% LaTex2e \documentclass[12pt]{article} \usepackage{amssymb,latexsym} \usepackage{hyperref} \pagestyle{myheadings} \def\header{A.Miller\hfill Model Theory M776 \hfill \today\hfill } \markboth\header\header \newcount\probno \probno=0 \def\prob#1{\advance\probno by 1 \par \bigskip \noindent{\bf Problem \the\probno .} (#1) \par} \def\al{\alpha} \def\continuum{{\mathfrak c}} \def\elemsup{\succeq} \def\elemsub{\preceq} \def\ff{{\mathcal F}} \def\ka{\kappa} \def\dd{{\mathcal D}} \def\kk{{\mathcal K}} \def\ll{{\mathcal L}} \def\om{{\omega}} \def\poset{{\mathbb P}} \def\proves{\vdash} \def\rmiff{\mbox{ iff }} \def\su{\subseteq} \def\zz{{\mathbb Z}} \def\isom{\simeq} \def\pr{\prime} \def\ult{{\mathcal U}} \def\qq{{\mathbb Q}} %%%%%%%%%%%%% % % \def\la{\langle} % \def\ra{\rangle} % \def\reals{{\mathbb R}} % \def\rmand{\mbox{ and }} % \def\st{\;:\;} % such that % \newtheorem{theorem}{Theorem} % \newtheorem{lemma}{Lemma} % \newtheorem{cor}{Corollary}[theorem] %%%%%%%%%%%%% \newenvironment{theorem}{\par\bigskip\noindent{\bf Theorem}\em}{\par} \newenvironment{cor}{\par\bigskip\noindent{\bf Corollary}\em}{\par} \newenvironment{example}{\par\bigskip\noindent{\bf Example}\em}{\par} \begin{document} \begin{flushright} Spring 2009 \end{flushright} Homework problems are due in class one week from the day assigned (which is in parentheses). \begin{theorem} (Ehrenfeucht-Fr\"aisse 1960 \cite{ehrenfeucht}). If $M$ and $N$ are $\ll$-structures and $M\equiv_n N$, then $M$ and $N$ model the same $\ll$-sentences of quantifier depth $\leq n$. \end{theorem} \prob{1-21 W} For structures $M$ and $N$ in the language of pure equality, prove that $M\equiv_n N$ iff $||M||=||N||$ or ($||M||\geq n$ and $ ||N||\geq n$). \prob{1-21 W} Let $M$ be an equivalence relation with exactly one equivalence class of size $n$ for each positive integer $n$ and no infinite classes. Let $N$ be the same, except in addition it has one infinite equivalence class. Use Ehrenfeucht games to prove that $M\equiv N$. \begin{theorem} (Ehrenfeucht-Fr\"aisse 1960). If $\ll$ is a finite language which contains only predicate symbols and constant symbols, then for every $n\in\om$ there exist a finite set $F_n$ of $\ll$-sentences each with quantifier depth $\leq n$ such that for any two $\ll$-structures $M$ and $N$, if ($M\models \theta \rmiff N\models \theta $) for every $\theta$ in $F_n$, then $M\equiv_n N$. \end{theorem} \prob{1-26 M} \noindent Let $L_n$ be a linear order of size $n$ and $L_\infty=\om+\om^*$ where $\om^*$ is the order type of the negative integers. (a) Prove that for every $n<\om$ there is an $N<\om$ such that $L_k\equiv_n L_\infty$ for all $k>N$. (b) Use the part (a) to prove that the linear orders $\om$ and $\om+\om^*+\om=\om+\zz$ are elementarily equivalent. (c) Use part (b) to prove that $(\om, S)$ and $(\om+\zz,S)$ are elementarily equivalent where $S$ is the successor operation, $S(x)=x+1$. \begin{theorem} (Cantor 1880) If $M$ and $N$ are countable $\ll$-structures, then $M\simeq N$ iff $M\equiv_\infty N$. \end{theorem} \prob{1-26 M} Let $T$ be a $\ll$-theory such that $T$ has no finite models and $\ll$ is countable. Prove that the following are equivalent: \begin{enumerate} \item $T$ is $\om$-categorical \item $M\equiv_\infty N$ for every pair of models $M$ and $N$ of $T$. \end{enumerate} Hint: You may use without proof a consequence of the Ryll-Nardzewski Theorem, namely if $M$ is a model of $T$ and $a_1,\ldots,a_n$ a tuple from $|M|$, then $$Th(M,a_1,\ldots,a_n)$$ is $\om$-categorical. You may also use the Downward-Lowenheim-Skolem-Tarski Theorem. \begin{theorem} (Carol Karp 1965). Given $\ll$-structures $M$ and $N$ the following are equivalent: \begin{enumerate} \item $M$ and $N$ satisfy the same $\ll_{\infty,\om}$ sentences \item $M\equiv_\infty N$ \end{enumerate} \end{theorem} \prob{1-28 W} Let $\kk$ be a class of $\ll$-structures. \par\noindent Prove that $\kk$ is EC iff both $\kk$ and $\overline{\kk}$ are ${\rm EC}_{\Delta}$. \begin{theorem} (\L os-Tarski 1955) A first-order theory $T$ is $\forall$-axiomatizable iff the models of $T$ are closed under taking substructures. \end{theorem} \begin{cor} The class of models of a sentence $\theta$ is closed under taking substructures iff $\theta$ is logically equivalent to a $\forall$-sentence. \end{cor} \begin{cor} The class of models of a sentence $\theta$ is closed under taking superstructures iff $\theta$ is logically equivalent to a $\exists$-sentence. \end{cor} \prob{2-02 M} Show that if a first-order theory $T$ is preserved by taking superstructures, then it can be axiomatized by existential sentences, i.e. $\exists$-sentences. Hint: Suppose $M\models (\exists-sent)\cap T$. Prove that $Th_{\forall}(M)\cup T$ is consistent. \begin{theorem} (Elementary Chain Lemma Tarski-Vaught 1957) If $$M_0\elemsub M_1\elemsub M_2\elemsub M_3 \elemsub \cdots$$ is a chain of elementary substructures and $$N=\bigcup_{n<\om}M_n$$ then $M_k\elemsub N$ for all $k<\om$. \end{theorem} \begin{theorem} (Chang-\L os-Suszko 1959) A first-order theory $T$ is axiomatizable by $\forall\exists$-sentences iff the models of $T$ are closed under chains of substructures. \end{theorem} \prob{2-04 W} (Directed Unions) Suppose $\dd$ is a directed set of $\ll$-structures and $M_\infty=\bigcup\dd$. Prove: \par (a) Every $\forall\exists$-sentence which is true in every $M\in\dd$, is true in $M_\infty$. \par (b) If for every $M\su N$ in $\dd$ we have $M\elemsub N$, then $M\elemsub M_\infty$ for every $M\in\dd$. \prob{2-04 W} (Direct Limits). Let $\poset=(\poset,\leq)$ be a poset (partially ordered set), $(M_p:p\in\poset)$ a family of $\ll$-structures, and $j_{pq}:M_p\to M_q$ be maps for each $p\leq q$ in $\poset$. State the appropriate conditions on $\poset$, these structures, and these maps, so as to naturally generalize problem above (a) and (b). \prob{2-06 F} Show that $T=Th(\qq,\leq,S)$ where $S$ is the successor function is finitely axiomatizable. Warning: it is not categorical in any power. \begin{theorem} (Lowenheim 1915) If $T$ is a theory in countable language and has a model, then it has a countable model. \end{theorem} \begin{theorem} (Lowenheim-Skolem) If $T$ is an $\ll$-theory which has an infinite model, then $T$ has models of all cardinality $\ka\geq |\ll|+\om$. \end{theorem} \begin{theorem} (Upward-Downward Lowenheim-Skolem-Tarski 1950s) See \par \url{http://www.math.wisc.edu/~miller/old/m776-97/preq.pdf} \end{theorem} \begin{theorem}({\L}os-Vaught Test 1954) If $T$ is an $\ll$-theory which has no finite models and is $\ka$-categorical for some $\ka\geq |\ll|+\om$, then $T$ is complete. \end{theorem} \begin{theorem} (McKinsey 1943) A first-order theory $T$ is axiomatizable by universal Horn sentences iff the class of models of $T$ is closed under substructure and products. \end{theorem} \prob{2-09 M} Prove that the class of well-orderings is not ${\rm PC}_{\Delta}$ but its complement is. \begin{theorem} (Keisler Sandwich 1960) An $\ll$-theory $T$ is $\exists\forall$-axiomatizable iff for any $\ll$-structures $M_1\su M_2\su M_3$ with $M_1\elemsub M_3$, if $M_1\models T$, then $M_2\models T$. \end{theorem} \prob{2-11 W} Let $M_1$ and $M_2$ be $\ll$-structures. Prove that $M_1\equiv M_2$ iff there are $\ll$-structures $N_1$ and $N_2$ such that $M_1\elemsub N_1$, $M_2\elemsub N_2$, and $N_1\isom N_2$. \begin{theorem} (Lyndon 1959) A first-order theory $T$ is axiomatizable by positive sentences iff the class of models of $T$ is closed under homomorphic images. \end{theorem} {\it \noindent {\bf Key Lemma}. Suppose $B\models Th_{POS}(A)$ then \par (a) there exists $B^\pr\elemsup B$ and $f:A\to B^\pr$ such that $$(B,f(a))_{a\in |A|}\models Th_{POS}(A,a)_{a\in |A|}$$ \par (b) there exists $A^\pr\elemsup A$ and $g:B\to A^\pr$ such that $$(B,b)_{b\in |B|}\models Th_{POS}(A^\pr,g(b))_{b\in |B|}$$} \prob{2-13 F} (a) Prove that $$B\models Th_{POS}(A) \rmiff A\models Th_{\neg POS}(B)$$ (b) Find $A$ and $B$ such that $B\models Th_{POS}(A)$ but $A\not\models Th_{POS}(B)$. \prob{2-13 F} Prove Key Lemma part (b). \begin{theorem} (Craig's Interpolation Lemma 1957) Suppose $\theta_1$ is an $\ll_1$-sentence, $\theta_2$ is an $\ll_2$-sentence, and $\proves \theta_1\to \theta_2$. Then there exists $\rho$ an $\ll_1\cap \ll_2$-sentence such that $\proves \theta_1\to \rho$ and $\proves \rho\to \theta_2$. \end{theorem} \prob{2-16 M} (Prove) Suppose $T_0$ is a complete $\ll_0$-theory, $T_1$ is a complete $\ll_1$-theory, and $\ll=\ll_0\cap \ll_1$. Then: $T_0\cup T_1$ is consistent iff $(T_0\cap (\ll-sent))\cup (T_1\cap (\ll-sent))$ is consistent. \prob{2-16 M}(Millar) (Disprove) Suppose $T_i$ (for $i=1,2,3$) is a complete consistent $\ll_i$-theory. Then: $T_1\cup T_2\cup T_3$ is consistent iff $T_i\cup T_j$ is consistent for all $i$ and $j$. \prob{2-18 W} Suppose that $M$ is an infinite $\ll$-structure, $\leq$ is a binary relation symbol in $\ll$, and $\leq^M$ is a linear order with no greatest element. Prove there exists $N\elemsup M$ with $||N||\leq \om_1+|\ll|+||M||$ and the cofinality of $\leq^N$ is $\om_1$. \begin{theorem} (Beth Definability) With respect to theories, implicitely definable implies explicitely definable. \end{theorem} \begin{theorem} (Addison 1960 \cite{addison60}) Let $\ll$ be a language containing at least one constant symbol. Suppose $\theta_0$ is a universal $\ll$-sentence and $\theta_1$ an existential $\ll$-sentence such that $\proves\theta_0\to\theta_1$. Then there exists a quantifier free $\ll$-sentence $\rho$ such that $\proves \theta_0\to\rho$ and $\proves \rho\to\theta_1$. \end{theorem} \prob{2-20 F} Suppose $\ll$ is language containing at least one relation or operation symbol but no constant symbols. Show there exists $\theta_0$ a universal $\ll$-sentence and $\theta_1$ an existential $\ll$-sentence such that \begin{enumerate} \item $\theta_0\to\theta_1$ is a logical validity, \item $\theta_1$ is not a logical validity, and \item $\neg\theta_0$ is not a logical validity. \end{enumerate} Show that there is no such pair of sentences in the language of pure equality. \begin{theorem} (Shoenfield 1960 in \cite{addison62}, \cite{shoenfield} p. 97) Suppose $\theta_0$ is a $\forall\exists$-$\ll$-sentence and $\theta_1$ is an $\exists\forall$-$\ll$-sentence such that $\proves\theta_0\to\theta_1$. Then there exists an $\ll$-sentence $\rho$ which is a boolean combination of existential and universal sentences such that $\proves \theta_0\to\rho$ and $\proves \rho\to\theta_1$. (Similar result holds for higher prenex classes.) \end{theorem} \prob{2-23 M} Let $R$ be a binary relation symbol. Note that $$\exists x \forall y \; R(x,y) \to \forall y \exists x\; R(x,y)$$ Prove that there does not exist a sentence $\rho$ which is a boolean combination of existential and universal sentences and interpolates between them. Hint: Consider $R$-structures in which every finite $R$-structure embedds. \begin{theorem} (Rabin 1959 \cite{rabin}, \cite{bell} p. 136.) There is a complete theory $T$ in a language of size continuum, which is categorical in power $\om$ and has no model of size $\kappa$ with $\om<\kappa<|2^\om|$. \end{theorem} \prob{2-25 W} Give an example of a theory $T$ with arbitrarily large finite models but no model of cardinality $\kappa$ with $\om\leq \kappa <|2^\om|$. \prob{2-25 W} Suppose that the continuum $|2^\om|$ is larger than $\aleph_\om$. Prove that for every $A\su \om$ there is a first order theory $T_A$ such that for every $n<\om$ $\;\;T_A$ has a model of cardinality $\om_n$ iff $n\in A$. \bigskip Open Question. Can we find $T_A$ which is complete? \bigskip \begin{theorem} (\L os 1955) For any $f_1,\ldots,f_n\in \prod_iA_i$ and formula $\theta$ \par\noindent $\prod_iA_i/\ult\models \theta([f_1],\ldots,[f_n])$ $\;\;$ iff $\;\;$ $\{i\in I: A_i\models \theta(f_1(i),\ldots,f_n(i))\}\in\ult$. \end{theorem} \prob{2-27 F} Prove \L os's Theorem for ultraproducts $\prod_{n\in\om}A_n/\ult$ and the language $\ll(Q_{c^+})$ where $Q_{c^+} x$ is the quantifier which means ``There are more than continuum many $x$ such that''. \begin{theorem} (Keisler unpublished see \cite{chang} p.472) If $T$ is a first-order theory with a model of size $\kappa\geq \om$, then for every $\lambda\geq \kappa^\om$ $\;\;T$ has a model of size $\lambda$. \end{theorem} \begin{theorem} (Keisler 1959) (CH) If $A\equiv B$ are countable and $\ult$ is a nonprincipal ultrafilter on $\om$, then $A^\om/\ult\isom B^\om/\ult$. \end{theorem} \begin{theorem} (Morley, Vaught 1962) If $A$ and $B$ are $\kappa$-saturated models of size $\kappa$, then $A\isom B$. \end{theorem} \prob{3-04 W} Suppose $A$ is an $\ll$ structure and $\leq$ is a binary relation symbol in $\ll$ such that $A$ reducted to $\leq$ is a linear order of uncountable cofinality. Prove: \par (a) There exists a proper elementary extension $B\elemsup A$ such that $|A|$ is cofinal in $|B|$, i.e., no new elements come at the end. \par (b) There exists elementary extensions $B\elemsup A$ of arbitrarily large cardinality such that $|A|$ is cofinal in $|B|$. \begin{theorem} (Hausdorf 1936 see \cite{kunen}) There are $2^\continuum$ many ultrafilters on $\om$. \end{theorem} \prob{3-06 F} Prove there exists $f_\al:\om\to\om$ for $\al<\continuum=|2^\om|$ such that for any $F\in [\continuum]^{<\om}$ and $s:F\to\om$ there exists $n<\om$ such that $f_\al(n)=s(\al)$ for all $\al\in F$. \prob{3-06 F} Prove that for any infinite cardinal $\ka$ there are $2^{2^\ka}$ ultrafilters on $\ka$. \begin{theorem} (Morley-Vaught 1962) If $\ka\geq |\ll|+\om$ and $A$ an $\ll$-structure of cardinality $2^\ka$, then there exists $B\elemsup A$ of cardinality $2^\ka$ which is $\ka^+$-saturated. \end{theorem} \begin{theorem} (Vaught) Let $T$ be theory in a countable language. Then the following are equivalent: \begin{enumerate} \item $T$ has a countable $\om$-saturated model \item $T$ has a countable weakly-saturated model \item $S_n(T)$ is countable for all $n$ \end{enumerate} \end{theorem} \begin{theorem} (Vaught) A structure $A$ is $\om$-saturated iff it is weakly saturated and $\om$-homogenous. \end{theorem} \begin{theorem} (Vaught) If $\ll$ is countable and $A$ is a countable $\ll$-structure, then there exists a countable $\om$-homogeneous $B\elemsup A$. \end{theorem} \prob{3-11 W} Suppose $T$ is a consistent $\ll$-theory with only infinite models. Suppose $\leq$ is a binary relation symbol in $\ll$ such that $T\proves$``$\leq $ is a linear order''. Prove that every $\om_1$-saturated model has cardinality at least continuum. \begin{theorem} (Vaught's Two Cardinal) If a theory $T$ in a countable language with a distinguished predicate $U$ admits $(\ka,\ka^+)$, then it admits $(\om,\om_1)$. \end{theorem} \begin{theorem} (Henkin-Orey 1954) If $T$ is a consistent theory in a countable language and $(\Sigma_n:n<\om)$ are nowhere dense partial types, then $T$ has a model omitting all $\Sigma_n$. \end{theorem} \prob{3-13 F} Suppose that $T$ is an $\ll$-theory and $S_n(T)$ is countable. Prove there exists a countable $\ll_0\su\ll$ such that every $\ll$-formula $\theta(x_1,\ldots,x_n)=\theta(\overline{x})$ there exists an $\ll_0$-formula $\theta_0(\overline{x})$ such that $T\proves \forall \overline{x}\;\; \left(\theta(\overline{x}) \;\leftrightarrow\; \theta_0(\overline{x})\right) $. \begin{theorem} (Henkin-Orey 1954) If $T$ is an $\om$-complete consistent theory, then $T$ has an $\om$-model. If $T$ is complete and has an $\om$-model, then $T$ is $\om$-complete. \end{theorem} \prob{3-23 M} Prove or Disprove. Suppose $T$ is an $\ll$-theory where $\ll$ is countable and $\Sigma_n$ for $n<\om$ are partial types. Suppose for every $N<\om$ that $T$ has a model omitting $(\Sigma_n:n\om$ that $T$ has a model $A$ of size $\ka$ such that for every countable $X\su |A|$, the structure $(A,c)_{c\in X}$ realizes only countably many types. \begin{theorem} (Erdos-Rado) $\beth_n^+(\ka)\to (\ka^+)^{n+1}_\ka$. \end{theorem} \begin{example} (Sierpinski) $2^\om\not\to (3)^2_\om$ $\;\;\;\; 2^\om\not\to (\om_1)^2_2$. \end{example} \prob{4-15 W} Suppose $T$ is a countable first-order theory with an infinite model. Prove that there exists countable models of $T$, $A(X)$ for $X\su\om$, such that for any $X,Y\su \om$ $$X\su Y \rmiff A(X)\elemsub A(Y).$$ \def\ga{\gamma} \begin{theorem} (Vaught's two cardinals far apart) Suppose $T$ is a countable theory with distinguished predicate $U$. Suppose for every $n$ there is a $\ka\geq\om$ such that $T$ has a model of type $(\beth_n(\ka),\ka)$. Then for every $\ga\geq \ka\geq \om$, $T$ has a model of type $(\ga,\ka)$. \end{theorem} \begin{theorem} (Morley) The Hanf Number of $\ll_{\om_1,\om}$ is $\beth_{\om_1}$. \end{theorem} \begin{theorem} (Silver, Erdos, Rowbottom) Let $\ka_0$ be the least $\ka$ such that $$\ka\to(\om)^{<\om}_2$$ Assume $\ka_0$ exists, then \begin{enumerate} \item $\ka_0$ is strongly inaccessible. \item The Hanf Number of $\ll_{\om_1,\om_1}$ is at least $\ka_0$. \item There are unboundedly many weakly compact cardinals less than $\ka_0$, however $\ka_0$ is not weakly compact. \end{enumerate} \end{theorem} \prob{4-22 W} Let $\ka_1$ be least such that $\ka_1\to(\om+1)^{<\om}_2$. Prove $\ka_1>\ka_0$. Extra credit: prove it is strongly inaccessible. \begin{theorem} (Morley) If a countable first-order theory is categorical in some uncountable power, then it is categorical in all uncountable powers. \end{theorem} \prob{4-27 M} Let $T_n=Th(\om^\om,P_s)_{s\in\om^{\leq n}}$ where $P_s$ is the unary predicate $$P_s=\{x\in \om^\om\::\; s\su x\}.$$ Prove that \par (a) rank$_{T_n}(x=x)=n+1$. \par (b) Give an example of a $T$ such that rank$_{T}(x=x)=\om$. \begin{theorem} (Shelah) Suppose $T$ is a countable theory. \par (a) If there exists $\ka\geq\om$ such that $T$ is $\ka$-stable, then $T$ is $\ka$-stable for every $\ka$ such that $\ka^\om=\ka$. \par (b) $T$ is unstable iff $T$ has the order property. \end{theorem} \begin{thebibliography}{99} \bibitem{addison60} Addison, J. W.; The theory of hierarchies. 1962 Logic, Methodology and Philosophy of Science (Proc. 1960 Internat. Congr.) pp. 26--37 Stanford Univ. Press, Stanford, Calif. \bibitem{addison62} Addison, J. 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