When
A
A
and
B
B
are countable dense subsets of
R
\mathbb {R}
, it is a well-known result of Cantor that
A
A
and
B
B
are order-isomorphic. A theorem of K.F. Barth and W.J. Schneider states that the order-isomorphism can be taken to be very smooth, in fact the restriction to
R
\mathbb {R}
of an entire function. J.E. Baumgartner showed that consistently
2
ℵ
0
>
ℵ
1
2^{\aleph _0}>\aleph _1
and any two subsets of
R
\mathbb {R}
having
ℵ
1
\aleph _1
points in every interval are order-isomorphic. However, U. Abraham, M. Rubin and S. Shelah produced a ZFC example of two such sets for which the order-isomorphism cannot be taken to be smooth. A useful variant of Baumgartner’s result for second category sets was established by S. Shelah. He showed that it is consistent that
2
ℵ
0
>
ℵ
1
2^{\aleph _0}>\aleph _1
and second category sets of cardinality
ℵ
1
\aleph _1
exist while any two sets of cardinality
ℵ
1
\aleph _1
which have second category intersection with every interval are order-isomorphic. In this paper, we show that the order-isomorphism in Shelah’s theorem can be taken to be the restriction to
R
\mathbb {R}
of an entire function. Moreover, using an approximation theorem of L. Hoischen, we show that given a nonnegative integer
n
n
, a nondecreasing surjection
g
:
R
→
R
g\colon \mathbb {R}\to \mathbb {R}
of class
C
n
C^n
and a positive continuous function
ϵ
:
R
→
R
\epsilon \colon \mathbb {R}\to \mathbb {R}
, we may choose the order-isomorphism
f
f
so that for all
i
=
0
,
1
,
…
,
n
i=0,1,\dots ,n
and for all
x
∈
R
x\in \mathbb {R}
,
|
D
i
f
(
x
)
−
D
i
g
(
x
)
|
>
ϵ
(
x
)
|D^if(x)-D^ig(x)|>\epsilon (x)
.