Dean Derkson’s Brute force argument
October 9, 2010
The continuum hypothesis states that there is no set whose cardinality is strictly between that of the Integers and that of the Real numbers.
My Brute force argument:
Is there a countably infinite set that can be put into 1 to 1 correspondence with the Real numbers?
Let’s start smaller. Can a countable set correspond with [0, 1)?
Let’s try Brute Force to make one:
Let’s make a list of fractions and start at 0/1. Lets call this Set 1
(Note we could write this as 0.0000000..)
Then we will add {1/10, 2/10, 3/10, 4/10, 5/10, 6/10, 7/10, 8/10, 9/10}. This is Set 2
This with the first element cover all possible combinations of digits at the first decimal place.
We can then continue and add 90 more elements.
{1/100, 2/100, 3/100... 9/100
11/100, 12/100, 13/100 … 19/100
21/100 … 29/100
31/100 … 39/100
.
.
.
91/100 … 99/100} This is Set 3
So now we have covered all the possible combinations of digits on the second decimal place.
We can continue this, making sets like Set n for all the positive integers n.
(Set 1 … Set n) will contain elements with all possible combinations of digits for the first n-1 decimal places.
This sequence of elements is countably infinite. We can make a bijection between each member on this list and the positive Integers.
In [0,1) there are Real numbers also. Lets pick an irrational one and call it C.
(C could be something like Pi/4)
Note: that we could even pick a rational one like 1/3.
As we can see C is not an element of any Set 1, Set2 … Set 100:
Lets assume that C is not an element of any Set n where n is a positive Integer.
(After all irrational numbers cannot be written as simple integer fractions. Proof elsewhere)
As the irrational number is irrational then at some point it must diverge from all possible elements on this list. Where does this divergence start? Where is the point of first divergence?
There must exist a first decimal point on C, let’s call it Cm at position m, Where C1 ...Cm-1 is the same as some element in Set1 … Set m but at Cm it is different from all 9 new possible elements in Set1 to Set m+1. Cm can not be {1,2,3,4,5,6,7,8, or 9}(the decimal combination C1 to Cm-1, 0 is in some Set m or below). But Cm must be one of the digits. We cant get the next one as Cm is the first. This is not possible.
Therefore: We have a contradiction:
C is an element of some Set n.
C is arbitrary.
C is an element of some Set n.
C is arbitrary.
Therefore any number C belonging to the Reals on [0,1) is an element of some
Set n.
There is no set whose cardinality is strictly between the Integers and the Reals.
QED
QED
What are we left with then:
Are proofs in infinity different from proofs dealing with “static” numbers?
What about Cantor’s diagonal proof? Is there another kind of number in the reals that is not rational, or irrational.
