Mobi's blog

[mobius]

PHILOSOPHY 407 WITH PROFESSOR(HOBO) MOBIUS

I will get back to the biblical reviews but first I had some other ideas I wanted to lay out;

chance = ignorance

This begins a series (which I probably won't finish) I've been wanting to do for a while explaining my personal philosophies on life. It's kind of necessary to discuss things in a certain order however, starting with the simpler concepts so the more complicated ones can be understood. I may do research but not much (which means I'll probably do a lot). These are intended to be thought experiments without getting too technical. Some of these things at least at the beginning may seem very simple but it will get more complicated as we go along.

I begin this by making entries each of which will focus on a concept. I explain my stance on these concepts and how I think they relate to us as humans or human society and separately to the physical universe. A few terms I will be using (possibly abusing but I don't care);
immaterial: means lacking relevance or importance; not made of matter, not physically real.
abstract: for the purpose of this blog, an "abstract" is a creation of the human mind. It's a word or set of phrases with meaning, but it's more than that; it describes and relates to many layers and structures in the mind. Abstracts however have no meaning what-so-ever without a human in the picture. A quick example is the word "tree". Trees exist without people there to see them, but the word itself means nothing to the tree without humans. Trees don't talk like us; trees have been around for thousands of years before we started speaking "tree" and likely could exist for thousands more if we disappeared.

Today I will attempt to define a deceptively simple concept: Chance and randomness.  What is it?

If you roll an ordinary 6 sided die; what is the outcome? Is it random?

You'd probably say definitively: "yes"

But let's look at it more closely;
You may say "I don't know the outcome" or its "one of 6 possible outcomes." We can calculate with probability a more detailed answer. But we cannot say with certainty what the outcome will be. So, the statement "I don't know" still holds true. It is still uncertain, even if uncertain within a limited boundary. We call this outcome "random".
But what if we could know?
What if we could calculate exactly how the die will roll and where it will land? If we knew or could somehow predict the exact motions of your arm, the die, the table it rolls on etc... it's conceivable that we could know what the outcome would be with every roll. Whether or not this is realistic is immaterial. The point is that: in this exercise; I can know with exact certainty the outcome 
Would you call this random?
Since you know what the answer is going to be, we say no. Because there is a pattern, or there is "order".

Take now the example of computer games and programs; when programming a game where you want a random element (like dice rolling). Since the computer programmer is dictating in a very direct and precise manner how this game works (inherently this is how programming works for the most part); they necessarily calculate everything with 100% (or as close as they can to that) accuracy. It's (as far as I know atm) impossible to create a 'true' random program, hence the name 'pseudo-random'. What is usually created is an algorithm with a huge number of 'seeds' which generate many different patterns so complex an ordinary person cannot predict them on their own.
Another simpler way of getting this randomness is to simply remove yourself from the deciding process. Example: pick up a menu and pick a food item at random. What do you usually do? Close your eyes and put your finger down on a random spot. You've eliminated your vision so your hand was less guided. If you couldn't limit your senses you'd have a very difficult time *not* making a conscious decision and feeling honest with yourself as calling it random.
So, going back to the original question: what is randomness?
It appears that if we have answers, have knowledge and can predict or understand outcomes, we don't call it random. If we lack knowledge or understanding or simply remove ourselves from understanding the whole picture; the result is we don't know how the outcome was arrived at, thus we call this outcome random or chance.
Thus in my opinion, chance and ignorance are essentially one and the same. Randomness is an abstract; a mental construct we use to 'explain away' our ignorance of the matter at hand. Randomness is immaterial to the universe.

[Simon]

For game programming, pseudorandomness with seeded algorithm is unpredictable enough.

If security matters, you can buy stronger randomness. Random.org takes it from atmospheric noise (weather measurements) and even better would be atomic decay, that's as random as you can get.

It appears that if we have answers, have knowledge and can predict or understand outcomes, we don't call it random. If we lack knowledge or understanding or simply remove ourselves from understanding the whole picture; the result is we don't know how the outcome was arrived at, thus we call this outcome random or chance.

Who is "we"? Many say that atomic decay is random. Some say that Downward Reduction is random, but I claim that learning some principles gives a huge edge on Downward Reduction.

-- Simon

[mobius]

Sorry I forget to respond to this way back;

For game programming, pseudo randomness with seeded algorithm is unpredictable enough.

If security matters, you can buy stronger randomness. Random.org takes it from atmospheric noise (weather measurements) and even better would be atomic decay, that's as random as you can get.

It appears that if we have answers, have knowledge and can predict or understand outcomes, we don't call it random. If we lack knowledge or understanding or simply remove ourselves from understanding the whole picture; the result is we don't know how the outcome was arrived at, thus we call this outcome random or chance.

Who is "we"? Many say that atomic decay is random. Some say that Downward Reduction is random, but I claim that learning some principles gives a huge edge on Downward Reduction.

-- Simon

"we" is everyone https://www.dictionary.com/browse/chance?s=t
for our purposes (our very limited lives) pseudo-randomness is of course good enough. I argue in light of the truth (the truth I'm arrogantly asserting) there is only pseudo-randomness or 'fate'. Fate in this sense; by my definition is just an act that is normal and we understand why it happened, and how and everything about it (at least that matters for the game we're playing).

Downward Reduction is not random. First there is skill then different strategies which cause unexpected behavior; but then highly unexpected things may happen due to lag, which is another form of "chance"; that is we cannot predict it accurately at present, but in theory we could.
So in a summary; it can be thought of as pseudo-random.

I don't really like discussing this issue at length as it really seems like some people either 'get it' or they don't. There's no in between or arguing that seems very useful. There's a truth that I became aware of at some point in my life; and since then I cannot unlearn it. It's not an opinion or a belief. It's a deeply embedded truth. But what is truth? I don't know

Anyway just for the record;
I believe that chance, purpose, free will, choice, control, progress, and more are all concepts that exist only in our minds. They serve a purpose and are 'real' in the sense that they exist in our society. But our society itself is only a thing of the mind. The deeper question that really gets to the point and to interesting territory is;

What is the mind?

The road to answer that question is, I assure you, far stranger than you can imagine. I'll make a blog post on this eventually.

--------

On time, part 3.
https://www.lemmingsforums.net/index.php?topic=2910.msg63579#msg63579

Since reading a lot about various things have enlightened me a bit on this topic; I'd like to return to it. If you're really interested I highly recommend the book "Time Machines" by Paul Nahin.

I haven't changed my opinion on my # 1 or 3 explanations. But in my #2 explanation; I now believe this explanation is not so silly or impossible.

The main problem/confusion is the difference between affecting the past and altering it. You can conceivably travel backward in time and affect things (maybe you build one of the great pyramids), but you didn't change anything. You build the pyramid; you always did, there always was a pyramid. There are no 'multiple timelines' here.* There is just one timeline, with you going back to the past and affecting it. So nothing is 'changing'. Everything is as everyone would've remembered.
For our purposes in real life; this means that either nobody's ever time traveled in history; or we aren't able to tell that it's happened.
(possible explination for building of the pyramids and other impressive ancient structures?) If you want to read more on that; I recomend looking into things like the Roman Temple of Jupiter or the 'stone of the pregnant woman'.

An excellent example is the movie 12 Monkeys.

Spoiler

A man (Bruce Willis) has a dream about his childhood and seeing a man in a airport. He is sent into the past to attempt to stop a deadly virus from spreading and wiping out mankind. This is of course what happens in his history and he's attempting to 'change' the past. He fails, ultimately and in the end accidentally helps the bad guy escape and kill most of humanity. It turns out that he himself (that is; his older self) is the man he saw in an airport as a child.
History played out exactly as he remembered; nothing changed. But he did affect his own past.

*I feel like the invocation of multiple time lines brings in the multiple universe theory again. Which as I stated in my ealier post; I find nothing wrong with; just that it doesn't really explain the difficulties with time travel itself.]

This all also suggests (perhaps demands) a different view of time. Somewhere I think I said that it feels like we don't have the brain capacity to really understand this; or require a new way of thinking in order to. I may have recently gained a very different view point, just the type which can explain this. But it is not easy to articulate into words.


Part 4; with thoughts and questions on the nature of time and consciousness itself coming soon.

[nin10doadict]

There's basically two big views of time travel: Either you can go to the past and mess with things and it changes the present/future, or you can't because the present is already a result of the stuff you messed with in the past.
Either way can make for some good stories.

As for whether time travel will ever become a reality, I doubt it. I don't think we were designed to be dealing with such things. Or maybe it already is possible and nobody knows because the present is a result of all the time travelers messing with the past?

[ccexplore]

If you believe there can be multiple parallel timelines, it could also be that whenever you try to travel into the past, you inevitably always end up in a different parallel timeline than the one you originated from.  In which case from your POV you could change the past to change your present/future in that destination timeline, but from POV of everyone else in your original timeline, you simply disappeared when you time traveled.

Looking at it from a more practical standpoint:
  - A lot of theoretical time travel device based on current knowledge of physics, tends to be that you can't actually any further into the past than the moment the device first became operational.  This at least limits the kinds of paradoxes you could run into.  (It would also neatly answer the question "if time traveling into the past is possible, why haven't we observed any such time travelers yet?")
  - It may be very difficult to control the location in space you end up in.  Consider that the earth and solar system is in constant motion around the galactic center, which in turn also moves relative to other galaxies in the local supercluster, plus the fact that the universe itself is also expanding (and apparently accelerating in the rate of expansion), the fact is that if you want to time travel into the past but still stay somewhere on earth, you might also have to actually travel quite some distance in space, since where you were in the past is actually quite some distance away from where you were in the present due to all that motion.
  - And of course, we don't really know what the energy requirements are for doing time travel, and more generally the finance requirements.  It's rather hard to justify a trip to the past if it costs the entire world's total GDP to do so, or if it requires consuming so much fuel for energy that there's none left for the world to survive!  On the flip side, it could mean that time traveling may possibly only be feasible on a small scale like sending tiny bits of subatomic particles, rather than entire humans for example.

[mobius]

As I'm still reading this book "Time Machines" I'm learning new interesting things constantly and feel inadequate to make a blog about this quite yet. In the meantime;
here's another interesting topic.

The Omnipotence Problem

I first heard this on TV, probably a Science show like "Through the Wormhole" or the like. I've read the argument in different ways and this is a simplified version of it.

Consider the thought experiment:

Let's say we want (and are somehow able) to build a super computer and we want to make it omnipotent. As in; it knows everything, from the beginning of the universe to the end. Every detail of every atom, event in time etc, everything.

Assume it's 'brain' will work at least in some fashion similar to our computers today; that is; data is written to some sort of disc or object that must take up physical space in the universe. Never mind that this space is incredibly small, by even decades old computer standards data on a disc is tiny. That point is that it is finite. If this is the case; then building this omnipotent super computer is impossible.

Here's why:
First there is the fact that the universe may be infinite. And if it is; then this computer must also be infinite in size in order that it's brain containing the data be large enough to hold all the data, meaning it would fill up the universe. This could lead to some interesting arguments but we can ignore this really, because there is another reason this is impossible, even if the universe is not infinite;
-In order for the computer to be able to contain data on everything; in must also contain data on itself.
-Every piece of data that is information on itself; is added material to itself; which it in turn must have information on; if it will be truly omnipotent. Therefore you must add more material to store that data, which is must know about so you must add more, and so on. An infinite recursion. Meaning you could never achieve true omnipotence.

A possible argument against this:
After one iteration of this (storing data on the data recursion) doesn't it become redundant? I guess the question comes down to: Is it safe to assume that from that point on it will always be identical therefore you can 'cheat' and forgo storing that data?
The question also comes down to the difference between power and omnipotence. Obviously a super computer of this nature will be extremely powerful; but everything is relative. The interesting take away point is that truly, honestly knowing everything, at least in this way, is impossible.

[mobius]

I've managed keep up a steady routine of exercising and meditating everyday after work for several weeks now. IT feels good to have a routine; that builds discipline. But what good does it do if the habits themselves don't do anything else for you? Exercising and meditation both take months-years to make noticeable changes in you.

So many intense and draining things happened in 2018. When I looked in the mirror recently I thought I suddenly looked noticeably older. Maybe that was just my imagination though.

I spent near 400$ on upgrading my PC and after figuring everything out it may take several more weeks yet till I have a working PC up and running.

My new years resolution is that I'm not going to spend any money on games, books, movies or any media until I've read/used at least 50% of the stuff I already have which I haven't looked at yet.

[mobius]

mobius On Time part 1

Think of two events in the past year in your life. Which came first? Now ask yourself how do you know which came first. The answer may seem obvious "I just know". But I'll bet the only reason you know is because you're comparing the two memories. If you have nothing to compare it to; it becomes difficult.

Think of some examples; for myself; I recently was looking at my phone and at a text message I sent to a friend. I was shocked when I saw that the date of the text was almost a year ago. It was fresh in my mind; before seeing the date I was thinking I had sent that... maybe a few months ago? Surely not that long ago. But now I know; now it's there, cemented in my mind. And any illusion that it was a few months ago is gone. But when I reflect on the event itself it still feels fresh in my mind. When I really think about it; I don't feel any kind of obvious  chronological identifier that goes with these or any memories. Every time I'm thinking about the past the only way I have of ordering events is to compare multiple events. And it is possible to mix them up and confuse ourselves very easily. Especially as people get older and our memories get "poorer".
Don't people often say things like "Feels like she was just born yesterday... now she's ten years old."
I've had arguments with my parents about this; they asserted that my sister got married after this other person moved into town. I swore it was the other way around. After looking at pictures the truth was rectified.

I think most will agree with this at least: the brightest memories are often not the most recent. The memories that are best recalled are the most important or emotional. Most of you reading this probably don't remember what you ate for lunch a week from today. (Unless you eat the same thing everyday, in which case you may not really remember; you're just coming to a logical conclusion ) But you may remember what you ate for lunch on the first day of school or at your's or some relatives wedding.

My point is that is seems to me like our sense of past is not very stable at all, even though people like to say it is. We tend to think of time and life as being very linear and simple, past to present to future. But the closer you look at the universe it is not so linear, not so simple. What makes up your past is not just your memories either but emotional and logical patterns/habits and customs you garner as you live. A person can be injured and lose their entire memory of even who they themselves are (amnesia) yet still function like a normal person (more or less).

The other minor thing I want to discuss briefly here is memory of dreams. According to a neurologist I listened to they think during sleep the chemical that is responsible for making stable memories is not being produced; hence why we rarely are able to remember our dreams, or they are quickly forgotten. The dream state seems to be focused on pumping out imagery and feelings but not on logic and memory.

Yet have you ever been doing something during the day and suddenly the memory of a long distant dream comes up? This suggests to me that the actual data memory of that dream was always there (never lost) but the access to it or ability to recall it was lost until found again. This happens with real life memories as well; suggesting that we may always contain all memory of everything that has ever happened to us but not necessarily the ability to recall it on demand. Which makes sense when you think about it; memory of everything without exclusion in theory could take up a lot of brain power and leave no room for other activity. It would make doing simple everyday tasks difficult; having to parse through everything just to find the important things. Indeed this is in simple terms what autism is. People like the famous 'Kim Peak' who have incredible memory (supposedly of every book he's ever read) have trouble socially and taking care of themselves in society without help.

Next class we will discuss the various theories of the concept of time, time travel and the passage of time versus the opposing so called 'block universe' theory.

[Simon]

Judging time of a memory: Very nice observation and examples, thanks.

I believe this problem comes from a substitution fallacy, i.e., from replacing the hard question
A: When did X happen?
with the easier question
B: How intense is the memory of X?
...then answering B, but wording that answer as if it were an answer to A, then believing in this reworded position and even defending it in an argument.

With memories, you often can rectify by relating X and Y each to a third memory Z, and then making two logical arguments why X must come before Z and why Z must come before Y. If you're lucky to find such a Z.

More examples of substitution of questions:

Parent want to buy something nice for their kid, and they realize that the kid likes computer games, and thus decide to buy a computer game. But instead of buying game X that makes the kid happiest, the parents buy game Y because the parents imagine (kid playing X happily) and (kid playing Y happily), then the parents decide that the parents will be happier themselves watching the kid play Y happily. Even at the risk of never seeing the kid happy with Y.

People are familiar with the spawn interval expressed as release rate -- which even holds water in an argument -- but then assert that the release rate is also simpler than the raw spawn interval. This is especially surprising when the same people correctly assess simplicity elsewhere, e.g., that it's bad to have two values of release rate to mean the same spawn interval, despite familiarity.

-- Simon

[mobius]

Graham's number

Let's have some fun with really big numbers.
Suppose you took the four vertices of a square. It may be obvious but important that we're dealing with two dimensions so it has four vertices (2^2). Draw lines connecting all possible pairs of every vertex; you'll find there are 6 in total (four on the 'outside' and two diagonals).

Now you can draw each of the 6 lines one of two colors (red or blue in my example). And there are a bunch of different combinations you can have there. You could draw many of these square representing each combination. See example A.

So what? Well there are some interesting configurations you could create... but not with this square; we need more dimensions. A three dimensional cube has 8 vertices. And now there are 28 possible line segments. Now color each line one of two colors same as before.

The whole point of this exercise is to avoid the following configuration:
A square (contained somewhere within this cube) with all 6 vertices the same color. See example B.

In three dimensions (a cube) it is possible to avoid this. Is it avoidable in larger dimensions? There is the 'four dimensional cube' or a Tesseract. See the below picture (Which I believe is correct but keep in mind only a representation). Here there are 120 different line connections. How many different ways can you color these?

A simple calculation but a not so simple answer: 2^120= 1,329,227,995,784,915,872,903,807,060,280,344,576

Already a number that's difficult to express by computers. To put this number into perspective here's a list of some 'ordinary numbers'

one thousand = 1,000
one million = 1,000,000
billion = 1,000,000,000
trillion = 1,000,000,000,000
quadrillion = 1,000,000,000,000,000
quintillion = 1,000,000,000,000,000,000
sextillion = 1,000,000,000,000,000,000,000
septillion = 1,000,000,000,000,000,000,000,000
octillion = 1,000,000,000,000,000,000,000,000,000

Astronomers by the way, estimate the number of stars in the known universe is around 70 sextiliion...
The above number is larger than all of these. But back to the problem; the question is can you avoid a flat (in a plane) square of all connected vertices with the same color within the tesseract? The answer is yes. It is known that it's avoidable in fact in 5 dimensions, all the way up to 12. So is it always avoidable?
The answer is no.
If the dimension is large enough you cannot avoid this configuration.

Let's take 13 dimensions. In a 13 dimensional cube you'd have 8192 vertices.
8192*8191/2 = 33,550,336 line segments. So possible configurations of red/blue lines in this example would be 2 to the power of 33,550,336... A number that most computers cannot compute. The answer to 13 is not presently known. But the point at which it must happen that it cannot be avoided, that is, the upper bound, is known; Graham's number.

How big is Graham's number? It's pretty big. So big that a different type of notation must be used to help describe it. Arrow notation describes powers of powers. Here's how it works:

3↑3 means 3 to the 3rd power (3 cubed) = 27
3↑↑3 means 3 to the power of three, to the power of three (in other words 3 to the 27 power) = 7,625,597,484,987. Every new arrow adds another exponent on top of the exponent, a tower of exponents so to speak.

So what is 3↑↑↑3? Simple; we already know two arrows mean 7 trillion, so this means 3 to the power of 7 trillion. What number is that? Well the answer contains 3.6 trillion digits. So I can't write it here. There is no specific name for this number or any numbers in this realm for that matter. This kind of number cannot even be written accurately in scientific notation (x*10^n). Already this is a number that is not really comprehendible by humans because we have no point of reference for such numbers. We're used to dealing with single and double digits in our lives. We may even struggle with mere triple digits. And 3↑↑↑3 isn't even close to the realm of Graham's number.

Now take 3↑↑↑↑3. Once again we can skip steps because we already know that this essentially means 3 to the power of ((3↑↑↑3)a 3.6 trillion digit number). So this number has many, many, many times more digits than the previous 3.6 trillion digit number. For every single arrow added to this operation it puts you in a whole other world of size so to speak. And every arrow raises it by a degree far greater than the previous increase. And still, we're not even in the ballpark of Graham's number.

3↑↑↑↑3 represents the number of arrows in between another pair of threes. This gives you another number of even greater earth shattering epicenes. Remember just 3↑↑↑3 is a number that cannot be written down. The number we're looking at is gotten to by doing an operation of 3 followed by a number of arrows equal to (3↑↑↑↑3) So; insane orders of magnitude greater than 3↑↑↑↑3. This is by the way far greater than a googol or a googolplex.

The answer to that then gives you another ridiculous number. This then is the number of arrows to use in the next step, between another pair of threes. Repeat this process 64 times. This is Graham's number.

This number is so large that (based on current knowledge) there is not room in the finite universe to store the number via digital data nor time in the universe's lifetime to write it out. In other words there are more particles in the known universe than digits in this number.

Amazingly while most of the actual digits are unknown the last several are (.......195387); because any power of three's final digits are very predictable.

Even getting to Graham's number is mind-boggling. Of course it's nothing compared to infinity. But that's a topic for another entry. What's your favorite gigantic number?

[mobius]

I've been LPing Nessy's levelpack on youtube. Been on a bit of a break lately but I'll get back to it eventually.

https://www.youtube.com/watch?v=rpVIVPo7Jak&t=108s

[WillLem]

Even getting to Graham's number is mind-boggling. Of course it's nothing compared to infinity. But that's a topic for another entry. What's your favorite gigantic number?

I love the Googolplex: a 1 followed by a Googol zeroes! A Googol is a 1 followed by a hundred zeroes, which can be written thus:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

With a Googolplex, there are less particles in the universe than there are 0s following the 1; like Graham's number, you couldn't write it down or even store it in a computer. It's smaller than Graham's number though.

Speaking of infinity, I've learned in recent years that there are different types of infinity, and whilst they are labelled in all sorts of ways, the two that make the most sense to me are these:

Divergent Infinity - an infinite series in which the individual elements have a finite limit. For example:

1, 2, 3, 4, 5, 6, 7, 8, 9... (a finite difference of 1 between each number)

2, 4, 6, 8, 10, 12, 14, 16... (a finite difference of 2 between each number)

1, 3, 5, 7, 9, 11, 13, 15... (a finite difference of 2 between each number)

Convergent Infinity - an infinite series in which the individual elements have an undefined limit which tends towards zero or infinity. For example:

1, 1, 2, 3, 5, 8, 13, 21... (each number is the previous two added together, providing differences between each number which tend towards infinity)

0.1, 0.01, 0.001, 0.0001, 0.00001, 0.000001... (the difference between the numbers in this series gets infinitely smaller, tending towards zero)

0.1, 0.11, 0.111, 0.1111, 0.11111, 0.111111... (the difference between the numbers in this series gets infinitely larger, tending towards infinity)

It helps to explain why there is an infinite series of whole numbers (1, 2, 3, 4, 5, 6, 7, 8, 9...) but there is also and infinite number of subdivisions between each whole number (0.1, 0.11, 0.111, 0.1111, 0.11111...)

A thought I've always found interesting is: where do you start counting between 0 and 1?

0.0000000000000000000000000000000000000000000000000000000000000000000000000000001? Well, why not

0.00000000000000000000000000000000000000000000000000000000000000000000000000000001?

0.(0 recurring) is one of my favourite infinite numbers, for this reason. It never even starts counting towards 1, it just exists in all its glory as a divergently infinite number.

[Proxima]

Convergent Infinity - an infinite series in which the individual elements have an undefined limit which tends towards zero or infinity. For example:

1, 1, 2, 3, 5, 8, 13, 21... (each number is the previous two added together, providing differences between each number which tend towards infinity)

0.1, 0.01, 0.001, 0.0001, 0.00001, 0.000001... (the numbers in this series get infinitely smaller, tending towards zero)

0.1, 0.11, 0.111, 0.1111, 0.11111, 0.111111... (the numbers in this series get infinitely larger, tending towards infinity)

Hmm, I think you've misunderstood something here. The last sequence you mention converges to a limit of 0.1(recurring) = 1/9. Every number in the sequence is larger than the last, but no number in the sequence exceeds 1/9, although they get as close to it as you like -- this is the precise mathematical definition of a limit.

We say that a series is divergent (or tends to infinity) if it does not converge to a limit. The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...) is an example: given any positive integer n, there is a Fibonacci number greater than n.

[mobius]

On the topic of infinity;

Cantor's Diagonal Argument which I've finally understood

There are different kinds of infinity. And as paradoxical as it sounds some are bigger than others.

There is countable infinity. These are all numbers that can be listed;

whole numbers; 1,2,3... integers; 0, 1, -1, 2, -2.... even fractions and decimals; 1/1, 1/2, 1/3...

even though some of these lists may seem larger than others (e.g. the list of integers at first appears like it must be twice as large as whole numbers remember that both are infinite in size; thus in that sense at least; they're the same.

but all real numbers cannot actually be listed like this. Real numbers include "everything" (considered by mathematicians today) integers, fractions, irrational numbers, transcendental numbers (pi and e...) And there is proof of this:

Begin by listing numbers arbitrarily;

0.12137195...
0.22114366...
0.31100212...
0.00176143...

Now we're going to make a number by taking numbers from this above list by taking a diagonal line through the list.

0.12137195...
0.22114366...
0.31100212...
0.00176143...

0.1217...

Now according to the rules of this concept the above number should actually appear in the list somewhere even if we haven't written it down yet. Remember this list is infinite.
But now we're going to make a new number by making up a rule and applying it to the number we made diagonally;

for every 1 we change it to a 2 and anything else is changed to a 1. So our new number would be;

0.2121...

Now the crazy part to consider is that this above number does not appear on the list. Think about it; it cannot, you can compare it to every number (even though this theoretical list is infinite). It's not the first number because it's different in the first digit, nor the second number because it's different in that digit and so on. We've purposefully changed every digit of this diagonally created number and thus have changed exactly one digit in every single number in the list.
Therefore you cannot list this number in this way so it is classified as a whole other type of infinity.

[WillLem]

Convergent Infinity - an infinite series in which the individual elements have an undefined limit which tends towards zero or infinity. For example:

1, 1, 2, 3, 5, 8, 13, 21... (each number is the previous two added together, providing differences between each number which tend towards infinity)

0.1, 0.01, 0.001, 0.0001, 0.00001, 0.000001... (the numbers in this series get infinitely smaller, tending towards zero)

0.1, 0.11, 0.111, 0.1111, 0.11111, 0.111111... (the numbers in this series get infinitely larger, tending towards infinity)

Hmm, I think you've misunderstood something here. The last sequence you mention converges to a limit of 0.1(recurring) = 1/9. Every number in the sequence is larger than the last, but no number in the sequence exceeds 1/9, although they get as close to it as you like -- this is the precise mathematical definition of a limit.

We say that a series is divergent (or tends to infinity) if it does not converge to a limit. The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...) is an example: given any positive integer n, there is a Fibonacci number greater than n.

I know what you mean. I think the misunderstanding here is that I'm saying the difference between the numbers in the series tends towards infinity. The series itself, as you've quite rightly pointed out, converges to a limit of 0.(1 recurring).

It's probably something in my generally non-mathematical way of understanding mathematics that's caused me to word it in a slightly confusing way!

EDIT: I've now edited my original post to be a bit clearer.