2007 | OriginalPaper | Buchkapitel
Higher Dimensions
Erschienen in: Walks on Ordinals and Their Characteristics
Verlag: Birkhäuser Basel
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The reader must have already noticed that in this book so far, we have only considered functions of the form
f
: [
θ
]
2
→
I
or equivalently sequences
$$ f_\alpha :\alpha \to I\left( {\alpha < \theta } \right) $$
of one-place functions. To obtain analogous results about functions defined on higher-dimensional cubes [
θ
]
n
, one usually develops some form of
stepping-up procedure
that lifts a function of the form
f
: [
θ
]
n
→
I
to a function of the form
g
: [
θ
+
]
n
+1
→
I
. The basic idea seems quite simple. One starts with a coherent sequence
e
α
:
α
→
θ
(
α
<
θ
+
) of one-to-one mappings and wishes to define
g
: [
θ
+
]
n
+1
→
I
as follows:
10.1.1
$$ g\left( {\alpha _0 ,\alpha _1 , \ldots ,\alpha _n } \right) = f\left( {e\left( {\alpha _0 ,\alpha _n } \right), \ldots ,e\left( {\alpha _{n - 1} ,\alpha _n } \right)} \right). $$