Question

Let $\mu$ be any infinite cardinal, and define a collection $N\subset[\mu]^\mu$ to be, maximal almost disjoint (MAD) over $\mu$, iff

1. $\forall\{A,B\}\in[N]^2$ $( A\cap B \in [\mu]^{<\mu})$
2. $\forall X\in[\mu]^\mu \exists A\in N$ $( X \cap A \in [\mu]^\mu)$

My questions are as follows: when $\mu$ is singular

1. What is known about MAD families (or any other combinatorial structures, like, towers, SFIP families without pseudo-intersection, etc) over $\mu$?
2. Are such families degenerate in the sense that an infinite family can have cardinality below $\mu$?
3. Is there any connection between such constructs on $\mu$ and the corresponding constructs on $cf(\mu)$?

The main point I really want to know is this: Is it possible to add new subsets to an arbitrary singular cardinal without adding new subsets the the cardinals below it?

Side Request: I've been told there are forcing constructions which will add an order type $\omega$ cofinal sequence to a cardinal with cofinality $\omega$, can anyone point me in the correct direction with a book or article?

Answer 1

Theorem If $0^{\sharp}$ does not exist and $\lambda$ is a singular cardinal, then any forcing adding subsets to $\lambda$ necessarily adds subsets to a cardinal below $\lambda$.

Proof: Let $\mathbb{P}$ be a partial order in the ground model and $G \subseteq \mathbb{P}$ be $V$-generic. Without loss of generality, we may assume that $\mathbb{P}$ is a partial order on a cardinal so that its elements are all ordinals. Also let $\vec{s} = \langle s_{\alpha}| \alpha < \text{cof}(\lambda)\rangle$ be a cofinal sequence in $\lambda$ in the ground model and $\dot{A}$ a $\mathbb{P}$-name for a subset of $\lambda$ in $V[G]$. Now suppose that $V$ and $V[G]$ agree on the bounded subsets of $\lambda$. Then for all $\alpha$, we have $a_{\alpha} = s_{\alpha} \cap \dot{A}_G \in \mathcal{P}(\lambda)^{V}$ so for every $\alpha < \text{cof}(\lambda)$, there will be:

$p_{\alpha} \in G$ such that $p_{\alpha} \Vdash \dot{A} \cap \check{s_{\alpha}} = \check{a_{\alpha}}$

Because $V$ is a definable class in $V[G]$, the forcing relation for $V$ is definable in $V[G]$, and we may therefore choose such a $\text{cof}(\lambda)$-sequence of conditions $p_{\alpha}$ below a condition forcing that $\dot{A}$ is a name for a subset of $\lambda$. Let $S_A = \{p_{\alpha}| \alpha < \text{cof}(\lambda)\} \in V[G]$ be such a set of ordinals. Now $S_A$ is a set of ordinals having size $\text{cof}(\lambda)$ in $V[G]$ so by the nonexistence of $0^{\sharp}$, it follows from Jensen's covering lemma that there is a constructible set $C$ of size $\theta = \max\{\omega_1, \textrm{cof}(\lambda)\}$ in $V[G]$ covering $S_A$. Then since $\lambda$ is singular, $\theta < \lambda$ whereby $C \in L \subseteq V$ will also have size $\theta$ in $V$ by virtue of the fact that a poset adding no new subsets to $\theta$ cannot collapse any cardinals below $\theta^{++}$. But now $S_{A} \subseteq C$ must also be in $V$ because otherwise $V[G]$ would be adding a subset of $\theta$ induced by $f''S_{A}$ where $f: C \rightarrow \theta$ is a bijection in the ground model. However, then $V$ can construct $A$ from $S_A$ in the ground model since

$A = \bigcup\{a \in \mathcal{P}(\lambda)| p \in S_{A} \land s_{\alpha} \in \text{range}(\vec{s}) \land p \Vdash \dot{A} \cap \check{s_{\alpha}} = \check{a}\}$. $\Box$

In particular, this shows that if $V$ is a forcing extension of $L$, then we cannot add a subset to a singular cardinal without adding a subset to a cardinal below it. I don't have an answer for what happens when $0^{\sharp}$ does exist, but at least this shows that your very interesting question is closely tied to the existence of certain large cardinals.

Answer 2

In question 2, perhaps I am missing something, but it seems that such families can have cardinality below $\mu$: just take $M = \{\mu\}$. That satisfies your definition of MAD and has 1 element. (Generally, it's easy to get small MAD families--the trick is constructing big ones, right? Or were you wondering if there are $\mu$ for which every MAD family is smaller than $\mu$?)

On your main point, I don't have an answer, but here is at least an observation. Let $\mu$ be singular and let $\kappa = cf(\mu)$. Let $P$ be a notion of forcing that adds new subsets to $\mu$ without adding subsets to any cardinals below $\mu$. Then $P$ is not $\kappa$-distributive (and hence not $\kappa$-closed). (Here, $\kappa$-closed is in the sense of Jech, not Kunen: $P$ is $\kappa$-closed if for $\lambda\leq\kappa$, descending $\lambda$-sequences in $P$ have lower bounds. See p. 228 of Jech 3rd millennium edition.) Proof: Let $\lambda_\alpha$, $\alpha<\kappa$, be cofinal in $\mu$. Let $x\subseteq\mu$ be in $V[G]$ but not $V$. Define $f:\kappa\to V$ by $f(\alpha) = x\cap \aleph_{\lambda_\alpha}$. We cannot have $f\in V$, otherwise $x$ would be in $V$. By Theorem 15.6 of Jech, then, $P$ cannot be $\kappa$-distributive.

As for your side request, if $\mu$ has cofinality $\omega$, there is at least one easy way to add order type $\omega$ cofinal sequences into $\mu$. Let $\lambda_n$, $n<\omega$, be a cofinal increasing sequence in $\mu$. Use any notion of forcing that adds new reals without collapsing any cardinals you care about (like adding Cohen reals). If $x\subseteq \omega$ is new, then $\lambda_n$ for $n\in x$ is a new cofinal sequence in $\mu$.

I'm not a set theorist and I haven't eaten in several hours, so that should all be taken with a grain of salt.

Answer 3

With regards to 2 and 3, if you have a MAD family of size $\kappa$ on $cf(\mu)$ then you have a MAD family of size $\kappa$ on $\mu$. (So, for example there would be a 'degenerate family' of the form you mention in 2 on $\aleph_\omega$ if there is a MAD family of size $\aleph_1$ on $\omega$).

Let me give the argument for $\mu=\aleph_\omega$. Suppose $\mathcal{A}$ is a MAD family on $\omega$. For each $A\subseteq\omega$, let $B(A)\subseteq\aleph_\omega$ be the union of the intervals $I_n=[\aleph_n,\aleph_{n+1})$ such that $n\in A$. Let $\mathcal{B}$ be all the $B(A)$, for $A\in\mathcal{A}$. We claim that $\mathcal{B}$ is MAD.

The only interesting thing to check is the maximality. Suppose that $C\subseteq\aleph_\omega$ has cardinality $\aleph_\omega$. Let $n_0 < n_1 < n_2 < \ldots $ be a sequence such that $|I_{n_{k+1}}\cap C|\geq\aleph_{{n_k}+2}$. (The +2 is there to make sure the $n_{k+1}$ we find must be bigger than $n_k$). Letting $X=\{n_k:k\in\omega\}$, there is $Y\in\mathcal{A}$ with $X\cap Y$ infinite. Then $B(Y)\cap C$ has cardinality $\aleph_\omega$.

As for general references to MAD families on singular cardinals, it looks like you should check out Erdos, Hechler's "On Maximal Almost-Disjoint Families over Singular Cardinals" http://www.renyi.hu/~p_erdos/1975-21.pdf and the more recent Kojman, Kubis and Shelah "On Two Problems of Erdos and Hechler: New Methods in Singular MADness" http://arxiv.org/abs/math/0406441