% LaTeX2e % Remark 3.4 A Dedekind finite Borel set % A. Miller August 2012 \documentclass[12pt]{article} \usepackage{amssymb} \pagestyle{myheadings} %\def\header{\today} \def\header{A. Miller \hfill } \markboth\header\header \def\col{{\mathbb Col}} \def\force{\forces} \def\forces{{\;\Vdash}} \def\group{{\mathcal H}} \def\la{\langle} \def\name#1{\stackrel{\circ}{#1}} \def\nn{{\mathcal N}} \def\om{\omega} \def\proof{\par\noindent Proof\par\noindent} \def\qed{\par\noindent QED\par\bigskip} \def\ra{\rangle} \def\res{\upharpoonright} \def\rmand{\mbox{ and }} \def\rmor{\mbox{ or }} \def\sm{{\setminus}} \def\st{\;:\;} % such that \def\su{\subseteq} \newtheorem{theorem}{Theorem}%[section] \newtheorem{prop}[theorem]{Proposition} \newtheorem{cor}[theorem]{Corollary} \newtheorem{define}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \begin{document} \begin{center} {\large Remark 3.4\\ A Dedekind Finite Borel Set} \end{center} \begin{flushright} Arnold W. Miller\\ August 2012 \end{flushright} \def\address{\begin{flushleft} Arnold W. Miller \\ miller@math.wisc.edu \\ http://www.math.wisc.edu/$\sim$miller\\ University of Wisconsin-Madison \\ Department of Mathematics, Van Vleck Hall \\ 480 Lincoln Drive \\ Madison, Wisconsin 53706-1388 \\ \end{flushleft}} Asaf Karagila pointed out that Remark 3.4 in my paper, A Dedekind finite Borel set \cite{ded}, directly contradicts Theorem 3.3 (c) in Does GCH imply AC locally? by A. Kanamori and D. Pincus \cite{kanamori}. Neither of the papers provides a proof.\footnote{ Aki Kanomori tells me that Theorem 3.3 (c) was incorrectly stated. Instead of $\nn$ it should be the Solovay-type inner model $\nn_0\subseteq \nn$. This is the model $HOD(^\omega ON)$ of $\nn$. See page 155 of {\bf Consequences of the Axiom of Choice} by Rubin and Howard who call $\nn_0$ Truss's model.} Here we give a proof of Remark 3.4 \cite{ded}. The Feferman-Levy model $\nn$ is described in Cohen \cite{cohen} p.143, Jech \cite{jech} p.142, and in the on-line only appendix to Miller \cite{longbor}. We use the notation from \cite{longbor}. $\col$ is the Levy collapse of $\aleph_\om$ , $\col_n$ the collapse of $\aleph_k$ for $kn$. Such an $n$ must exist because for any $x$ there exists $n$ such that $x\in F_n$ and hence $\pi_m(x)\in F_n$ for all $m$. It is easy to check that $h$ maps $B$ onto $2^\om$. \begin{prop} In the Feferman-Levy model $\nn$ the set $B$ has the property that there is no one-to-one map taking $2^\om$ into $B$. \end{prop} \proof Suppose for contradiction that there is in $\nn$ a one-to-one map $f:2^\om\to B$. Define $$C_n=\{x\in B\st \forall m>n \;\;\pi_m(x)=\pi_n(x)\}$$ and note that $B=\bigcup_nC_n$ and each $C_n$ is countable. Let $$p\forces \name{f}:2^\om\to \name{B} \mbox{ is one-to-one }$$ and suppose that $p\in\col_n$ and $H_n$ (the subgroup of $\group$ which is the identity on $\col_n$) fixes $\name{f}$. Since $C_n$ is countable and $f$ is one-to-one, we may choose $N>>n$ and $q\leq p$ such that $$q\forces {f}^{-1}(\name{C}_n)\su \name{F}_N.$$ Now let $\name{x_N}$ be a canonical name for a real which codes the generic collapse of $\aleph_N$. The important property it has is that $x_N\in F_{N+1}\sm F_{N}$ and its values are completely determined by $N^{th}$ coordinate of the generic filter and not any earlier ones, e.g., the $\leq n$ coordinates. Take $r\leq q$ such that $$r\forces \name{f}(\name{x}_N)=\name{y}, \pi_n(\name{y})=\tau, \rmand \tau\in \name{F}_{n+1}\sm \name{F}_n$$ where $\tau$ is a $\col_{n+1}$-name for an element of $2^\om$. For $G$ $\col$-generic over $M$ and containing $r$ note that $\tau^{G_{n+1}}\in M[G_{n+1}]\sm M[G_{n}]$. So we can find $k$ and $r_0,r_1\leq r$ such that \begin{itemize} \item $r_0$ and $r_1$ are the same on every coordinate except $n$, \item $r_0\forces \tau(k)=0$, and \item $r_1\forces \tau(k)=1$. \end{itemize} Now let $\rho\in\group$ be an automorphism of $\col$ which fixes every coordinate except $n$ and makes $\rho(r_0)$ and $r_1$ compatible. To do this move the domain of $r_0\res(\{n\}\times\om)$ disjoint from the domain of $r_1$. Since $\rho\in H_n$ it fixes $\name{f}$, i.e., $\rho(\name{f})=\name{f}$. Since $\rho$ fixes the $N^{th}$ coordinate $\rho(\name{x}_N)=\name{x}_N$. But this means that $$\rho(r_0)\forces \pi_n(\name{f}(\name{x}_N))(k)=0$$ and $$r_1\forces \pi_n(\name{f}(\name{x}_N))(k)=1$$ which contradicts that $\rho(r_0)$ and $r_1$ are compatible. \qed \begin{thebibliography}{99} \bibitem{cohen} Cohen, Paul J.; {\bf Set theory and the continuum hypothesis.} W. A. Benjamin, Inc., New York-Amsterdam 1966 vi+154 pp. \bibitem{jech} Jech, Thomas J.; {\bf The axiom of choice.} Studies in Logic and the Foundations of Mathematics, Vol. 75. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. xi+202 pp. \bibitem{kanamori} Kanamori, A.; Pincus, D.; Does GCH imply AC locally?, Paul Erdos and his mathematics, II (Budapest, 1999), 413-426, Bolyai Soc. Math. Stud., 11, János Bolyai Math. Soc., Budapest, 2002. http://math.bu.edu/people/aki/7.pdf \bibitem{longbor} Miller, Arnold W.; Long Borel hierarchies, Math Logic Quarterly, 54(2008), 301-316. http://www.math.wisc.edu/$\sim$miller/res/longbor.pdf \bibitem{ded} Miller, Arnold W.; A Dedekind Finite Borel Set, Arch. Math. Logic 50 (2011), no. 1-2, 1--17. http://www.math.wisc.edu/$\sim$miller/res/ded.pdf \end{thebibliography} \address \end{document}