By Lisa Jacobsen

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Since α β · αγ = α β+γ for ordinals α, β, and γ (as mentioned in the previous section), we have ω · ω ω = ω 1 · ω ω = ω 1 +ω = ω ω . Thus we can omit powers of ω in products to the left of “much larger” powers. Omitting redundant terms in sums and products leads to a notation for ordinals known as the Cantor normal form. 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.

Axiom 2. There is a set with no members, called the empty set. Axiom 3. For any sets X and Y, there is a set whose only members are X and Y. (This set, { X, Y }, is called the unordered pair of X and Y. Note that, when Y = Z, Axiom 1 gives {Y, Z } = {Y }. ) The ZF axioms are the following. Axiom 4. For any set X there is a set whose members are the members of members of X. (In the case where X = {Y, Z }, the members of members of X form what is called the union of Y and Z, denoted by Y ∪ Z. ) Axiom 5.

9 shows the curve for 1 For an up-to-date report on The Method, with some interesting mathematical speculations, see Netz and Noel (2007). ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 22 1. 9. Graph of the polynomial y = x5 − x + 1. the actual example y = x 5 − x + 1. Since the curve passes from below to above the x-axis, and the x-axis has no gaps, the curve necessarily meets the x-axis. That is, there is a real value of x for which x5 − x + 1 = 0. Similarly, any odd-degree polynomial equation has a real solution, and the fundamental theorem of algebra follows.

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Analytic theory of continued fractions III by Lisa Jacobsen
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