집합론

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