The word was left for you

we

Chapter 5 An introduction to Infinity


Chapter 5 An introduction to Infinity

“That is a useful allegory for getting a better understanding of what is going on, while getting it wrong in every detail.”
Terry Pratchet about the Cabinet of Curiosities

The possibility of one thing containing the things that are outside is an important idea from the Bible.

We know that we live in Him and He in us, because He has given us of his Spirit.
1 John 4:16
It is also a fun feature of Dr Who's TARDIS that it appears bigger on the inside than the box containing it.  To my mind a great feature for parking your space-time ship in small spaces.

 It is an important concept.  It may seem impossible to have things appear bigger on the inside than on the outside.  However, it is possible for them to have the same amount of space, in fact they always do: that space is infinite!

Some doubt whether the physical Universe can contain God.  However, Christians believe that Jesus was the Godhead in human form. If that is so then creation can contain God.

It is possible for us to think of some parts of the whole of existence that is the whole of the abstract existence, most of which is outside the physical Universe and those thoughts are contained in a model of the Universe in our brains.  So if we can think of everything within the whole of abstract existence then depending how clearly we understand it the whole of abstract existence might  exist within the physical Universe.

We know that we live in Him and He in us, because He has given us of his Spirit.
1 John 4:16
The Bible states that God is in us as we are in him through the Holy spirit, that is the Biblical Word.  This section explains how logically and mathematically this might be true.  Assuming you are interested in following the mathematical argument.

I am afraid Infinities are mathematical concepts and if you want to understand them you will need to understand a little maths, in which case read on.  Otherwise you can take it on faith that there are as many numbers between -1 and infinity as there are between 1 and 2 and 0 and 4 even though these ranges of numbers contain each other, the number of numbers in all of these ranges is exactly the same, infinite!
If you are not interested in maths take a skip and a jump to chapter 6.

Now for the more formal explanation:

Please note 1/x maps every single positive real number between [1, ∞) to a single number between (0, 1].
1->1
10->0.1
100->0.01
1000->0.001
1000000-> 0.000001
1000000000-> 0.000000001
1000000000000->0.000000000001
etc.
and vice versa.

The square bracket is inclusive and a round bracket is exclusive.
 The symbol ∞ means infinity and is not a number but a limit. for every finite number n number there is another number n + 1: 1+ 1= 2; 2 +1 = 3 etc There is no ∞ + 1. "∞)" just means carry on without limit.  Because only finite numbers are included it means 1/n can never reach 0. So you can get as close as you like to 0 but never reach it so it is excluded too.

The same is true for every negative between [-1, 0) mapping back out to (-∞, -1]. It follows that there are as many numbers between 1 and ∞ as between 0 and 1.


Given the centre of a range C and a size S;

To map from inside to outside or vice versa use S2/(X-C)+C.

 e.g. in the specific case of y = 1/x:

 C = 0 and S = 1

Notice in the specific case C = 0 which removes the need for subtraction and addition moving the centre up and down. Similarly 12 is 1. So there is no need for the squaring to get the size correct.

For a more general example C = 2 and S =3

So the inner range is 6 across and centred around the number 2 [-1, 5].

So the number 6 which is slightly outside the range should be mapped into the range

9/(6 – 2) + 2 = 9/4 + 2 = 2.25 + 2 = 4 .25

Coming back out

9/(4.25 – 2) + 2  = 9/2.25 + 2  = 4 + 2  = 6

On the upper boundary it maps to itself.

9/(5 – 2) + 2 = 9/3 + 2 = 3 + 2 = 5

On the lower boundary it maps to itself.

9/(-1 – 2) + 2 = 9/-3 + 2 = -3 + 2 = -1

Below the bottom limit from 4 comes into the range

9/(-4 – 2) + 2 = 9/-6 + 2 = -3/2 + 2 = (4-3)/2 = 1/2

And back out again

9/(1/2 – 2) + 2 = 9/-3/2 + 2 = -18/3 + 2 = -6 +2 = 4

Using this general function just by changing the parameters C and S it can be seen that there are as many numbers between 0 an 0.0001 as the are between 0.1 and 3 or in any and every range of numbers.

Specifically there are as many numbers between 0 and 1 which equals the number of numbers otherwise known as the cardinality between 1 and ∞. 

There is also a 1 to 1 mapping y = x + 1.

So there are the same number of numbers between [0, 1] as there are between [1 and 2] except that the range between 1 and 2 is a part of the range 1 and ∞.  So all of these ranges have the same number of numbers (or points) otherwise known as cardinality. In this case the number of points is uncountably infinite, that is we cannot put a label to each number or point on a geometric line or graph. It is generally thought that there are in fact an infinite number of cardinalities but we do not need to go into that here.

Welcome to the bizarreness of infinity and the number system we use every day. 1 is as infinitely far away from zero as it is from infinity.

This is just the surface of bizarreness. When you get into applied physics it is seriously bizarre.  What it does show is that there is as much detail within your finger tip as there is outside it.  Before physicists scream Quantum mechanics and p-branes limits access to the detail, I suggest you read chapter 8 where I ask the question "What is in the Calabi Yau Manifolds?"

A religious allegory:

Note that the generalization of function, maps the whole of infinity into your thoughts and when looking at infinity you are reaching out to God, which defies all understanding, and maps to the origin or the centre of your thoughts. From there you can build your house upon on the rock, where you can see him and he can see you.

As Terry Pratchet says, after a certain comment about the cabinet of curiosities, “That is a useful allegory for getting a better understanding of what is going on, while getting it wrong in every detail.” 

So is this allegory. It is technically completely wrong.  It does give the correct image that focusing your life on the infinite will give you a sure foundation to build your life upon.  It also expresses very clearly just how exactly you can see things with the right sort of mathematics.  However, the mathematics used in this example and the barrier of the Planck length (the smallest length we might someday theoretically measure) does not allow you to see anything at all.