We consider saturation properties of ideals in models obtained by forcing with countable chain condition partial orderings. As sample results, we mention the following. If
M
[
G
]
M[G]
is obtained from a model
M
M
of GCH via any
σ
\sigma
-finite chain condition notion of forcing (e.g. add Cohen reals or random reals) then in
M
[
G
]
M[G]
every countably complete ideal on
ω
1
{\omega _1}
is
ω
3
{\omega _3}
-saturated. If "
σ
\sigma
-finite chain condition" is weakened to "countable chain condition," then the conclusion no longer holds, but in this case one can conclude that every
ω
2
{\omega _2}
-generated countably complete ideal on
ω
1
{\omega _1}
(e.g. the nonstationary ideal) is
ω
3
{\omega _3}
-saturated. Some applications to
P
ω
1
(
ω
2
)
{\mathcal {P}_{{\omega _1}}}({\omega _2})
are included and the role played by Martin’s Axiom is discussed. It is also shown that if these weak saturation requirements are combined with some cardinality constraints (e.g.
2
ℵ
1
>
(
2
ℵ
0
)
+
)
{2^{{\aleph _1}}} > {({2^{{\aleph _0}}})^ + })
), then the consistency of some rather large cardinals becomes both necessary and sufficient.