Saturday, February 21, 2004

# 2/21/2004 09:41:00 pm - Starting on Undecidability

I am reading the first section of Enderton's "Chapter 3: Undecidability" and trying to finish Homework Assignment 10. While attempting to answer a question* in the assignment, I searched the web for the definition of "descending chain" (I was hoping to understand the term better) and came upon a file which contained a solution to the question I was trying to solve! From the address of the file, I was able to find a webpage from the University of Ottawa that was using Enderton's "A Mathematical Introduction to Logic" as a textbook for a logic course. The page contained answer files to some problems found in Enderton's book! This was what I have been looking for since I started attempting problems in Enderton's book. It's nice to have model solutions as a reference. I have provided the link to this page in my mathematical logic page.

*The question is:

Consider a language with a two-place predicate symbol **<**, and let B = (N;<) be the structure consisting of the natural numbers with their usual ordering. Show that there is some A elementarily equivalent to B such that **<**^{A} has a descending chain. (That is, there must be a_{0},a_{1},... in |A| such that < a_{i+1},a_{i} > Î **<**^{A} for all i.) Suggestion: Apply the compactness theorem.

To do: Complete Homework Assignment 10.

Random comment: This progress update page is a good thing for me because it helps me to stay focused on acheiving my goal.

Monday, February 02, 2004

# 2/02/2004 09:35:00 pm - Models of Theories

I have just read Section 2.6 (Models of Theories) of

Enderton's book. I skipped the last two theorems in this section because it makes references to parts of mathematics that I do not understand, e.g. algebraically closed fields. According to my tutor (I managed to get a logician from the Mathematics Department at NUS to teach me mathematical logic for free!) YY, I can go on to Enderton's chapter on Undecidability ( This chapter contains the proofs of Godel's Incompleteness Theorems! Finally!) after I'm done with Section 2.6. Although the chapter on Undecidability will make reference to Section 2.7 (which I have not read), I should come back to it later since it is important to move on and get a sense of progress at this point in time. Yippee!

To do: Write about Skolem's paradox and the limitations of first order logic.

Random gripe: Why do most logic textbooks have the word "elementary" or "introduction" in them when clearly there is nothing elementary (in the sense of simple) about the subject matter the author is introducing in his book? One would also expect "introduction" books to be gentle primers, but not so for the logic books I've come across. Maybe this has to do with the nature of the beast called "mathematical logic".

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