In this course, we have used the weekly forum discussions to delve into Set Theory’s applications, asking such questions as: How is Set Theory used in the different branches of mathematics? Is it a common or uncommon technique? Is it used to cut complexities of arguments? Increase the transparency of proofs? Solve problems not otherwise easily (or time effectively) solvable? Unfortunately, the weekly forum discussions by their nature have provided only limited opportunity to learn about how Set Theory is used in different branches of mathematical sciences.

With that in mind, for this writing project we are going to delve in more depth into this issue.

Please choose one Set Theory related topic. You can choose any of the topics we’ve covered in this course, or even a topic that we did not cover. As long as it relates to Set Theory, it is fine. For a quick list of the topics we’ve covered, just look at the titles of the subsections of each chapter in the textbook or supplementary reading material provided throughout the course.

Then pair your chosen topic with an example from various branches of mathematical sciences of how that topic can be applied to solve problems. Examples include (but are by no means limited to):

- Theory of Syntactic Structures and Languages
- Computability of Undecidability
- Proof Theory
- Graphs
- Trees
- Model Theory

There are numerous examples of Set Theory applications scattered throughout our textbook: in the first introductory pages of each chapter, in selected chapter sections (e.g., 1.10), and contained within the description of many homework problems located throughout the chapter. You can also find examples at any of the websites mentioned in the weekly forum descriptions, or at websites you research and find yourself.

Once you have picked a topic and application, you are ready to do research and write up your findings. For your chosen “Topic + Application” combination, be ready to answer the following questions:

Describe the Set Theoretic topic in general. Give a brief overview of the mathematics involved. If you are writing about (for example) finite versus infinite axiomatizations, then your topic is comparisons of different axiomatizations of Set Theory and you would write an overview of it, much as you would find in a textbook.

Describe the full range of applications for your chosen topic. Although the rest of your paper will be devoted to the single application of your chosen topic, for this section the idea is to paint a picture of the full range of applications that are possible. For example, logic and set theory can be used anywhere that you have inductive and recursive reasoning, and this occurs not only in abstract mathematics but also in linguistics, theory of computation, automata, and the like.

Be sure to list each of the different application areas and briefly (a sentence or two for each different application) indicate how your chosen topic is used for each type of application. For example, in linguistics is used to model how human beings acquire speech and are able to communicate and make conventions or agreements.

Now it is time to zero in on the specific “topic + application” combination that you have chosen. In this part of the paper, you should describe how the specific topic you chose is applied to the area you have chosen. Essentially this is the same idea as item 2 above, but because you are focusing here on only your one single chosen application, you can go into it in much more depth and detail than you did for the brief overviews you wrote in item 2.

Continuing with our example, you might describe in detail how Set Theory is used within linguistics. What kinds of fundamental issues regarding human mind and intelligence is it used for? Could the linguistic problems be addressed any other way, and if so what is the advantage of using set theoretic concepts? To what extent, if any, is the process computerized and what changes, if any, are made to it in order to accommodate computerization? These particular questions might not apply to your own particular topic+application, but I am sure you can think of many similar aspects to discuss once you have done the research.

The last part of the paper centers on doing some illustrative conclusions.

The total paper should be approximately 3-5 pages in length (not counting the cover page, references page, or any exhibits), and include a full set of references (including the textbook if you use it, as I assume you will). APA format requires that the font used should be Arial or Times New Roman, 10 or 12 point only; the pages be double spaced; and the margins should be no more than one inch all around.

Be sure to fully reference your paper, including all references you used (textbook, other books and all websites).

Please format the paper in APA style. You can find an online APA style guide in the APUS Library. Navigate to APUS Log in » Online Library: CampusGuides » Library Tutorial Center »Writing Help or open the Citation page and then select the APUS APA Style Guide. Another excellent APA resource is found by googling “The Owl at Purdue” (OWL is the online writing center at Purdue University) – there you will find not only a style guide but a sample paper that you can use as a model for what your paper should look like.