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?



Singular Cardinals, and A Strange Question.

> 
> **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.


