Anzeige
Erschienen in:
2007 | OriginalPaper | Buchkapitel
Higher Dimensions
Erschienen in: Walks on Ordinals and Their Characteristics
Verlag: Birkhäuser Basel
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
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
$$ g\left( {\alpha _0 ,\alpha _1 , \ldots ,\alpha _n } \right) = f\left( {e\left( {\alpha _0 ,\alpha _n } \right), \ldots ,e\left( {\alpha _{n - 1} ,\alpha _n } \right)} \right). $$