MHF 4102 - AXIOMATIC SET THEORY                                          FLORIDA INT'L UNIV

            PROJECTS 1 & 2                                                                                                              FALL 2011

 

           Pick two of the following 4 topics and write a 4 to 6 page (1.5 spacing, 1” L/R margins) paper
            on each of the two topics you choose.  By a 4-page paper, we mean 4 complete pages and this
            includes the references. If you have to exceed 6 pages, you may do so, but try to avoid making
            the paper too long or technical.   The paper should be aimed at students with your background
            and you may assume that the intended audience is knowledgeable about Calculus and epsilon -
            delta proofs and is familiar with the standard notations of Mathematics.  The paper should be a
            continuous essay with a blank line between paragraphs & the references not  on a separate page.

THE  FOUR  TOPICS :

1.  The Axiom of Choice, the Banach-Tarski Paradox, and their relevance to Mathematics.

2.  Construction of the Nonstandard Real (Hyper-real) Numbers and its applications.

3.   Non-well-founded Sets (Hypersets) and a few elementary applications. 

4.   Sets of Uniqueness for Trigonometric series and the origin of infinite ordinals & cardinals.

 

      THE SOURCES you may wish to consult are:

Standard Encyclopedias such as Encyclopedia Brittanica & Mathematical Encyclopedias

Handbooks on Mathematical Logic or Set Theory & Mathematical Journals

Textbooks on Set theory, Mathematics, & Mathematical Logic

Wolfram Math World (http://mathworld.wolfram.com/) 

Wikipedia  (http://en.wikipedia.org/wiki/Main_Page)

 

            Be careful with on-line information (like #5 ) as they are not considered reliable but they can lead

you to reliable sources.     The kind of things that you are expected to include in your paper are:

1.      An introductory paragraph that describes in a non-technical way the topic under discussion.

2.      The history of the topic under discussion and the people involved in its solution.

3.   Some details of the situation and the standard approach to the topic

4.   The key ideas or a sketch of some proofs concerning the topic, when possible

5.     Brief mention of alternative approaches to the standard one, whenever these exists

6.     Any applications arising from the ideas under consideration

7.     Any unsolved problems that still remain about the topic in question.

8.     A concluding paragraph about the significance & importance of the topic under discussion.

             9.   Include at least 5 different sources & indicate where these sources were used in your essay.

You should only quote things when they are particularly relevant, very revealing, and nicely said.

Otherwise, just express everything in your own word in the way you understand them. 

 

            ONE LAST THING – never include any statement in your paper that you do not understand.