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From the PREFACE. THE e-book of this tract has been behind schedule through various factors, and i'm now forced to factor it with no Dr Riesz's assist in the ultimate correction of the proofs. This has at any price, one virtue, that it provides me the possibility of announcing how unsleeping i'm that no matter what worth it possesses is due regularly to his contributions to it, and particularly to the very fact, that it comprises the 1st systematic, account of his attractive concept of the summation of sequence via 'typical means'.

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To encode a number x in (0, 1) by a sequence of 0s and 1s we make successive bisections of the interval, each time choosing the subinterval in which x lies. If x lies in the left half, write down 0; if the right half, write down 1. Then bisect the subinterval in which x lies, and repeat the process. The resulting sequence of 0s and 1s is called the binary expansion of x. 8 shows how we find the binary expansion of 1/3. After the first bisection, 1/3 lies in the left half, so we write down 0. After bisecting the left half, 1/3 lies in the right half, so we write down 1.

This notation is needed only for ordinals less than ε 0 , that is, ordinals less than some member of the sequence ω ωω, ω, ωω , ωω ωω , ... Such an ordinal α necessarily falls between two terms of this sequence, and example will suffice to show how we obtain its notation. ω • Suppose that ω ω < α < ω ω . It follows that α falls between two ω terms of the following sequence with limit ω ω : ωω , ωω , 2 ωω , 3 ... • Suppose ω ω < α < ω ω . It follows that α falls between two terms 3 of the following sequence with limit ω ω : 2 3 ωω , ωω 2 2 ·2 , ωω 2 ·3 , ...

Fortunately, in this book we need names mainly for ordinals up to ε 0 , and a nice uniform system of such names exists. 6. Perhaps the most concrete way to realize countable ordinals is by sets of rational numbers, with the natural ordering. Indeed, we can stick to sets of rational numbers in the unit interval (0, 1). The ordinal ω, for example, is the ordinal number of the set 12 , 34 , 78 , . . 1 shows a picture of this set of rationals, marked by vertical lines whose height decreases towards zero as we approach ω.

### An Explicit Approach To Elementary Number Theory by stein

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