to prove that the power set of n has a higher cardinal number than n we must show that it is not...
TRANSCRIPT
To prove that the power set of N has a higher cardinal number than N
we must show that it is not possible to establish a one to one
correspondence between the two:
Imagine that all subsets of N are listed below:
,4,2,16
,4,3,15
,5,3,14
,6,5,43
,6,5,22
,6,4,11
To prove that the power set of N has a higher cardinal number than N
we must show that it is not possible to establish a one to one
correspondence between the two:
,4,2,16
,4,3,15
,5,3,14
,6,5,43
,6,5,22
,6,4,11
Now number them:
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
The infinitely many natural numbers are listed here:
x is under a member
o is under a nonmember
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
Draw a diagonal
Suppose we have a numbered list of sets of natural numbers
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
Reverse the elements on the diagonal
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
The green set is different from every set listed!
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
The green set is different from set 1 because the first element is different
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
The green set is different from set 2 because the second element is different
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
The green set is different from set 3 because the third element is different
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
The green set is different from set 4 because the fourth element is different
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
For every k , the k th element in the green set differs from the k th element in the set numbered k. It is impossible to count the power set of the set of natural numbers!
,5,4,3
,6,2,1
,6,5,4,2,16
,4,3,15
,6,5,3,14
,6,5,43
,6,5,22
,6,4,11
654321
oxxxoo
xoooxx
xxxoxx
ooxxox
xxoxox
xxxooo
xxooxo
xoxoox
tes
No matter how you try to line up the subsets on the right to count them,you can use this method to produce a set (like the green one) that is not on the list!