Honors Interdisciplinary Seminar (175)

 Axiom Systems and Independence

Assignment following Week #9 (Due 11/16 in class) 
(Section 004 - Due 10/5 in class)

1. There are four assigned readings for next week:
   • Read the excerpt from William Dunham on the discovery of non-Euclidean geometry.  That reading can be found on ereserve.
   • Read the excerpt from Michael Henle on the history of the parallel postulate.  That reading can be found on ereserve.  The Henle reading provides background material similar to Dunham's, but with a slightly different perspective.
   • Read the excerpt from Nagel /Newman's Gödel's Proof on the profound results of Gödel.  This excerpt can be found on ereserve.
   • Read the first section of Peter Suber's essay, "50 Years Later, The Questions Remain; Kurt Gödel in Blue Hill."  (If you read the first 6 paragraphs, that should be sufficient.)

   (Remember that if you print any of these excerpts, the reproduction is not to be used for any purpose other than private study, scholarship, or research. If you use a photocopy or reproduction for purposes in excess of fair use, then you may be liable for copyright infringement.)

2. Complete the Study Questions below.
3. Complete Response Paper #1.  This requires two steps:

   1.       By Thursday, November 12 at 11:00 am, email me the introductory paragraph for your essay.  Do exactly one of the following:  (a) highlight the thesis statement in your introduction in bold-face type or (b) write a note explaining why your introduction does not include a thesis statement.

   2.       Finish the paper for class on November 16.   Parameters for the Response Papers can be found HERE, and possible topics for your essay can be found HERE.

4. Read any emails I send you, and respond if requested.
5. There may be a brief quiz at the start of our next class period.

Study Questions 

Please carefully write out your answers to these questions.  Make a copy of your answers and be prepared to hand that copy in at the start of class.  Look at the Study Questions Main Page for general guidelines on study questions in Steve's section.

1. What was the biggest historical question surrounding Euclid's 5th postulate?  Explain.  How was this historical question eventually resolved?  

2. List at least five mathematicians who worked on resolving the big question surrounding Euclid's 5th postulate.

3. Both Dunham and Henle list several statements that are equivalent to Euclid's parallel postulate.  What does "equivalent" mean?  (If you aren't sure exactly, based on the readings, then look at other sources.  The first paragraph of a Wikipedia entry gives a pretty reasonable explanation.

4. Give a careful statement of "Playfair's Postulate."  Draw a picture and explain the postulate in your own words.
   Then list several other statements that are equivalent to the parallel postulate.

5. We say that a statement can be disproved if the statement can be proven false.  A statement is said to be independent in an axiom system if it can be neither proved nor disproved from the axioms.
   Using our axiom system for "People and Clubs," decide if each of the statements can be proven, disproven, or is independent of the axioms.

   1. There could be three people living in Hilbert.
   2. There could be four people living in Hilbert.
   3. There are four people living in Hilbert.
   4. If there are n people living in Hilbert, then there could be a club with n-1 people.
   5. There are nine people living in Hilbert.
6. What does Gödel have to say about independence as it pertains to axiom systems?  Carefully explain.
7. What does consistency of an axiom system mean?  What does Gödel have to say about the issue of consistency?  Carefully explain.
8. Glance through Euclid's propositions in Book I again.  See if you can find the first proposition that uses the Fifth postulate in its proof.  (I recommend using Joyce's online Elements for this).
9. Now try to find all of the propositions in Book I that use the Fifth postulate in the proof.  Include a proposition on this list if it uses the Fifth postulate directly, or if it uses a proposition that uses the Fifth postulate.

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