1.1.4 - Jacob's Ladder: A Journey towards the infinite

1.1.4

Jacob's Ladder: A Journey towards the infinite

" Jacob had a dream in which he saw a stairway resting on the earth, with its top reaching to heaven, and the angels of God were ascending and descending on it, and there above itstood the LORD "

-- Genesis 28:12

Previous Article: 1.1.3 A Definition of The Counting Numbers

Jacob's Ladder

The ladder in Jacob's dream, found in genesis, I think is a perfect metaphor for the infinite. While we can not hope to reach the infinite (or heaven) by climbing it's sacred steps we implicitly accept that the ladder exists as a complete object, and that infinity (the LORD) exists somewhere at the other end.

In this metaphorical sense, when we count we can think of it as ascending Jacob's ladder towards heaven. While the LORD can certain instantly transport us to heaven, we can not hope to reach there ourselves. We can only approach it.

With that metaphor in mind let's attempt to construct our own Jacob's ladder by using the definition of counting numbers established in the previous article to define larger and larger counting numbers.

The only rule that is important is that we are not allowed to use infinity, and we must define a number explicitly, and can't use vague ideas ( such as all the grains of sands on the earth, or the largest number I could come up with in a day).

Towards the Infinite

In the previous article we stated that the size of any finite collection defines a counting number. Let's say we begin with the smallest counting number...

(A)

By placing a single object between the paratheses we define the smallest counting number, the only "small number". To get the next counting number we simply add another object, say another "apple"...

(AA)

We know by the definitions that this number is larger than the last. It is also the first "large number". To go further we simply add yet another object to obtain...

(AAA)

This is larger than all previous entries and is also the first "very large number". Yet we can define even larger numbers. We simply add another object and...

(AAAA)

...and we have a larger number still. Have we really pushed the envelop yet, or can we go much further? Obviously we can continue to add another object and another object to obtain a whole sequence of larger and larger numbers...

(AAAAA)

(AAAAAA)

(AAAAAAA)

(AAAAAAAA)

(AAAAAAAAA)

(AAAAAAAAAA)

etc.

However simply acknowledging that we can continue does not constitute an explicit definition, so we have to conclude that (AAAAAAAAAA) is still the largest number we have yet explicitly defined. To have a larger number still we must actually write it out.

Obviously I don't have to add a single A at a time. I can simply copy and paste the previous "champion number" and then insert many addition A 's. For example ...

(AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA)

We know this is larger than all the other numbers because I simply added objects to the previous collection. This is already much larger than any number that was defined in the previous article. Yet if we allow ourselves to use more than a single line we can go even further. Again I copy the previous entry and add to it to ensure the next entry is larger still...

(AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA)

It is quite easy to add even more lines...

(AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA)

In fact I can really speed up the process by simply copying and pasting whole lines of A 's , rather than entering A in A by A. Lets try it...

(AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA)

That's pretty huge for our discussion thus far. It almost fills up the entire page. Certainly this number is larger than a whole lot of other numbers. Yet we can go even further. Instead of copying and pasting a single line I can now copy and paste a whole page over and over again...

(AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA)

Wow! Now that is a huge number! And I've barely begun ! It didn't not take much effort to create this many A 's. That is only because of the capabilities of text editing of coarse, but can I continue this indefinitely? Well remember that in order to copy and paste text the computer has to store all this data. There is only a finite amount of memory in the computer and therefore there are pieces of text too long to be copied and pasted in a single copy and paste. It also takes time to copy and paste, and even though that time is quite small for the most part, repeat it enough times and it will take up any length of time. Therefore there are pieces of text that even given the vast text editing capabilities I have at my disposal would take too long for me to write out. Furthermore this webpage itself is limited in the amount of text it can contain. I would probably reach the limits of the web page long before I'd reach the theoretical limit of how many times I could copy and paste a piece of text.

Now saying this does not mean I have defined a number larger than the one above. Unless I explicitly define a larger one, the one above is still the largest yet defined. So how can I progress further then. Is my only choice to keep lengthening it more and more? Just for the sake of it I will make one more attempt to create a VERY VERY large number in this way (scroll down gradually to get the full effect)...

(AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA)

Now that's a huge number of A 's !!! As you can see this is quickly becoming a completely impractical and somewhat crazy excersize. Yet how else can we define numbers at this point other than explicitly defining them with a collection ?!

Actually there are ways to go further still. Let's say for argument sake we were to create such an "array of A's" such that every line had exactly the same number. Let's say every line contains...

(AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA)

... A 's.

Now imagine an "array of A's such as this where every line contains exactly that many A 's...

(AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

...

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA)

Now you may object that this is not an exact number because the number of lines is arbitrary. That problem is easily remedied however. Now we state that "Alpha" is a number containing as many lines as A 's in the largest number previously defined !

That's insanely huge !! Think about it. For every A there is now an entire line. When we can correspond many objects from a collection to each single object in another collection we know that this collection must be much larger than the other collection. Also note that the number "Alpha" is exact! It is therefore a well defined number even though we did not explicitly write it out in full. Just when we seemed to be reaching a dead end we have discovered a terrific way out ! Not only have we gotten "just alittle further" but we have gone way way beyond the previous entry, and with far less effort (and almost no copying and pasting I might add)

Just how far Jacob's ladder can we now climb? Can we go further with this reasoning?

Sure we can. Now that we can define "Alpha" we can define the number "Beta" as a number with alpha lines! Yikes!

Then we can define "Gamma" with beta lines, "Delta" with gamma lines, "Epilson" with delta lines, and so on. In general if N is the largest number we have yet defined we can define a larger by simply having a new number defined as having N lines.

Again stating this doesn't mean we have explicitly built such a sequence, but it does pave the way to one being constructed. The problem is that we need a way to keep track of these numbers. So far we have Alpha, Beta, Gamma, Delta, and the truly massive Epsilon ! It's so huge in fact, that I probably couldn't hope to write it out in full !

Are we going to have to continue to come up with new names in order to continue. Is there someway to automate this process?

In fact there are ways we can do this. Let "[A]" represent "Alpha", then let "[AA]" represent "Beta", let "[AAA]" represent "Gamma", "[AAAA]" represent "Delta", and "[AAAAA]" represent "Epilson".

Note the use of the block braces " [ " and " ] ". Also note that...

[A] =! (A)

[AA] =! (AA)

[AAA] =! (AAA)

[AAAA] =! (AAAA)

and

[AAAAA] =! (AAAAA)

In fact, [A] > (A) and [AA] > (AA) etc. What does [A] signify. Does the "A" stand for "apple"? No. In this case we are simply using these expressions as "names". In mathematics we will often use mathematical expressions to symbolize something. These expressions may have no deeper meaning than simply to provide a label or name to a concept. In this case these are ready made designations. Furthermore we can now extend much further than "Epilson".

We can define [AAAAAA] as having Epilson lines, [AAAAAAA] as having [AAAAAA] lines, [AAAAAAAA] as having [AAAAAAA] lines, ... ... ...

In general we can say that [A] is the smallest member of such a sequence. To obtain the next member simply add another A between the " [ ] " and then this number will be defined as having the same number of lines as the previous member of the sequence.

Now we can jump ahead to...

[AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA]

WOAH ! Pretty wild stuff. Just how large would this number be? Is it more than the number of people of the earth, the grains of sand on the earth, the molecules in the sun, or even the number of particles in the universe?! It might be pretty far up there already... yet we can go further still.

Yes because now we can have things like...

[AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA]

But wait we've seen this before. Now we can have more and more lines, wait, why not just have an alpha number of A's within block brackets.

In order to make such a thing easily expressible we will define the " [ ] " in a special way. We say that a collection can be written between the block brackets. When it is it takes the collection size ( or number) from it and uses it to generate a member in the alpha,beta, gamma sequence (let's call it the alpha sequence for short). It can also simply take a number with it's usual meaning.

In this way we can say that "Alpha" is the number "[A]". If we place this number within the block braces we have...

[[A]]

That is equivalent to Alpha A 's within block braces. But now we can use the trick again to have...

[[[A]]]

Now guess what, we can have a whole slew of block braces within block braces !!!

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[A]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Welcome to the true nature of Jacob's ladder ! It a ladder we can continuously move faster and faster on and yet never really get anywhere! Just when we have thought we have "skipped over all the unnecessary steps" we find that we are still simply going up a step at a time, and infinity is always the step beyond all steps no matter how fast our sequence !

Let's say we go beyond all of the steps of adding another set of block braces , brace by brace and instead jump ahead of this whole sequence by having ALPHA BRACKETS !

Then we would have ...