We define and investigate
ω
\omega
-cohesiveness, a strong notion of indecomposability for subsets of the integers and their isols. This notion says, for example, that if
X
X
is the isol of an
ω
\omega
-cohesive set then, for any integer
n
n
implies that, for some integer
k
,
⋅
(
X
−
k
n
)
≤
Y
k, \cdot (\begin {array}{*{20}{c}} {X - k} \\ n \\ \end {array} ) \leq Y
or
Z
Z
. From this it follows that if
f
(
x
)
∈
T
1
f(x) \in {T_1}
, the collection of almost recursive combinatorial polynomials, then the predecessors of
f
Λ
(
X
)
{f_\Lambda }(X)
are limited to isols
g
Λ
(
X
)
{g_\Lambda }(X)
where
g
(
X
)
∈
T
1
g(X) \in {T_1}
. We show existence of
ω
\omega
-cohesive sets. And we show that the isol of an
ω
\omega
-cohesive set is an
n
n
-order indecomposable isol as defined by Manaster. This gives an alternate proof to one half of Ellentuck’s theorem showing a simple algebraic difference between the isols and cosimple isols. In the last section we study functions of several variables when applied to isols of
ω
\omega
-cohesive sets.