973">[https://www.sciencedirect.com/topics/computer-science/set-theory Set Theory - an overview]</ref>
 +
# It is often said that set theory belongs to them simultaneously and forms their common link.<ref name="ref_4973" />
 +
# It was with Cantor 's work however that set theory came to be put on a proper mathematical basis.<ref name="ref_6e4f">[https://mathshistory.st-andrews.ac.uk/HistTopics/Beginnings_of_set_theory/ Set theory]</ref>
 +
# These papers contain Cantor 's first ideas on set theory and also important results on irrational numbers.<ref name="ref_6e4f" />
 +
# Inand Cantor published his final double treatise on sets theory.<ref name="ref_6e4f" />
 +
# It contains an introduction that looks like a modern book on set theory, defining set, subset, etc.<ref name="ref_6e4f" />
 +
# Set Theory is a branch of mathematics that investigates sets and their properties.<ref name="ref_47a8">[https://iep.utm.edu/set-theo/ Internet Encyclopedia of Philosophy]</ref>
 +
# The basic concepts of set theory are fairly easy to understand and appear to be self-evident.<ref name="ref_47a8" />
 +
# However, despite its apparent simplicity, set theory turns out to be a very sophisticated subject.<ref name="ref_47a8" />
 +
# Sections 1 and 2 below describe the “naïve” principles of set theory that were used and developed by Cantor.<ref name="ref_47a8" />
 +
# In set theory the natural numbers are defined as the finite ordinals.<ref name="ref_56ed">[https://plato.stanford.edu/entries/set-theory/basic-set-theory.html Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)]</ref>
 +
# The axioms of set theory imply the existence of a set-theoretic universe so rich that all mathematical objects can be construed as sets.<ref name="ref_c8d5">[https://plato.stanford.edu/entries/set-theory/ Set Theory (Stanford Encyclopedia of Philosophy)]</ref>
 +
# Also, the formal language of pure set theory allows one to formalize all mathematical notions and arguments.<ref name="ref_c8d5" />
 +
# In order to avoid the paradoxes and put it on a firm footing, set theory had to be axiomatized.<ref name="ref_c8d5" />
 +
# A property is given by a formula \(\varphi\) of the first-order language of set theory.<ref name="ref_c8d5" />
 +
# Set theory has its own notations and symbols that can seem unusual for many.<ref name="ref_1306">[https://www.mbacrystalball.com/blog/2015/10/09/set-theory-tutorial/ Problems, Formulas, Examples]</ref>
 +
# For this reason it is often said that set theory provides a foundation for mathematics.<ref name="ref_03cb">[https://warwick.ac.uk/fac/sci/maths/undergrad/ughandbook/year3/ma3h3/ MA3H3 Set Theory]</ref>
 +
# Unrestricted set formation leads to various paradoxes (Russell, Cantor, Burali-Forti), thereby motivating axiomatic set theory.<ref name="ref_03cb" />
 +
# Because of its abstract nature, the influence of set theory exists behind the scenes of many other branches of mathematics.<ref name="ref_13bf">[https://towardsdatascience.com/set-theory-history-overview-c98bac98f99c Set Theory — History & Overview. Part I — What Is Set Theory & Why Is It…]</ref>
 +
# … every mathematician who wishes to refresh his knowledge of set theory will read it with pleasure.<ref name="ref_9e90">[https://www.springer.com/gp/book/9783540440857 Set Theory - The Third Millennium Edition, revised and expanded]</ref>
 +
# Thomas Jech’s Set Theory contains the most comprehensive treatment of the subject in any one volume.<ref name="ref_9e90" />
 +
# These notes for a graduate course in set theory are on their way to becoming a book.<ref name="ref_bba6">[http://www.math.toronto.edu/weiss/set_theory.html Downloading "Set Theory"]</ref>
 +
# Unfortunately, like several other branches of mathematics, set theory has its own language which you need to understand.<ref name="ref_0a79">[https://www.skillsyouneed.com/num/set-theory.html Simple Set Theory]</ref>
 +
# The set theory is very important in order to understand data and databases.<ref name="ref_4991">[https://www.sqlshack.com/learn-sql-set-theory/ Learn SQL: Set Theory]</ref>
 +
# If we’re talking from the perspective of the set theory, you can look at each table as one set.<ref name="ref_4991" />
 +
# We talked a lot about the set theory so far, and now it’s time for some practice.<ref name="ref_4991" />
 +
# It is not possible to discuss functions sensibly without using the language and ideas of elementary set theory.<ref name="ref_6fb1">[https://amsi.org.au/ESA_Senior_Years/SeniorTopic2/2b/2b_2content_1.html Set theory]</ref>
 +
# Structural set theory thus looks very much like type theory.<ref name="ref_40fc">[https://ncatlab.org/nlab/show/structural+set+theory structural set theory in nLab]</ref>
 +
# These are the basic ideas behind set theory.<ref name="ref_4e84">[https://www.storyofmathematics.com/sets-set-theory Sets & Set Theory]</ref>
 +
# A closely related branch is Set Theory, which provides a simple, uniform background in which to do virtually all mainstream mathematics.<ref name="ref_1b4a">[https://math.vcu.edu/research/logic-and-set-theory/ Virginia Commonwealth University]</ref>
 +
# Thus, here we briefly review some basic concepts from set theory that are used in this book.<ref name="ref_4131">[https://www.probabilitycourse.com/chapter1/1_2_0_review_set_theory.php Set Theory Review]</ref>
 +
# The second part of the course will be devoted to more advanced topics in set theory.<ref name="ref_2d0c">[https://www.math.uni-hamburg.de/home/loewe/2018-19-I/mastermath_settheory_2018_content.html MasterMath Set Theory 2018]</ref>
 +
# As a consequence, no prior knowledge of axiomatic set theory is assumed.<ref name="ref_2d0c" />
 +
# The hybrid nature of set theory as a mathematical research area and the foundations of mathematics.<ref name="ref_2d0c" />
 +
# Directed graphs as models of the language of set theory.<ref name="ref_2d0c" />
 +
# There are a number of different versions of set theory, each with its own rules and axioms.<ref name="ref_7819">[https://mathworld.wolfram.com/SetTheory.html Set Theory -- from Wolfram MathWorld]</ref>
 +
# Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo.<ref name="ref_7547">[https://www.elsevier.com/books/foundations-of-set-theory/fraenkel/978-0-7204-2270-2 Foundations of Set Theory, Volume 67]</ref>
 +
# This book provides an introduction to axiomatic set theory and descriptive set theory.<ref name="ref_90c9">[https://www.worldscientific.com/worldscibooks/10.1142/11324 Set Theory and Foundations of Mathematics: An Introduction to Mathematical Logic]</ref>
 +
===소스===
 +
<references />

2020년 12월 17일 (목) 05:16 기준 최신판

  • Meyries, Martin. “Infinity - A Simple, but Not Too Simple Introduction.” arXiv:1506.06319 [math], June 21, 2015. http://arxiv.org/abs/1506.06319.
  • Dauben, Joseph W. ‘The Trigonometric Background to Georg Cantor’s Theory of Sets’. Archive for History of Exact Sciences 7, no. 3 (1 January 1971): 181–216. doi:10.1007/BF00357216.
  • Doyle, Peter G., and Cecil Qiu. ‘Division by Four’. arXiv:1504.01402 [math], 6 April 2015. http://arxiv.org/abs/1504.01402.

노트

  1. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.[1]
  2. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s.[1]
  3. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment.[1]
  4. Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams.[1]
  5. But not only was the basis of set theory shaken by Rusell's antinomy; logic itself was endangered.[2]
  6. It is often said that set theory belongs to them simultaneously and forms their common link.[2]
  7. It was with Cantor 's work however that set theory came to be put on a proper mathematical basis.[3]
  8. These papers contain Cantor 's first ideas on set theory and also important results on irrational numbers.[3]
  9. Inand Cantor published his final double treatise on sets theory.[3]
  10. It contains an introduction that looks like a modern book on set theory, defining set, subset, etc.[3]
  11. Set Theory is a branch of mathematics that investigates sets and their properties.[4]
  12. The basic concepts of set theory are fairly easy to understand and appear to be self-evident.[4]
  13. However, despite its apparent simplicity, set theory turns out to be a very sophisticated subject.[4]
  14. Sections 1 and 2 below describe the “naïve” principles of set theory that were used and developed by Cantor.[4]
  15. In set theory the natural numbers are defined as the finite ordinals.[5]
  16. The axioms of set theory imply the existence of a set-theoretic universe so rich that all mathematical objects can be construed as sets.[6]
  17. Also, the formal language of pure set theory allows one to formalize all mathematical notions and arguments.[6]
  18. In order to avoid the paradoxes and put it on a firm footing, set theory had to be axiomatized.[6]
  19. A property is given by a formula \(\varphi\) of the first-order language of set theory.[6]
  20. Set theory has its own notations and symbols that can seem unusual for many.[7]
  21. For this reason it is often said that set theory provides a foundation for mathematics.[8]
  22. Unrestricted set formation leads to various paradoxes (Russell, Cantor, Burali-Forti), thereby motivating axiomatic set theory.[8]
  23. Because of its abstract nature, the influence of set theory exists behind the scenes of many other branches of mathematics.[9]
  24. … every mathematician who wishes to refresh his knowledge of set theory will read it with pleasure.[10]
  25. Thomas Jech’s Set Theory contains the most comprehensive treatment of the subject in any one volume.[10]
  26. These notes for a graduate course in set theory are on their way to becoming a book.[11]
  27. Unfortunately, like several other branches of mathematics, set theory has its own language which you need to understand.[12]
  28. The set theory is very important in order to understand data and databases.[13]
  29. If we’re talking from the perspective of the set theory, you can look at each table as one set.[13]
  30. We talked a lot about the set theory so far, and now it’s time for some practice.[13]
  31. It is not possible to discuss functions sensibly without using the language and ideas of elementary set theory.[14]
  32. Structural set theory thus looks very much like type theory.[15]
  33. These are the basic ideas behind set theory.[16]
  34. A closely related branch is Set Theory, which provides a simple, uniform background in which to do virtually all mainstream mathematics.[17]
  35. Thus, here we briefly review some basic concepts from set theory that are used in this book.[18]
  36. The second part of the course will be devoted to more advanced topics in set theory.[19]
  37. As a consequence, no prior knowledge of axiomatic set theory is assumed.[19]
  38. The hybrid nature of set theory as a mathematical research area and the foundations of mathematics.[19]
  39. Directed graphs as models of the language of set theory.[19]
  40. There are a number of different versions of set theory, each with its own rules and axioms.[20]
  41. Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo.[21]
  42. This book provides an introduction to axiomatic set theory and descriptive set theory.[22]

소스

  1. ↑ 1.0 1.1 1.2 1.3 Set theory
  2. ↑ 2.0 2.1 Set Theory - an overview
  3. ↑ 3.0 3.1 3.2 3.3 Set theory
  4. ↑ 4.0 4.1 4.2 4.3 Internet Encyclopedia of Philosophy
  5. ↑ Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)
  6. ↑ 6.0 6.1 6.2 6.3 Set Theory (Stanford Encyclopedia of Philosophy)
  7. ↑ Problems, Formulas, Examples
  8. ↑ 8.0 8.1 MA3H3 Set Theory
  9. ↑ Set Theory — History & Overview. Part I — What Is Set Theory & Why Is It…
  10. ↑ 10.0 10.1 Set Theory - The Third Millennium Edition, revised and expanded
  11. ↑ Downloading "Set Theory"
  12. ↑ Simple Set Theory
  13. ↑ 13.0 13.1 13.2 Learn SQL: Set Theory
  14. ↑ Set theory
  15. ↑ structural set theory in nLab
  16. ↑ Sets & Set Theory
  17. ↑ Virginia Commonwealth University
  18. ↑ Set Theory Review
  19. ↑ 19.0 19.1 19.2 19.3 MasterMath Set Theory 2018
  20. ↑ Set Theory -- from Wolfram MathWorld
  21. ↑ Foundations of Set Theory, Volume 67
  22. ↑ Set Theory and Foundations of Mathematics: An Introduction to Mathematical Logic
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