Sunday, June 9, 2013

Ordinal numbers

Like all surreal numbers, ordinal numbers can be described by a transfinite sequence of terms that are ordered by order of magnitude. The predecessors of the ordinal number $\omega^\omega$ can all be described by a sequence of ordinal monomials $m \omega^n$ totally ordered by the value of the exponent $n$.

The predecessors of the ordinal number $\epsilon_0$ contain all the ordinal numbers that can be represented by exponential polynomials. Such ordinal numbers can be represented by sequences of ordinal numbers raised to the power of some other ordinal sequence such as $w^{w^2+2} + w^2 + 2$.

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