Solve the Continuum Hypothesis and get $1,000,000 and a job

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.

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

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.

Land

 
Real estate, or land, is real. It is something all people need. You need a place to put your feet. Sure there are still some nomadic peoples wandering around on the planet but that simply means they don't really own anything. They just wander around with the shirts on their back. Real estate is something people need, will pay for, and will fight and die for if necessary. Most other forms of money come from real estate.
Methods of making money

Farming (the next oldest profession)
Renting (selling land for a time)
storing something for others (like renting but people don't get total access to your land)
Mining (Now you need mineral rights)
camping (buy and wait till the value goes up)
Some people here in Korea buy a piece of land and put in a 10 story concrete structure. Then when an apartment block wants to build next door they have to pay for the sunshine they would deprive your building of.

Of course the main thing to do here is to get land people want or get people to want your land. There are parts of the coast of Chile which are really dry right next to the ocean. I would like to get a portable desalination plant, one that could run on solar power, and ship it to a piece of desert and then create my own oasis. The land value would rise. I could start a farm. Then a resort, as it is so pretty on the ocean.

My first post

 
We all want to get rich for our own reasons. Here in this blog I am going to post some of my thoughts on the subject. I have a good job that gives me time in my life to write and do other things and, in my spare time, I think about making money. Here are my thoughts on that subject. Of course for the first few weeks, months, and perhaps even years no one will read this. Whatever then this is my own place where I can put my thoughts down for review from time to time.