Research Topic in Graph Theory or Non-Well-Founded Set Theory

In summary, Dragonfall chose to do research on Fourier Analysis and found it to be a good subject to study.
  • #1
Dragonfall
1,030
4
I'm doing to come up with a subject in either of them to do either an "independent study" or "project" on, the former is a course which simply requires you to learn the subject and the latter is "independent study" + a x-page paper. Unfortunately I don't know either subject too well so I can't come up with anything specific enough. Can anyone make a suggestion?
 
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  • #2
non well-founded set theory and use it to build up the foundation for non-standard analysis because I found graph theory to be sort of boring. But, some think non-standard analysis is virtually useless as history has never provided a situation when a theorem in standard analysis could only be proven by non-standard analysis techniques. Nevertheless, it would be cool to really understand it if you have an interest in analysis and if you have a interest in mathematical logic then this is also obviously a good choice (but in that regard so is graph theory as it is an example of descriptive set theory). As far as I know one of the most basic objects in model theory is an Ultrafilter and I am sure you would get a really good understanding of these in non well-founded set theory. Depends on your other interest really.
 
  • #3
That's the problem with non-well-founded set theory: nobody in my department seems to care, or want to do it. The problem with graph theory though is that since I know very little of it, the questions I have are either well-known hard problems or easy to solve problems, nothing in the middle that I could work on.
 
  • #4
well, if you ask for material on non well founded set theory, then there's jonh barwise's vicoious circle for first glance (you can view it for free from stanford), and you can search tom forster from cambridge university, I think he's a major reasercher is in this field.
 
  • #5
I can't find the book by Barwise that you said. Are you sure you have the title right?

I've decided to do it on non-well-founded set theory (my other alternative is Fourier Analysis, but I decided against it). Unfortunately there is only ONE book which deals with the subject, namely "Non-Well-Founded sets" by Azcel. Does anyone know of other books (preferably textbooks) which deal with the subject?
 
  • #6
JON BARWISE AND LAWRENCE MOSS,
"VICIOUS CIRCLES. ON THE MATHEMATICS OF NON-WELLFOUNDED PHENOMENA."
Stanford: CSLI Publications, 1996
Lecture Notes Number 60
x + 390 pp. ISBN 1-57586-009-0 (hardback) or 1-57586-008-2 (paperback)

There's a review of it captured on Project Euclid from "Modern Logic", Volume 8, Number 1/2; don't know if this link will work for you : http://projecteuclid.org/DPubS/Repo...ew=body&id=pdf_1&handle=euclid.rml/1081878069
.
 
  • #7
Dragonfall, what did you end up doing and can you share the results of your research?
 

1. What is graph theory?

Graph theory is a branch of mathematics that studies the properties and relationships of graphs, which are mathematical structures used to model pairwise relationships between objects.

2. How is graph theory used in real-world applications?

Graph theory has many practical applications, such as in computer science, social networks, transportation networks, and operations research. It can be used to model and analyze complex systems, identify patterns and connections, and optimize networks and processes.

3. What is non-well-founded set theory?

Non-well-founded set theory is a mathematical theory that extends traditional set theory by allowing sets to contain themselves as elements. This allows for the creation of infinite sets and can be useful in modeling certain phenomena, such as circular definitions and self-referential structures.

4. How does non-well-founded set theory differ from traditional set theory?

The main difference between non-well-founded set theory and traditional set theory is that non-well-founded sets can contain themselves as elements, while traditional sets cannot. This allows for the study of more complex and abstract structures, but also introduces challenges and paradoxes that do not arise in traditional set theory.

5. What are some applications of non-well-founded set theory?

Non-well-founded set theory has been used in various areas of mathematics, such as topology, logic, and category theory. It has also been applied in computer science, particularly in the study of programming languages and formal verification. Additionally, it has been used in the study of philosophical concepts, such as self-reference and infinite regress.

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