2
The original definition of o-minimality form mathematical
logic (model theory) in this context is as follows. We say
that f:
n
m
is o-minimal (over the ordered field of
reals) if and only if every subset of that is (first
order) definable in ( ,<,0,1,+,•,f) is the union of
finitely many open intervals and finitely many points.
More generally, we say that V is collectively o-minimal ( )
if and only if V obeys MULTIVARIATE ( ) and every subset of
that is (first order) definable in the structure M =
(
,<,0,1,+,•,f,g,h,.
..), where {f,g,h,.
..} = V, is a finite
union of open intervals and points.
We prove the following.
THEOREM A. f:R
n
R
m
is o-minimal (
) if and only if f lies
in some special rich class ( ). More generally, V is
collectively o-minimal ( ) if and only if V is a subset of
some special rich class ( ). Furthermore, if V is
collectively o-minimal ( ) then there is a least rich class
(
) containing V, and it is special.
1. RICH CLASSES OF REAL FUNCTIONS.
Let f:
n
m
. A zero selector of f is a g with the
property in the ZERO SELECTION ( ) clause.
We now fix a rich class V of real functions. We let M =
(
,<,+,•,f,g,h.
..), where V = {f,g,h,.
..}.
LEMMA 1.1. There exist f
1
,...,f
8
:
V and f
9
,...,f
12
:
4
such that
i. f
1
(x) = 1/x if x ≠ 0.
ii. f
2
(x) = 0 if x ≠ 0; 1 otherwise.
iii. f
3
(0) = 0; f
3
(x) = 1 elsewhere.
iv. x
≥
0
´
(f
4
(x) = sqrt(x) f
4
(x) = -sqrt(x)).
v. f
5
(x) = 0
´
x
≥
0.
vi. f
6
(x) = 0
´
x > 0.
vii. f
7
(x) = 0 if x
≥
0; 1 otherwise.
viii. f
8
(x) = 0 if x > 0; 1 otherwise.
ix. f
9
(x,y,z,w) = z if x
£
y; w otherwise.
x. f
10
(x,y,z,w) = z if x < y; w otherwise.
xi. f
11
(x,y,z,w) = z if x ≠ y; w otherwise.
xii. f
12
(x,y,z,w) = z if x = y; w otherwise.