# Logic1109 - Generalized induction 0 1 2 Generalized...

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Generalized induction First we define a component relation between elements of an arbitrary set. C(x, y): x is a component of y. Ex1. The set N = {0, 1, 2, 3. …}, C(x, y) if x < y Ex2. The same set N, but now C(x, y) if x+1 =y. Ex3. The set A = {p, ~p, ~~p, ~~~p, …}, C(X, Y) if ~X = Y.
Descending chain is a finite sequence (x1, x2, …, xn) or an infinite sequence (x1, x2, …, xn, …) such that each term other than x1 is a component of the previous term. From the previous examples, (n, n-1, …, 2, 1, 0) and (~~~p, ~~p, ~p, p) are des- cending chains respectively. G In the first case if C(x, y) is defined to be x < y, you may skip some numbers.

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16.2 Suppose all descending chains are finite. Then it follows: an element x is not a component of itself. Otherwise, there is an infinite descending chain (x, x, …, x, x, … ). No two elements x and y are not compon- ents of each other. Otherwise, there is an infinite descending chain (x, y, x, y, …, x, y, … ).
16.3 The set N = {0, 1, 2, …} C(x, y) if x+1 =y (y is called the

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Logic1109 - Generalized induction 0 1 2 Generalized...

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