By Erwin Kreyszig
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Extra info for Solutions manual for Advanced engineering mathematics 8ed
A cut is intuitively a separation of the rational numbers into a lower set L and an upper set U, as if by an infinitely sharp knife. 8. Historical Background 23 such that every member of L is less than every member of U. Cuts ( L, U ) represent both rational and irrational numbers as follows: • If L has a greatest member, or U has a least member, say r, then ( L, U ) represents the rational number r. • If L has no greatest member and U has no least member, then ( L, U ) represents an irrational number.
Another example is ω · 3, which represents the left-to right ordering 1, 4, 7, ... 2, 5, 8, ... 3, 6, 9, ... Conversely, any well-ordering of the positive integers is represented by what we call a countable ordinal—a countable set α with properties that generalize those of the particular ordinals mentioned above: α is wellordered by the membership relation, and any member of a member of α is a member of α. The countable ordinals go inconceivably far beyond the ordinal ε 0 that we struggled to reach in the last section.
The Axiom of Choice 39 The formal statement of the axiom of choice is quite simple: Axiom of choice. For any set X of nonempty sets S, there is a function choose( S) (a “choice function for X”) such that choose(S) is in S for each set S in X. 9. Here is how we use the axiom of choice to get out of our present quandary. For each countable limit ordinal α, let x α be the set of all increasing sequences with limit α, and then let X be the set of all such x α . A choice function F for X gives an increasing sequence F ( x α ), with limit α, for each countable limit ordinal α.