Surreal Number - "To Infinity ..."

"To Infinity ..."

Let there be an ordinal ω greater than the natural numbers, and define S_ω as the set of all surreal numbers generated by the construction rule from subsets of S_*. (This is the same inductive step as before, since the ordinal number ω is the smallest ordinal that is larger than all natural numbers; however, the set union appearing in the inductive step is now an infinite union of finite sets, and so this step can only be performed in a set theory that allows such a union.) A unique infinitely large positive number occurs in S_ω:

S_ω also contains objects that can be identified as the rational numbers. For example, the ω-complete form of the fraction 1/₃ is given by:

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The product of this form of 1/₃ with any form of 3 is a form whose left set contains only numbers less than 1 and whose right set contains only numbers greater than 1; the birthday property implies that this product is a form of 1.

Not only do all the rest of the rational numbers appear in S_ω; the remaining finite real numbers do too. For example

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The only infinities in S_ω are ω and -ω; but there are other non-real numbers in S_ω among the reals. Consider the smallest positive number in S_ω:

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This number is larger than zero but less than all positive dyadic fractions. It is therefore an infinitesimal number, often labeled ε. The ω-complete form of ε (resp. -ε) is the same as the ω-complete form of 0, except that 0 is included in the left (resp. right) set. The only "pure" infinitesimals in S_ω are ε and its additive inverse -ε; adding them to any dyadic fraction y produces the numbers y±ε, which also lie in S_ω.

One can determine the relationship between ω and ε by multiplying particular forms of them to obtain:

ω · ε = { ε · S₊ | ω · S₊ + S_* + ε · S_* }.

This expression is only well-defined in a set theory which permits transfinite induction up to . In such a system, one can demonstrate that all the elements of the left set of ω · ε are positive infinitesimals and all the elements of the right set are positive infinities, and therefore ω · ε is the oldest positive finite number, i. e., 1. Consequently,

1/_ε = ω.

Some authors systematically use ω−1 in place of the symbol ε.