Fall 2014

Email: (My last name)@math.wisc.edu

Office: 513 Van Vleck Hall

Office Hours: Please just drop by.

Class Room: B211

- 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 $\leq 22$).
- 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 $\leq 43$)
- 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,
- 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:- Show that if $X$ is any finite or co-finite subset of $\omega$, then $X$ is a spectrum.
- Show that the set of even numbers is a spectrum.
- Show that the set of odd numbers is a spectrum.
- Show that for any $n,m\in \omega$, the set of integers congruent to $n$ mod $m$ is a spectrum.
- Show that the set of powers of primes is a spectrum.
- Show that the set of powers of 17 is a spectrum.
- 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?