The $\textit{cofinality}$ of a partially ordered set $\left( P,\leq \right)$, written $cof(P)$, is the smallest cardinality of a subset $T$ of $P$ that is unbounded in $P$, i.e. for every element $p\in P$ there is a larger element $q\in T$ such that $p\leq q$.

A $\sigma -ideal$ $I$ of $\mathbb{R}$ is a collection of subsets of $\mathbb{R}$ that is closed under taking subsets and countable unions, i.e. $U\in I, V\subset U \Rightarrow V\in I$ and $U_{n}\in I, n<\omega \Rightarrow \cup _{n<\omega }U_{n} \in I$.

We can now define the cofinality $cof(I)$ of a $\sigma$-ideal I as the cofinality of the partial order $\left( I,\subseteq \right)$.

We can prove some neat results such that if the ideal I is nonprincipal (i.e. it is not the set of all subsets of some subset $U\subset \mathbb{R}$), then its cofinality is at least uncountable.

My question is: are there any $\sigma$-ideals of $\mathbb{R}$ whose cofinality is greater than continuum, the cardinality of $\mathbb{R}$?

The cofinality of a partially ordered set $\left( P,\leq \right)$, written $cof(P)$, is the smallest cardinality of a subset $T$ of $P$ that is [EDIT: cofinal] in $P$, i.e. for every element $p\in P$ there is a larger element $q\in T$ such that $p\leq q$.

A $\sigma$-ideal $I$ of $\mathbb{R}$ is a collection of subsets of $\mathbb{R}$ that is closed under taking subsets and countable unions, i.e. $U\in I, V\subset U \Rightarrow V\in I$ and $U_{n}\in I, n<\omega \Rightarrow \cup _{n<\omega }U_{n} \in I$.

We can now define the cofinality $cof(I)$ of a $\sigma$-ideal I as the cofinality of the partial order $\left( I,\subseteq \right)$.

We can prove some neat results such that if the ideal I is nonprincipal (i.e. it is not the set of all subsets of some subset $U\subset \mathbb{R}$), then its cofinality is at least uncountable.

My question is: are there any $\sigma$-ideals of $\mathbb{R}$ whose cofinality is greater than continuum, the cardinality of $\mathbb{R}$?

EDIT: thanks for the terminology and markup tips :)