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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, n1, …, 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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This note was uploaded on 02/04/2010 for the course HSS Hss105 taught by Professor Yeelee during the Fall '09 term at Korea Advanced Institute of Science and Technology.
 Fall '09
 Yeelee

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