\documentclass[12pt]{report} \usepackage{oldlfont} \usepackage{amssymb} \usepackage{latexsym} % Set left margin - The default is 1 inch, so the following % command sets a 1.25-inch left margin. %\setlength{\oddsidemargin}{0.25in} \setlength{\oddsidemargin}{0in} % Set width of the text - What is left will be the right margin. % In this case, right margin is 8.5in - 1.25in - 6in = 1.25in. %\setlength{\textwidth}{6in} \setlength{\textwidth}{6.5in} % Set top margin - The default is 1 inch, so the following % command sets a 0.75-inch top margin. \setlength{\topmargin}{-0.25in} % Set height of the header \setlength{\headheight}{0.3in} % Set vertical distance between the header and the text \setlength{\headsep}{0.2in} % Set height of the text \setlength{\textheight}{9in} % Set vertical distance between the text and the % bottom of footer \setlength{\footskip}{0.1in} %\markboth{TFA groups}{TFA groups} % Set the beginning of a LaTeX document \begin{document} \def\Ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\sim$}}}{}} \def\ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\scriptscriptstyle\sim$}}}{}} \def\R{{\mathbb R}} \def\V{{\mathbb V}} \def\N{{\mathbb N}} \def\n{{\mathbb N}} \def\Q{{\mathbb Q}} \def\Z{{\mathbb Z}} \def\C{{\mathbb C}} \def\O{{\cal O}} \def\e{\epsilon} \def\h{{\hat{X}}} \def\B{{\cal B}} \def\n{\noindent} %\def\Box{\square} %\def\box{\square} % Redefine "plain" pagestyle \makeatletter % `@' is now a normal "letter' for LaTeX \renewcommand{\ps@plain}{% \renewcommand{\@oddhead}{\textrm{Your Header}\hfil\textrm{\thepage}}% \renewcommand{\@evenhead}{\@oddhead}% \renewcommand{\@oddfoot}{}% empty footer \renewcommand{\@evenfoot}{\@oddfoot}} \makeatother % `@' is restored as a "non-letter" character \Huge \title{\Huge { PREWELLORDERS AND INNER MODEL THEORY}} % Enter your title between curly braces \author{{\Huge Greg Hjorth}\\{\LARGE greg@math.ucla.edu}} % Enter your name between curly braces \date{\Huge April 10, 1999\\ Vegas} % Enter your date or \today between curly braces \maketitle \pagestyle{myheadings} \markright{\large Prewellorders} % Set to use the "plain" pagestyle %\pagestyle{plain} \n{\bf $\S$0. Quick recall of the definitions} \bigskip In what follows I will undertake the customary identification of $\R$ with $2^\N$ -- the space of all infinite sequences of zeroes and ones. \bigskip \bigskip \n {\bf 0.1 Definition} A subset of ${\mathbb R}^n$ is {\it Borel} if it appears in the $\sigma$-algebra generated by the open sets. \bigskip \bigskip A set $A\subset{\mathbb R}^n$ is {\it analytic }, or $\Ubf{\Sigma}^1_1$, if there is some Borel $B\subset {\mathbb R}^{n+1}$ such that $A$ is equal to the projection of $B$:- namely, \bigskip \bigskip $$A=\{\vec x\in {\mathbb R}^n:\exists y\in \R (\vec x ^\smallfrown y\in B)\}.$$ \bigskip \bigskip A set $C\subset \R^n$ is {\it coanalytic}, or $\Ubf{\Pi}^1_1$, if its complement is analytic. \newpage \null \bigskip \bigskip We then proceed to iterate these definitions, thereby obtaining the {\it projective heirarchy.} \bigskip \bigskip \bigskip \bigskip \bigskip \bigskip A set $A\subset{\mathbb R}^n$ is $\Ubf{\Sigma}^1_{k+1}$ if there is some $\Ubf{\Pi}^1_k$ set $C\subset {\mathbb R}^{n+1}$ such that $A$ is the projection of $B$. \bigskip \bigskip A set $C \subset \R^n$ is $\Ubf{\Pi}^1_k$ if its complement $\R^n\setminus C$ is $\Ubf{\Sigma}^1_k$. \newpage \n {\bf 0.2 Definition} Let $\kappa$ be an ordinal. A subset $T\subset \kappa^{<\N}\times 2^{<\N}\times 2^{<\N}\times \cdots 2^{<\N}$ is said to be a {\it tree} if it is closed under subsequences \bigskip \bigskip -- in the sense that if $(u, s_1, \cdots , s_n)\in T$ then for any smaller $l$ we have $$(u\mid_l, s_1\!\mid_l, \cdots , s_n\!\mid_l)\in T.$$ \bigskip \bigskip \bigskip \bigskip A subset $A$ of $\R^n$ ($\sim (2^\N)^n$) is said to be {\it $\kappa$-suslin} if there is some tree $T\subset \kappa^{<\N}\times 2^{<\N}\times 2^{<\N}\times \cdots 2^{<\N}$ such that $A$ is the {\it projection} of the set of branches through $T$ \bigskip \bigskip -- in the sense that $(x_1,..., x_n)\in A$ if and only if there is some $f\in \kappa^\N$ such that for all $k\in \N$ $$(f\!\mid_k, x_1\!\mid_k, \cdots x_n\!\mid_k) \in T.$$ \newpage The big theorem here is that for $\kappa=\Theta^{L(\R)}$ the sup of the definable in $L(\R)$ prewellorders of $\R$ we have \bigskip \bigskip \n {\bf 0.3 Theorem} (Moschovakis\footnote{\Large See for instance Y.N. Moschovakis, {\bf Descriptive set theory,} North-Holland Publishing Company, Amsterdam, 1980.}) Assume the axiom of projective determinacy.\footnote{\Large Or some manner of large cardinal assumptions: See John Steel's talk from this conference, or {\it Projective determinacy} by D.A. Martin and J.R. Steel, {\bf Proceedings of the National Academy of Sciences}, U.S.A. vol. 85(1988), pp. 6582--6586.} Then every projective set is $\kappa$-suslin in a very nice way. \bigskip In particular, the tree exists in $L(\R)$. \bigskip \bigskip \bigskip \bigskip Classically it was shown that every $\Ubf{\Sigma}^1_1$ set is $\aleph_0$-suslin and every $\Ubf{\Sigma}^1_2$ is $\aleph_1$-suslin. \bigskip \bigskip Martin and Solovay showed that every $\Ubf{\Sigma}^1_3$ set is $(\aleph_\omega)^{L(\R)}$-suslin in $L(\R)$.\footnote{\Large D.A.Martin, R.M Solovay, {\it A basis theorem for $\Sigma^1_3$ sets of reals,} {\bf Annals of Mathematics}, vol. 89(1969), pp. 138-159. This is assuming sharps.} \newpage \n{\bf $\S$1. ``The suslin representation of a projective relation tells us everything"} \bigskip \bigskip One finds many theorems in descriptive set theory where a complete description of the possible characteristics of a relation is given by a envoking its suslin representation. For the purposes of this talk the most important such theorem is: \bigskip \bigskip \n {\bf 1.1 Theorem} (Kunen-Martin) Suppose $R\subset \R^2$ is $\kappa$-suslin and wellfounded.\footnote{\Large I.e. no infinite descending chain $(x_i)_{i\in \N}$ with $x_i Rx_{i+1}$ at each $i$.} Then the rank of $R$ is less than $k^+$. In particular, any prewellorder (that is to say an onto map $\pi:\R\rightarrow \alpha$ for some ordinal $\alpha$) in which the relation $$\pi(x)<\pi(y)$$ is $\kappa$-suslin must have length strictly less than $\kappa^+$. \newpage \null \bigskip \bigskip In most situation Kunen-Martin presents optimal bounds. For instance it is true that every $\Ubf{\Sigma}^1_1$ wellfounded relation has rank less than $\aleph_1 (=(\aleph_0)^+)$, and that $\aleph_1$ is the sup of the ranks of $\Ubf{\Sigma}^1_1$ wellfounded relations. Similarly $(\aleph_2)^{L(\R)}$ bounds the ranks of $\Ubf{\Pi}^1_1$ wellfounded relations, and under large cardinal or determinacy hypothesis this bound is optimal.\footnote{\Large From here until the finish of the talk I will simply assume that axiom of determinacy holds in $L(\R)$.} \bigskip \bigskip The bound is also optimal if we consider $\Ubf{\Sigma}^1_1$ or $\Ubf{\Pi}^1_1$ prewellorders (i.e. we prewellorder either a $\Ubf{\Sigma}^1_1$ or $\Ubf{\Pi}^1_1$ set with a relation $\leq$ that is respectively $\Ubf{\Sigma}^1_1$ or $\Ubf{\Pi}^1_1$). \bigskip \bigskip Equally for $\Ubf{\Sigma}^1_{2n+1}$ and $\Ubf{\Pi}^1_{2n+1}$. \bigskip \bigskip Moreover it is indeed true that $(\aleph_{\omega+1})^{L(\R)}$ is the precise bound for ranks of wellfounded Boolean-$\Ubf{\Pi}^1_2$ relations. \newpage \null \bigskip \bigskip \bigskip \bigskip Thus we may be superficially drawn to the conclusion that any question along these lines can be decided by either an appeal to Kunen-Martin, or the construction of a suitable counterexample to show that no improvement on their theorem is possible in the given situation. \newpage \n {\bf $\S$2. ``But it's not true!"} \bigskip \bigskip \n {\bf 2.1 Theorem} (Jackson; late 1980's\footnote{\Large{\it Partition properties and well-ordered sequences,} {\bf Annals of Pure and Applied Logic,} vol. 48(1990), pp. 81--101.}) Every $\Ubf{\Pi}^1_2$ prewellorder has rank less than $(\aleph_2)^{L(\R)}$. \bigskip In fact there is no strictly increasing or decreasing $(\aleph_2)^{L(\R)}$-sequence of $\Ubf{\Pi}^1_2$ sets in $L(\R)$. \bigskip \bigskip \bigskip \bigskip This is much better than the bound of $(\aleph_{\omega+1})^{L(\R)}$ predicted by Kunen-Martin. \bigskip \bigskip \bigskip \bigskip Steve Jackson also obtained appropriate generalizations to $\Ubf{\Pi}^1_{2n}$ prewellorders for every $n$. \newpage This still left open: \bigskip \bigskip \bigskip \bigskip \n {\bf 2.2 Question} Can we obtain a similar bound for prewellorders only ``slightly more complicated than $\Ubf{\Pi}^1_2$"? for instance of the form $$x\leq y$$ if and only if $$L[x, y, z]\models \psi(x, y, z)$$ for some fixed $z\in \R$ and some predetermined formula $\psi$. \bigskip \bigskip \bigskip \bigskip \n {\bf 2.3 Question} Can there be an $\aleph_2^{L(\R)}$-sequence of distinct $\Ubf{\Sigma}^1_2$ sets in $L(\R)$? \newpage \n {\bf $\S$3. Connections with inner model theory} \bigskip \bigskip \n {\bf 3.1 Theorem} (Hjorth; 1995\footnote{\large{\it Two applications of inner model theory to the study of $\Ubf{\Sigma}_1^2$ sets,} {\bf Bulletin of Symbolic Logic,} vol. 2(1996), pp.94--107.}) In $L(\R)$ there is no $\aleph_2^{L(\R)}$-sequence of distinct sets in $\Ubf{\Sigma}^1_2$. \bigskip \bigskip \bigskip \bigskip In retrospect we could interpret the argument as doing the following: \leftskip 0.6in \bigskip \bigskip \n (i) Show that the distinctness of our sequence of $\Ubf{\Sigma}^1_2$ sets can be witnessed by conditions bounded below the least $<\delta$-strong in some $1-$small mouse. \n (ii) Show that this ordinal bound arrives at an ordinal less than $\aleph_2^{L(\R)}$ in some direct limit of iterations of this mouse. \n (iii) Show that this bounds the length of the sequence of distinct sets. \leftskip 0in \newpage \null \bigskip \bigskip\bigskip \bigskip\bigskip \bigskip \n {\bf 3.2 Definition} A set $A\subset \R^n$ is $\Ubf{\Gamma}_{1, k}$ if there is some $z\in\R$ and formula $\psi$ such that $A$ equals the set of $\vec x$ such that $$L[\vec x, z]\models \psi(\vec x, z, \aleph_0^{L(\R)}, \aleph_1^{L(\R)},\cdots, \aleph_k^{L(\R)}).$$ \bigskip \bigskip Thus the $\Ubf{\Gamma}_{1, 0}$ prewellorders are those alluded to at the end of $\S 2$ as ``slightly more complicated than $\Ubf{\Pi}^1_2$." \bigskip \bigskip \bigskip \bigskip Tony Martin has shown that: $$\bigcup_{k\in \N}\Ubf{\Gamma}_{1, k}=\bigcup_{k\in \N}\Game(\omega\cdot k- \Ubf{\Pi}^1_1).$$ \newpage \null \bigskip \bigskip \n {\bf 3.3 Theorem} (Woodin; c.1996) For each $k$, the sup of the $\Ubf{\Gamma}_{1, k}$ is strictly bounded below $\aleph_{\omega}^{L(\R)}$. \bigskip \bigskip \bigskip \bigskip \bigskip \bigskip Hugh Woodin's argument involved analysing where the tree representation for $\Sigma^1_3$ appears in HOD$^{L(\R)}$. \bigskip \bigskip \bigskip \bigskip This in turn makes it possible to determine that $\delta^M$ (the least $\delta$ which is Woodin in $L(M\cap {\mathbb V}_\delta)$) is sent to $\aleph_{\omega}^{L(\R)}$ in a certain direct limit of iterations of a suitable mouse $M$. \newpage \null \bigskip \bigskip \n {\bf 3.4 Theorem} (Hjorth; 1998) For each $k$ the sup of the ranks of the $\Ubf{\Gamma}_{1, k}$ prewellorders is exactly $\aleph_{2+k}^{L(\R)}$. \bigskip \bigskip \bigskip \bigskip \bigskip \bigskip \bigskip \bigskip The proof used a twist on the classical boundedness principle to obtain sharper bounds on the images of the ``responsible ordinals" in direct limits of the relevant mouse. \newpage \n {\bf $\S$4. So: What remains?} \null \bigskip \bigskip \bigskip \bigskip \n {\bf 4.1 Question} Can we nicely characterize the $\aleph_n^{L(\R)}$'s in HOD$^{L(\R)}$? \bigskip \bigskip \bigskip \bigskip \bigskip \bigskip Given a suitable mouse $M$ and a sufficiently universal direct limit of iterations, can we predict which ordinals go to the various $\aleph_n^{L(\R)}$'s? \newpage The methods of 3.3 suffice to give the precise bounds of $\ubf{\delta}^1_{2n}$ for prewellorders of the form $$x\leq y$$ iff $$C_{2n}(x, y, z)\models \psi(x, y, z),$$ but would appear to fall short of answering (for instance): \bigskip \bigskip \bigskip \bigskip \n {\bf 4.2 Question} For $k>1, n>1$, what are the precise bounds for $\Ubf{\Gamma}_{2n+1, k}$ prewellorders?\footnote{\Large Here I will define a $\Ubf{\Gamma}_{2n+1, k}$ set to be one unformily definable over the minimal iterable fine structural model for $2n$ Woodins, where we are allowed the first $k$ uniform indiscernibles as parameters. In some form this definition would appear due to Itay Neeman -- see {\it Optimal proofs of determinacy,} {\bf Bulletin of Symbolic Logic,} vol. 1(1995), pp. 327-339} \end{document}