Set theory: on the structure of the real line.

*(English)*Zbl 0834.04001
Wellesley, MA: A. K. Peters Ltd. ix, 546 p. (1995).

This book is about measure and category on the real line and set theory. It covers the results below as well as much more than can be mentioned here.

Let \(I\) be a nontrivial ideal of subsets of the reals, \(\mathbb{R}\). Define: \[ \begin{alignedat}{2} &\text{additivity}:\;&\text{add}(I) &= \min\{|F|:\;F\subseteq I,\;\bigcup F\not\in I\},\\ &\text{cofinality}: &\text{cof}(I) &= \min \{|F|:\;F\subseteq I,\;\forall A\in I\;\exists B\in FA\subseteq B\},\\ &\text{covering}: &\text{cov}(I) &= \min \{|F|:\;F\subseteq I,\;\bigcup F=\mathbb{R}\}, \\ &\text{non}: &\text{non}(I) &= \min \{|X|:\;X\subseteq \mathbb{R},\;X\not\in I\}. \end{alignedat} \] The last three cardinals are also known in the literature under the names, Covering (cof), Baire (cov), and Uniformity (non). The three latter abbreviations (cof, cov, non) were introduced by Cichoń and I think are an improvement over these earlier names used by Kunen and Miller.

For any \(\sigma\)-ideal \(I\) in \(\mathbb{R}\) which contains all singletons and doesn’t contain \(\mathbb{R}\), it is easy to see that add is \(\leq\) both cov and non; and both of these are \(\leq\text{cof}\).

The two main ideals are: \[ \begin{aligned} &{\mathcal N} \text{ (Null) the ideal of measure zero subsets of the real line, and}\\ &{\mathcal M} \text{ (Meager) the ideal of first category subsets of the real line}. \end{aligned} \] Two other cardinals which are closely related to these are \({\mathfrak b}\) and \({\mathfrak d}\), \[ \begin{alignedat}{2} &\text{ḏominating}:\;&{\mathfrak d}&=\min\{|F|:\;F\subseteq\omega^\omega,\;\forall f\in\omega^\omega\;\exists g\in F\;\forall^\infty n\;f(n)\leq g(n)\},\\ &\text{unḇounded}:&{\mathfrak b}&=\min\{|F|:\;F\subseteq\omega^\omega,\;\forall f\in\omega^\omega\;\exists g\in F\;\exists^\infty n\;f(n)\leq g(n)\}. \end{alignedat} \] The symbols \(\forall^\infty\), \(\exists^\infty\) stand for “for all but finitely many” and “there exist infinitely many”, respectively. These cardinals can also be characterized in terms of the \(\sigma\)-ideal \({\mathcal K}\), which is the \(\sigma\)-ideal generated by the compact subsets of \(\omega^\omega\). Namely \({\mathfrak d}=\text{add}({\mathcal K})=\text{non}({\mathcal K})\) and \({\mathfrak b}=\text{cof}({\mathcal K})=\text{cov}({\mathcal K})\). In this case the four cardinals collapse to two.

Rothberger proved that \[ \text{cov}({\mathcal M})\leq\text{non}({\mathcal N})\qquad\text{and} \qquad \text{cov} ({\mathcal N})\leq \text{non} ({\mathcal M}). \] Miller and Truss proved that \[ \text{add} ({\mathcal M})= \min \{\text{cov} ({\mathcal M}), {\mathfrak d}\}. \] Fremlin pointed out that a similar argument shows that \[ \text{cof} ({\mathcal M})= \max \{\text{non} ({\mathcal M}), {\mathfrak b}\}. \] Bartoszyński and, independently, Stern and Raisonnier showed that \[ \text{add}({\mathcal N})\leq \text{add}({\mathcal M}). \] It is easy to dualize their arguments to get \[ \text{cof}({\mathcal M})\leq \text{cof} ({\mathcal N}). \] The weaker result \(\text{add} ({\mathcal N})\leq {\mathfrak b}\) was earlier proved by Miller. Putting these facts together we get Cichoń’s diagram: \[ \begin{tikzcd} \operatorname{cov}(\mathcal N) \ar[r] & \operatorname{non}(\mathcal M) \ar[r] & \operatorname{cof}(\mathcal M) \ar[r] & \operatorname{cof}(\mathcal N)\\ & \mathfrak b \ar[u] \ar[r] & \mathfrak d \ar[u] & \\ \operatorname{add}(\mathcal N) \ar[uu] \ar[r] & \operatorname{add}(\mathcal M) \ar[u] \ar[r] & \operatorname{cov}(\mathcal M) \ar[u] \ar[r] & \operatorname{non}(\mathcal N)\ar[uu] \end{tikzcd} \]

The arrows mean \(\leq\). Cichoń’s diagram was also independently discovered by Jech. If we restrict to models of set theory where the continuum is \(\omega_2\), then no other relationships hold between these ten cardinals. This involves the construction of a number of forcing relations and generic models of set theory. The result was proved by Martin-Solovay, Kunen, Miller, Kamburelis, Krawczyk, Cichoń, Judah-Shelah, and Bartoszyński-Judah-Shelah. For the forcing arguments the main difficulty is often whether some property is preserved by iterated forcing. This book contains many results (mostly due to Shelah) that some particular property is preserved by the countable support iteration of proper forcings.

There are some interesting combinatorial equivalents of these cardinals. For example, Bartoszyński and Miller showed that: \[ \begin{aligned} \text{cov} ({\mathcal M}) &= \min \{|F|:\;F\subseteq \omega^\omega,\;\neg \exists g\in \omega^\omega\;\forall f\in F\;\exists^\infty n\;g(n) =f(n)\},\\ \text{non} ({\mathcal M}) &= \min \{|F|:\;F\subseteq \omega^\omega,\;\neg \exists g\in \omega^\omega\;\forall f\in F\;\forall^\infty n\;g(n)\neq f(n)\}. \end{aligned} \] For measure, Bartoszyński proved the following combinatorial characterizations. Let \[ {\mathcal C}= \{H:\;\omega\to [\omega ]^{<\omega}:\;\forall n\in \omega\;|H(n)|\leq n^2\}. \] Then \[ \begin{aligned} \text{add} ({\mathcal N}) &= \min \{|F|:\;F\subseteq \omega^\omega,\;\neg \exists H\in {\mathcal C} \forall f\in F\;\forall^\infty n\;f(n)\in H(n)\},\\ \text{cof} ({\mathcal N)} &= \min \{|F|:\;F\subseteq {\mathcal C},\;\forall f\in\omega^\omega\;\exists H\in F\;\forall^\infty n\;f(n)\in H(n)\}.\end{aligned} \] It was earlier shown by Miller that the cofinality of \(\text{cov} ({\mathcal M})\) cannot be countable (a fact which is immediate from the above characterization of \(\text{cov} ({\mathcal M})\)). Bartoszyński-Judah-Shelah showed that \[ \text{cof} (\text{cov} ({\mathcal M}))\geq \text{add} ({\mathcal N}). \] Recently it was shown by Shelah to be consistent that \(\text{cov} ({\mathcal N})= \aleph_\omega\).

Other topics which are taken up in this book are:

\(\bullet\) filters on \(\omega\) and their measurability or baireness,

\(\bullet\) strong measure sets, strongly meager sets and the Borel conjecture, and

\(\bullet\) projective sets \((\Sigma^1_n)\) and their measurability or baireness.

Let \(I\) be a nontrivial ideal of subsets of the reals, \(\mathbb{R}\). Define: \[ \begin{alignedat}{2} &\text{additivity}:\;&\text{add}(I) &= \min\{|F|:\;F\subseteq I,\;\bigcup F\not\in I\},\\ &\text{cofinality}: &\text{cof}(I) &= \min \{|F|:\;F\subseteq I,\;\forall A\in I\;\exists B\in FA\subseteq B\},\\ &\text{covering}: &\text{cov}(I) &= \min \{|F|:\;F\subseteq I,\;\bigcup F=\mathbb{R}\}, \\ &\text{non}: &\text{non}(I) &= \min \{|X|:\;X\subseteq \mathbb{R},\;X\not\in I\}. \end{alignedat} \] The last three cardinals are also known in the literature under the names, Covering (cof), Baire (cov), and Uniformity (non). The three latter abbreviations (cof, cov, non) were introduced by Cichoń and I think are an improvement over these earlier names used by Kunen and Miller.

For any \(\sigma\)-ideal \(I\) in \(\mathbb{R}\) which contains all singletons and doesn’t contain \(\mathbb{R}\), it is easy to see that add is \(\leq\) both cov and non; and both of these are \(\leq\text{cof}\).

The two main ideals are: \[ \begin{aligned} &{\mathcal N} \text{ (Null) the ideal of measure zero subsets of the real line, and}\\ &{\mathcal M} \text{ (Meager) the ideal of first category subsets of the real line}. \end{aligned} \] Two other cardinals which are closely related to these are \({\mathfrak b}\) and \({\mathfrak d}\), \[ \begin{alignedat}{2} &\text{ḏominating}:\;&{\mathfrak d}&=\min\{|F|:\;F\subseteq\omega^\omega,\;\forall f\in\omega^\omega\;\exists g\in F\;\forall^\infty n\;f(n)\leq g(n)\},\\ &\text{unḇounded}:&{\mathfrak b}&=\min\{|F|:\;F\subseteq\omega^\omega,\;\forall f\in\omega^\omega\;\exists g\in F\;\exists^\infty n\;f(n)\leq g(n)\}. \end{alignedat} \] The symbols \(\forall^\infty\), \(\exists^\infty\) stand for “for all but finitely many” and “there exist infinitely many”, respectively. These cardinals can also be characterized in terms of the \(\sigma\)-ideal \({\mathcal K}\), which is the \(\sigma\)-ideal generated by the compact subsets of \(\omega^\omega\). Namely \({\mathfrak d}=\text{add}({\mathcal K})=\text{non}({\mathcal K})\) and \({\mathfrak b}=\text{cof}({\mathcal K})=\text{cov}({\mathcal K})\). In this case the four cardinals collapse to two.

Rothberger proved that \[ \text{cov}({\mathcal M})\leq\text{non}({\mathcal N})\qquad\text{and} \qquad \text{cov} ({\mathcal N})\leq \text{non} ({\mathcal M}). \] Miller and Truss proved that \[ \text{add} ({\mathcal M})= \min \{\text{cov} ({\mathcal M}), {\mathfrak d}\}. \] Fremlin pointed out that a similar argument shows that \[ \text{cof} ({\mathcal M})= \max \{\text{non} ({\mathcal M}), {\mathfrak b}\}. \] Bartoszyński and, independently, Stern and Raisonnier showed that \[ \text{add}({\mathcal N})\leq \text{add}({\mathcal M}). \] It is easy to dualize their arguments to get \[ \text{cof}({\mathcal M})\leq \text{cof} ({\mathcal N}). \] The weaker result \(\text{add} ({\mathcal N})\leq {\mathfrak b}\) was earlier proved by Miller. Putting these facts together we get Cichoń’s diagram: \[ \begin{tikzcd} \operatorname{cov}(\mathcal N) \ar[r] & \operatorname{non}(\mathcal M) \ar[r] & \operatorname{cof}(\mathcal M) \ar[r] & \operatorname{cof}(\mathcal N)\\ & \mathfrak b \ar[u] \ar[r] & \mathfrak d \ar[u] & \\ \operatorname{add}(\mathcal N) \ar[uu] \ar[r] & \operatorname{add}(\mathcal M) \ar[u] \ar[r] & \operatorname{cov}(\mathcal M) \ar[u] \ar[r] & \operatorname{non}(\mathcal N)\ar[uu] \end{tikzcd} \]

The arrows mean \(\leq\). Cichoń’s diagram was also independently discovered by Jech. If we restrict to models of set theory where the continuum is \(\omega_2\), then no other relationships hold between these ten cardinals. This involves the construction of a number of forcing relations and generic models of set theory. The result was proved by Martin-Solovay, Kunen, Miller, Kamburelis, Krawczyk, Cichoń, Judah-Shelah, and Bartoszyński-Judah-Shelah. For the forcing arguments the main difficulty is often whether some property is preserved by iterated forcing. This book contains many results (mostly due to Shelah) that some particular property is preserved by the countable support iteration of proper forcings.

There are some interesting combinatorial equivalents of these cardinals. For example, Bartoszyński and Miller showed that: \[ \begin{aligned} \text{cov} ({\mathcal M}) &= \min \{|F|:\;F\subseteq \omega^\omega,\;\neg \exists g\in \omega^\omega\;\forall f\in F\;\exists^\infty n\;g(n) =f(n)\},\\ \text{non} ({\mathcal M}) &= \min \{|F|:\;F\subseteq \omega^\omega,\;\neg \exists g\in \omega^\omega\;\forall f\in F\;\forall^\infty n\;g(n)\neq f(n)\}. \end{aligned} \] For measure, Bartoszyński proved the following combinatorial characterizations. Let \[ {\mathcal C}= \{H:\;\omega\to [\omega ]^{<\omega}:\;\forall n\in \omega\;|H(n)|\leq n^2\}. \] Then \[ \begin{aligned} \text{add} ({\mathcal N}) &= \min \{|F|:\;F\subseteq \omega^\omega,\;\neg \exists H\in {\mathcal C} \forall f\in F\;\forall^\infty n\;f(n)\in H(n)\},\\ \text{cof} ({\mathcal N)} &= \min \{|F|:\;F\subseteq {\mathcal C},\;\forall f\in\omega^\omega\;\exists H\in F\;\forall^\infty n\;f(n)\in H(n)\}.\end{aligned} \] It was earlier shown by Miller that the cofinality of \(\text{cov} ({\mathcal M})\) cannot be countable (a fact which is immediate from the above characterization of \(\text{cov} ({\mathcal M})\)). Bartoszyński-Judah-Shelah showed that \[ \text{cof} (\text{cov} ({\mathcal M}))\geq \text{add} ({\mathcal N}). \] Recently it was shown by Shelah to be consistent that \(\text{cov} ({\mathcal N})= \aleph_\omega\).

Other topics which are taken up in this book are:

\(\bullet\) filters on \(\omega\) and their measurability or baireness,

\(\bullet\) strong measure sets, strongly meager sets and the Borel conjecture, and

\(\bullet\) projective sets \((\Sigma^1_n)\) and their measurability or baireness.

Reviewer: A.W.Miller (Madison)

##### MSC:

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |