Walks on Ordinals and Their Characteristics pp 289-312 | Cite as

# Higher Dimensions

Chapter

## Abstract

The reader must have already noticed that in this book so far, we have only considered functions of the form of one-place functions. To obtain analogous results about functions defined on higher-dimensional cubes [

*f*: [*θ*]^{2}→*I*or equivalently sequences$$
f_\alpha :\alpha \to I\left( {\alpha < \theta } \right)
$$

*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:$$
g\left( {\alpha _0 ,\alpha _1 , \ldots ,\alpha _n } \right) = f\left( {e\left( {\alpha _0 ,\alpha _n } \right), \ldots ,e\left( {\alpha _{n - 1} ,\alpha _n } \right)} \right).
$$

(10.1.1)

## Keywords

Relevant Object Countable Subset Regular Cardinal Stationary Subset Singular Cardinal
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Birkhäuser Verlag AG 2007