Math 770
Fall 2014

Instructor: Uri Andrews
Email: (My last name)@math.wisc.edu
Office: 513 Van Vleck Hall
Office Hours: Please just drop by.
Class Room: B211

Homeworks:

Homeworks will be regularly assigned and posted here with due-dates. If you do not yet have the textbook and need to know what the problems are, please drop me an e-mail.
  1. Due Friday Sept. 12: Exercises I.2.1, I.6.3, I.6.11, I.6.13, I.6.15, I.6.17, I.6.20 (i.e. all the exercises in the text on pages $\leq 22$).
  2. Due Wednesday Sept. 24: I.7.13, I.7.16, I.7.17, I.7.19, I.7.21, I.7.22, I.7.23, I.8.10, I.8.11, I.8.13, I.8.22, I.8.23 (i.e. all the exercises in the text on pages $\leq 43$)
  3. Due Friday Oct 3: I.9.6, I.10.5, I.11.3, I.11.6, I.11.12, I.11.15, I.11.19, I.11.21, I.11.22, I.11.23, I.11.24, I.11.25, I.11.28, I.11.29, I.11.31, I.11.33, I.11.34, I.11.35, I.11.36, I.11.37, I.11.38,
  4. Due Wednesday Nov. 5: Show that if $t(\bar{x})$ is a term and $s$ and $s'$ are $\mathcal{M}$-assignments which agree on $\bar{x}$, then $t^{\mathcal{M},s}=t^{\mathcal{M},s'}$. Also, show that if $\phi(\bar{x})$ is a formula and $s$ and $s'$ are $\mathcal{M}$-assignments which agree on $\bar{x}$, then $\mathcal{M}\models_s \phi$ if and only if $\mathcal{M}\models_{s'} \phi$.

    Call a model $\mathcal{M}$ spiffy if for every $a,b\in M$, there is an automorphism of $\mathcal{M}$ that moves $a$ to $b$. Let $T$ be a countable theory which has an infinite model which is spiffy. Let $\kappa$ be any infinite cardinal. Show that $T$ has a spiffy model of cardinality $\kappa$. (Note that this was essentially on the qual this past August.)

    Let $\mathcal{L}$ be the language of ordered rings (i.e. the language generated by symbols $+,\cdot, -, 0, 1, <$ in the way you expect). Let $\mathcal{N}$ be the structure $(\mathbb{N},+,\cdot,-,0,1,<)$. Show that there is a structure $\mathcal{M}$ so that $\mathcal{M}$ and $\mathcal{N}$ satisfy the same sentences and $M$ contains an infinite $<$-descending sequence. That is, there are elements $a_1,a_2,a_3,\ldots$ so that for each $i$, $\mathcal{M}\models a_i>a_{i+1}$.

    We say that a set $A\subseteq \omega$ is a spectrum if there is a language $\mathcal{L}$ and a sentence $\phi$ so that $n\in A$ if and only if there is a model of $\phi$ of size $n$ (we call this $A$ the spectrum of $\phi$). The spectrum problem (still open and apparently quite hard) is to characterize which subsets of $\omega$ are spectra. In particular, it is no known if the complement of a spectrum is a spectrum. The following are easier:
    1. Show that if $X$ is any finite or co-finite subset of $\omega$, then $X$ is a spectrum.
    2. Show that the set of even numbers is a spectrum.
    3. Show that the set of odd numbers is a spectrum.
    4. Show that for any $n,m\in \omega$, the set of integers congruent to $n$ mod $m$ is a spectrum.
    5. Show that the set of powers of primes is a spectrum.
    6. Show that the set of powers of 17 is a spectrum.
    7. Show that if $X$ and $Y$ are spectra, then $X\cup Y$ is a spectrum and $X\cap Y$ is a spectrum.

    Let $\phi$ be a sentence so that the spectrum of $\phi$ is infinite. Show that $\phi$ has an infinite model. If $\phi$ has an infinite model, must the spectrum of $\phi$ be infinite?
  • Take Home Final

    Source of Problems:

    A good source of problems can be found in the "E" section of old qualifying exams. They can all be found here. If you are a math graduate student, do consider taking either 771/773/776 and the logic qual. After this course, you'll be half way there!

    Links

    1. Itai Ben Yaacov's notes for 770
    2. Andres Caicedo's notes on proving compactness via the ultraproduct construction.