Will's Blog: Permission or Forgiveness?

[WillLem]

Topics:

Infinity (from post 1 onwards)
Expanding Earth Theory, Truth, Simulation Theory, Ice Cream and Mathematics...
Tetris
The Perfect Game Controller!
Retrobriteing
Paradoxes
Permission or Forgiveness?



On Infinity.

EDIT: Given the misunderstandings that have arisen from the initial version of this post, I just want to start this off by clarifying a couple of things:

1. I have nothing against mathematics, mathematicians, or indeed any school of abstract thought (actually, I happen to be subscribed to Numberphile on YouTube and I regularly watch and enjoy their videos, and find them very fun and educational). I am simply interested to understand it better, and have come up against questions to which the answers have caused further confusion, in some areas. In other areas, I now feel more enlightened thanks to people's explanations and the progress of this discussion.

2. I know that infinity is not a number. I am asking about the relation of the concept of infinity to the mathematical system of numbers.

---

In trying to understand infinity, we often use numbers to demonstrate its size and shape. For example, there are an infinite number of natural numbers given that it's always possible to add 1 to any natural number. However, there are also an infinite number of real numbers between 0 and 1.

The problem is, that in attempting to count from 0 to anything, you must first define an arbitrary "counting distance."

In counting from 0-10, for example, we might impose a "counting distance" of (1). So:

1 (1) 2 (1) 3 (1) 4 (1) 5 (1) 6 (1) 7 (1) 8 (1) 9 (1) 10

You could just as easily make the counting distance (2) or (5), though.

So, then, we might want to know what the distance between 0 and 1 is. How do we count that? Well, we could use a counting distance of (0.1). So:

0 (0.1) 0.1 (0.1) 0.2 (0.1) 0.3 (0.1) 0.4 (0.1) 0.5...

But, then what about the distance between 0 and 0.1. Now we need a yet smaller counting distance. Let's try (0.01):

0 (0.01) 0.01 (0.01) 0.02 (0.01) 0.03 (0.01) 0.04 (0.01) 0.05...

Because of this tendency towards smaller and smaller decimal numbers, mathematicians have concluded that there is a larger infinity between 0 and 1 than the size of an "infinite list" of natural numbers. In fact, the understanding seems to be that that the numbers between 0 and 1 are "unlistable", in that you can always generate a real number that isn't on a given list by simply changing one digit within any real number relative to its position on the list.

This may be true, but it's a truth that seems somewhat arbitrary, imposed as it is by the limitations of a digital counting system.

In order to define anything using numbers, it's necessary to choose a digital "counting distance," which is always arbitrary and finite when observed from analog reality.

Infinity is an analog concept, and is therefore incompatible with any digital counting system.

Hence, the mathematical notion of "different sized infinities" (such as Aleph 0, Aleph 1, etc) seems necessarily and conceptually flawed; defining their "size" in such a way succeeds only in placing yet another arbitrary finite value on the idea of infinity.

Infinity is not finite. It is in-finite.

Therefore, I would ask the question: is Mathematics the best tool we have for understanding and interacting with the infinite? Perhaps Music and Art are better, but could there even be something that we as humans can't even conceive that would truly allow us to comprehend the incomprehensible?

[Proxima]

Because of this tendency towards smaller and smaller decimal numbers, mathematicians claim that there is a larger infinity between 0 and 1 than the size of an "infinite list" of natural numbers. In fact, they claim that the numbers between 0 and 1 are "unlistable", in that you can always generate a real number that isn't on a given list by simply changing one digit within any real number relative to its position on the list. This may be true, but it's a truth that's arbitrary, imposed as it is by the limitations of a digital counting system.

No, that's not how it works.

Mathematicians have a precise definition that they use to say that two infinite sets are of the same size, or are of different sizes. Two sets are of the same size if they can be put into one-to-one correspondence: pairing every element of one set with a unique element from the other.

Code Select Expand

{England, France, Germany}
    |        |        |
    v        v        v
{magenta, orange, lavender}

This works for infinite sets as well, provided you can pair the elements in a patterened way so that it's clear the pattern continues for ever:

Code Select Expand

{1, 2, 3, 4,  5,  6...}
|  |  |  |   |   |
v  v  v  v   v   v
{1, 4, 9, 16, 25, 36...}

In particular, the process of pairing elements with the natural numbers {1, 2, 3...} is called counting, and any set for which this can be done is countable.

Now, despite the "tendency towards smaller and smaller decimal numbers", the terminating decimals between 0 and 1 are countable. By "terminating decimal" I mean a number like 7/25 whose decimal expansion (in this case 0.28) comes to an end.

Let's try to prove this. You can't list them in ascending numerical order, because before 0.1 would come 0.01, and before that would come 0.001, and before that... you can't even get started.

But you can list all the decimals of length 1, then all those of length 2, then all those of length 3, and so on. Because each list is finite, you can complete it and go on to the next.

However, some numbers are non-terminating, for instance 1/3 = 0.3333333.... Cantor's claim is that the set of all numbers between 0 and 1 is non-countable (too large to be counted). You cannot put them into a list, in any order, than includes every number.

This is not because our decimal system of notation is limited. It's obvious that I can't physically write an infinite list, but Cantor's claim is much stronger: even granting me the mathematically idealised power to create any list of numbers I like, in any notation, it is still impossible to list all numbers between 0 and 1.

This is proved by contradiction. First, suppose we have such a list. Every number has a decimal representation (maybe terminating, maybe not), so we imagine that the list is in decimal form. Take the first digit of the first number, the second digit of the second number, the third digit of the third number and so on, and change each digit to something else. This creates a new number that isn't anywhere on the list (it can't be, because it differs in at least one digit from every number on the list). Suppose I try to get round this by creating this "diagonal" number and adding it to the list? Well, I can't add it "at the end", because the list is infinite. If I add it at, for example, the 1000th place, that changes what number is in each place from that point onward, so we can create a new diagonal number that also isn't on the list.

Let me know if you have any more questions or need me to explain some of this better

[WillLem]

This is proved by contradiction. First, suppose we have such a list. Every number has a decimal representation (maybe terminating, maybe not), so we imagine that the list is in decimal form. Take the first digit of the first number, the second digit of the second number, the third digit of the third number and so on, and change each digit to something else. This creates a new number that isn't anywhere on the list (it can't be, because it differs in at least one digit from every number on the list). Suppose I try to get round this by creating this "diagonal" number and adding it to the list? Well, I can't add it "at the end", because the list is infinite. If I add it at, for example, the 1000th place, that changes what number is in each place from that point onward, so we can create a new diagonal number that also isn't on the list.

I understand this perfectly, I just don't think it's an effective way of interacting with infinity. The very idea of creating an infinite list of anything is absurd: if it's a truly infinite list, you will always be able to add something to that list that isn't already on it.

I understand the difference between listing integers and real numbers, as demonstrated by Hilbert's Infinite Hotel paradox and indeed Cantor's list of non-terminating decimals. This isn't what I'm disputing at all: these guys are far more mathematically sound than I am and have probably spent far more time thinking about this sort of thing, so all respect to their theories.

What I'm saying is that any attempt to demonstrate, interact with, or otherwise understand infinity which involves anything that is finite in its nature (such as digital numbers) is flawed.

The fact is, you can't make an infinite list: you can only infer an infinite list using ellipses, e.g.:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10...

Is not an infinite list. If implies that it could be infinite if it were to continue in the same manner, but who is to say that it does?

What if a revelation of the rest of this list manifested the following result:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 56.

Again, just so there's absolutely no misunderstanding here:

I understand that Cantor's discovery was that the "infinity" as represented by a list of non-terminating decimals seems to be larger than the "infinity" of, say, a list of natural integers with a "counting distance" of (1).

N.B. (I'm not sure what the actual mathematical term for a "counting distance" is...).

However... and this is my point: both lists would only go on forever if they actually existed, but they are imaginary, and can only ever be so. Neither adequately represents infinity as a concept. Let's suppose that it were possible to make such a list: which would be the longer list? If one of the lists is longer even by 1 digit, then the other is not an infinite list.

Here's another list:

1, 2, 3, 4, 5, 6, 7, 8, 9, orange, 10, 11, 12, 13, 14, 15, The Simpsons, 16, 17, January, 18...

What do the ellipses even mean?

[∫tan x dx]

I'm sorry bud, but what you've written here is complete nonsense. I'm going to give you a breakdown of what's wrong bit by bit.

In trying to understand infinity, we often use numbers to demonstrate its size and shape.

Nope. Mathematicians generally use set theory to discuss the kind of infinities you are talking about. We use sets of numbers to give examples of the results, so that it is somewhat easier to process.

The problem is, that in attempting to count from 0 to anything, you must first define an arbitrary "counting distance."

First of all, the notion of "counting distance" has no real meaning in terms of actual mathematics. If you want to get technical, when defining what "counting" is, then the Peano axioms (https://en.wikipedia.org/wiki/Peano_axioms) are a nice way to define the Natural numbers (denoted N = {0,1,2,...}). There are the natural numbers and there is addition, neither are used or defined in the way you are claiming. The Natural number known as "1", "one", "unity" (or whatever other notion you wish to describe it) is most certainly not arbitrary. Its existence is in fact a logical consequence of the axioms by which the Natural numbers are defined. The name we give to it, as well as the symbols we use to denote it are arbitrary, but what these symbols represent in a formal context is not arbitrary.

Because of this tendency towards smaller and smaller decimal numbers, mathematicians claim that there is a larger infinity between 0 and 1 than the size of an "infinite list" of natural numbers.

This is utterly wrong. Mathematicians know that the set of real numbers between 0 and 1 has a greater cardinality than the Natural numbers, but their arguments have nothing to do with what you've written here. In fact, this statement says very little other than saying that there is no smallest real number strictly greater than zero. In fact the sequences that you have given all feature Rational numbers. It is an established fact that the set of Rational numbers (denoted Q) is of the exact same cardinality as N, and both have smaller cardinalities than R.

In fact, they claim that the numbers between 0 and 1 are "unlistable"...

This is true, and I have to say that the term traditionally used to describe this - "Uncountable" - is not exactly the most illuminating. Unlistable is a better word for it in my opinion, since the elements of an unlistable set cannot be put into one-to-one correspondence with the Natural numbers. That is, they cannot be listed in their entirety.

... in that you can always generate a real number that isn't on a given list by simply changing one digit within any real number relative to its position on the list.

What you are referring to here is Cantor's Diagonal Argument (https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument). This is a famous mathematical proof which shows that there can be no bijection (one-to-one correspondence) between the Natural numbers and the Real numbers; the Real numbers have a larger cardinality.

This may be true, but it's a truth that's arbitrary, imposed as it is by the limitations of a digital counting system.

It is true. We've proved it to be true. It's not an arbitrary truth, any more than it is true that 1+1=2. The fact that we use the decimal counting system has absolutely nothing to do with whether or not it is true. I agree that the exact machinations of the proof in its common presentation does rely on base-10 notation, but there are several ways to restate and prove the result without referring to decimal notation. An analogous statement works in binary, ternary, base-67325469253, and indeed any base you care to think of. And if that's still not enough, then I can give a proof that does not rely on any system of "decimally" bits. You can use the continued fraction expansion (https://en.wikipedia.org/wiki/Continued_fraction) of a real number instead of its decimal expansion, and continued fractions do not require decimal points.

All of this is of course largely meaningless, since the proof still stands regardless of how you wish to represent numbers on paper. The proof simply says that no matter how you try to pair up the Natural numbers with the set (0,1), there will always be an infinite amount of real numbers left out.

In order to define anything using numbers, it's necessary to choose a digital "counting distance," which is always arbitrary and finite when observed from analogous reality.
False. Not that this statement actually has any rigorous meaning anyway. As stated before, mathematicians do not define numbers in this manner, and the term "counting distance" has no meaning in this context whatsoever. It is a term you have invented which bears no resemblance to any terminology used in a formal context. And mathematics is all about formal contexts.

Infinity is an analogous concept, and is therefore incompatible with any digital counting system.

This statement has no real meaning either. Are you trying to say that it is impossible to write down infinity as a number? Of course it is impossible; infinity is not a number. It is not an element of the Naturals, Rationals or Reals. The term "Infinity" is actually much more nuanced than you give it credit, and has various differing meanings based on the context in which it is being used.

For example, in analysis the term "as x tends to infinity" refers to the behaviour when the variable x grows without limit. In systems such as the surreal numbers or hyperreal numbers, infinity is actually used as a number, and you can perform arithmetical operations with it. The context that you are referring to in your post is set theory, however. Specifically the properties of cardinalities of sets.

Either way, your statement is once again meaningless.

Hence, the mathematical notion of "different sized infinities" (such as Aleph 0, Aleph 1, etc) is necessarily and conceptually flawed; defining their "size" in such a way succeeds only in placing yet another arbitrary finite value on the idea of infinity.

Nope. First off, the mathematical notation (denoting cardinalities as Aleph 0, Aleph 1, etc) has nothing to do with whether or not the theory is valid. We could have called them "Zoobie", "Clomb", "Spoozle" for all the difference that it makes. Which is none; the notation that humans assign to them is only a human-readable representation to the underlying mathematical objects.

You also seem to think that such objects are not definable, even though they most certainly are. Aleph 0 is defined as the cardinality of the Natural numbers. Aleph 1is the cardinality of the set of all countable ordinal numbers. These concepts are indeed well defined; mathematicians wouldn't use them otherwise. Furthermore, we have not "placed yet another arbitrary finite value on the idea of infinity". We have simply given names to the different cardinalities. There is a hierarchy amongst these objects, with aleph 0 being the smallest.

Infinity is not finite. It is in-finite.

This is completely redundant... Nobody was even saying this anyway???

Therefore, Mathematics is not the best tool we have for understanding and interacting with the infinite.

Wrong. Completely and utterly wrong. Mathematics is the BEST tool for understanding infinity.

Perhaps Music and Art are better...

Nope. Music and art lack the rigor and preciseness to even come close.

... there could even be something that we as humans can't even conceive that would truly allow us to comprehend the incomprehensible.

And this is where I think the core problem lies. You assume that infinity is this magical, mystical, inconceivable force of reality.

It really isn't.

Infinity is actually completely understandable from a mathematical context, as long as you understand the rules and know how to use them appropriately. It seems that you do not, however. Either way, what you have written here shows a complete lack of understanding of infinity from any formal point of view, as well as a poor understanding of mathematics in general.

Mathematicians have been working on this stuff for centuries. We know what we are doing. You do not.


I apologise if my rant comes off as being confrontational or overly hostile. But I cannot abide when people who clearly have no idea what they are talking about decry well established results in any given area. It's this kind of self assured superiority that causes flat earthers, and other nonsensical beliefs.

[Proxima]

What do the ellipses even mean?

That is a gold-standard excellent question, and deserves a much longer answer that I don't have time to attempt right now

However, I think it's not the main issue here.

It is of course true that you can't have an infinite list of numbers in the real world. It's equally true that you can't have a perfectly cubical object, or a perfectly straight line. When mathematicians talk about the properties of a cube, they mean an imaginary, idealised cube. You can't touch one or interact with one, but you can prove that the length of the diagonal is sqrt(3) times the length of the side.

Infinite lists fall into the same category. If you don't like my use of ellipses, suppose I say "the set of all natural numbers". That is a well-defined set: for any mathematical object, we know how to test whether it is a natural number or isn't. Then, by convention, I want to use "1, 2, 3, 4..." not to mean "some list of numbers beginning 1, 2, 3, 4" but specifically "the set of all natural numbers". Since the set is well-defined, we may want to talk about it; therefore it helps to have a notation allowing us to do so. It feels like maybe your problem is with the notation rather than the concept?

[WillLem]

First of all, the notion of "counting distance" has no real meaning in terms of actual mathematics.

You're correct here, this is a term I used for want of a more established technical term that I couldn't find whilst writing this post. To explain it, what I mean is the distance between any given number and the next number in a pattern or series of numbers.

For instance:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 has a "counting distance" of (1).

0, 2, 4, 6, 8, 10, 12, 14, 16, 18 has a "counting distance" of (2).

0, 0, 1, 2, 3, 5, 8, 13, 21 has a "counting distance" of (the previous two digits added to each other).

Mathematicians know that the set of real numbers between 0 and 1 has a greater cardinality than the Natural numbers, but their arguments have nothing to do with what you've written here.

Maybe I've misunderstood then: does "cardinality" refer to a group of numbers' tendency towards an infinite value? This is a term I'm unfamiliar with. It's always been my understanding that the Aleph groups are a way to try to define different infinite series?

An analogous statement works in binary, ternary, base-67325469253, and indeed any base you care to think of.

I know what you're saying, and counting systems can be extremely accurate. However, by its definition, an analogous system is continuous. The simplest example I can think of is a circle. Mathematics uses 3.14159 to approximate circular measurements, but pi is yet another example of an ultimately digital, finite number that's used to interact with an analogous shape that's infinite in its nature. Granted, it's accurate enough for everyday purposes, but it's still a digital representation of an analogous truth.

All of this is of course largely meaningless, since the proof still stands regardless of how you wish to represent numbers on paper. The proof simply says that no matter how you try to pair up the Natural numbers with the set (0,1), there will always be an infinite amount of real numbers left out.

But what exactly does this prove, other than: infinite lists are infinite?

It is a term you have invented which bears no resemblance to any terminology used in a formal context. And mathematics is all about formal contexts.

Hmm, this is probably the reason I was never any good at it!

Infinity is an analogous concept, and is therefore incompatible with any digital counting system.

This statement has no real meaning either. Are you trying to say that it is impossible to write down infinity as a number?

No, I'm not saying that, but I can see where the misunderstanding may have occurred. I suppose a better way to put it might be: infinity is a concept greater than that which can be accurately represented and/or understood using a digital counting system. But even that doesn't really do justice to what it is I'm thinking. It's difficult to put these things into words sometimes.

In systems such as the surreal numbers or hyperreal numbers, infinity is actually used as a number, and you can perform arithmetical operations with it. The context that you are referring to in your post is set theory, however. Specifically the properties of cardinalities of sets.

I've never heard of surreal numbers (or indeed hyperreal ones), that sounds interesting!

I do appreciate you clarifying the context of my post, though. I was only vaguely aware of set theory and number cardinality, and didn't really know what they meant. I have a better understanding of it now, and realise that this, indeed, is likely what I'm referring to as far as the maths goes. Regarding infinity though, I still believe that this is a concept that's ultimately beyond mathematics.

Either way, your statement is once again meaningless.

You've said this quite a lot in your post. I realise it was something of a rant, and clearly something you're passionate about, and ultimately it's good to have your engagement. However, since you feel quite happy to reiterate this statement again and again, I'll take issue with it: if you have failed to recognise the meaning in something that someone has taken the time to formulate and share, that's on you. Discussion often helps to reveal meaning, especially concerning topics that rely very heavily on terminology, level of education, abstract perspective, and/or ongoing investigation.

Hence, the mathematical notion of "different sized infinities" (such as Aleph 0, Aleph 1, etc) is necessarily and conceptually flawed; defining their "size" in such a way succeeds only in placing yet another arbitrary finite value on the idea of infinity.

Nope. First off, the mathematical notation (denoting cardinalities as Aleph 0, Aleph 1, etc) has nothing to do with whether or not the theory is valid. We could have called them "Zoobie", "Clomb", "Spoozle" for all the difference that it makes. Which is none; the notation that humans assign to them is only a human-readable representation to the underlying mathematical objects.

My issue isn't necessarily with the notation here, it's with the concept of "different sized infinities" or "different sized infinite groups". If something is infinite, it has no finite limit. Therefore how can it have a "size" or be put into a "group", as we understand it? If it can be grouped/sized, then it isn't infinite. I could be wrong about this, but it doesn't seem like I am. It's this aspect of it that I'm looking to discuss and understand better, really.

You also seem to think that such objects are not definable, even though they most certainly are. Aleph 0 is defined as the cardinality of the Natural numbers. Aleph 1is the cardinality of the set of all countable ordinal numbers. These concepts are indeed well defined; mathematicians wouldn't use them otherwise. Furthermore, we have not "placed yet another arbitrary finite value on the idea of infinity". We have simply given names to the different cardinalities.

OK, fair enough: cardinality is a term that's new to me. My understanding was that the Aleph groups referred to the infinite nature of what they are used to define. Clearly, I have misunderstood this.

Mathematics is the BEST tool for understanding infinity.

Why?

Music and art lack the rigor and preciseness to even come close.

Show me a hypothetically infinite list of seemingly meaningless numbers, and I'll initially feel confused until either someone explains them, or I apply pre-existing knowledge to make sense of them. Then, I'll probably "understand" the concept at least in a basic, cognitive sense. Show me a picture of a corridor that seems to go on forever, or a video of an endlessly spinning globe, or even just a picture of a circle, and I'll have a far better, deeper comprehension of what "infinity" is.

And this is where I think the core problem lies. You assume that infinity is this magical, mystical, inconceivable force of reality.

Perhaps this is true. But my point is more that I do not believe that mathematics is the best way to interact with it, whatever it is, whether it's a magical, mystical force of reality or a boring, easy-to-understand force of reality.

Either way, what you have written here shows a complete lack of understanding of infinity from any formal point of view, as well as a poor understanding of mathematics in general.

I'll grant you that, my understanding of mathematics is likely to be far inferior to yours. I am interested though, and I do try.

But I cannot abide when people who clearly have no idea what they are talking about decry well established results in any given area. It's this kind of self assured superiority that causes flat earthers, and other nonsensical beliefs.

By the same sword, I cannot abide when people hold onto "established results" so rigidly and stubbornly that they refuse to open their mind up to any possible alternative perspective, however reasonable/absurd it may be. So, at least we feel the same way about that, if only from different perspectives!

For the record, though: I am not a flat Earther! There is far too much evidence to prove that Earth is a sphere, if not a perfect one. However, I do like the idea of Antarctica being a huge ice wall surrounding the Earth, and explorers seeking to find what's on the other side of it. That kind of idea yields great stories!

I can see the opportunity to learn a lot here. Maths was never my strong point, and every time I venture towards it I find something I either don't agree with, don't understand, or that just seems like it's trying to use tangible, rigid concepts and theories to express the intangible. I don't dislike maths though, there's massive beauty and order in a lot of it, and I continue to find that attractive. I think my brain is just wired to question established formalities, and this can sometimes prevail in any internal dialog or learning process whether I like it or not!

[Proxima]

You're correct here, this is a term I used for want of a more established technical term that I couldn't find whilst writing this post. To explain it, what I mean is the distance between any given number and the next number in a pattern or series of numbers.

For instance:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 has a "counting distance" of (1).

The term you want is first differences. ("First" because we can then go on to talk about second differences, i.e. the distance between each first difference and the next, and so on. Thus, the sequence 1, 4, 9, 16, 25... has first differences of 3, 5, 7, 9... and second differences of 2, 2, 2, 2....)

Maybe I've misunderstood then: does "cardinality" refer to a group of numbers' tendency towards an infinite value? This is a term I'm unfamiliar with. It's always been my understanding that the Aleph groups are a way to try to define different infinite series?

"Cardinality" just means "size", as in the size of a set, and it's used because it's more specific, the word "size" having many possible shades of meaning.

The cardinality of a set is the number of elements it contains. For example, the set of all even primes has cardinality 1 (there is only one even prime, namely 2).

Aleph-0 is defined as the cardinality of the set of natural numbers. Note that I say "set" here and not "series". These are two different concepts. A set is a collection of objects, independent of ordering.

I know what you're saying, and counting systems can be extremely accurate. However, by its definition, an analogous system is continuous. The simplest example I can think of is a circle. Mathematics uses 3.14159 to approximate circular measurements, but pi is yet another example of an ultimately digital, finite number that's used to interact with an analogous shape that's infinite in its nature. Granted, it's accurate enough for everyday purposes, but it's still a digital representation of an analogous truth.

Again I think you are confusing notations with the objects being referred to. Let's take 1/3 first as a simpler example. The number 1/3 is well-defined: it's the result of dividing 1 by 3, or the number that represents how much of a cake you have if you divide it into three equal slices. I could also say "the number that is the solution to 3x - 1 = 0", and mathematicians would happy with that, but one should be wary of that sort of definition unless it's really clear that there is exactly one such number (which, in this case, there is).

To write 1/3 as a decimal, we first ask: How many times does 1/10 go into it? Three. The amount left over, (1/3 - 3/10), can be calculated precisely by fraction arithmetic, and is exactly 1/30. How many times does 1/100 go into this remainder? Three. And so on. This is just the standard way of converting any number to a decimal. The end result, 0.333333..., means that 1/3 is made up of three tenths, three hundredths, three thousandths, and so on.

I say "and so on" because there is a clear pattern here, but it's important to note that if you carry out this process literally, it would never end. You can never express 1/3 exactly as a sum of tenths, hundredths, thousandths and so on; there will always be a little bit left over. Still, as soon as we see that the pattern is "every digit is 3", we are happy to put "..." at the end. In this case, it can be proved that the process will in fact continue in the same way for ever; every single digit will always be 3; there are no hidden surprises a few million digits down the line.

That doesn't mean the number 1/3 is ill-defined, though! It just means it doesn't fit neatly into our decimal system.

Pi is a similar case, but a bit more complicated. There is a precise number, which we call pi, that is the ratio of the circumference to the diameter of any circle (or any number of other equivalent definitions). If you try to work out its decimal representation, you get "3.141592652589793238462643383279502884197169399375105820974944592307816406...." -- a higgledy piggledy jumble of digits with no pattern. That doesn't mean the number pi itself is ill-defined. It just means that when we say "pi" or use it in a formula, we mean the exact number, not any of the decimal, fractional or other approximations.

(On a side note: you mean "analogue". "Analogous" is a separate word that means "behaving in a similar way to something, so that you could draw an analogy between them".)

But what exactly does this [the diagonal argument] prove, other than: infinite lists are infinite?

It proves that some infinite sets are larger (have greater cardinality) than others. Again, think of the set itself -- the mathematical collection of all numbers between 0 and 1 -- not any physical list.

My issue isn't necessarily with the notation here, it's with the concept of "different sized infinities" or "different sized infinite groups". If something is infinite, it has no finite limit. Therefore how can it have a "size" or be put into a "group", as we understand it? If it can be grouped/sized, then it isn't infinite. I could be wrong about this, but it doesn't seem like I am. It's this aspect of it that I'm looking to discuss and understand better, really.

Please go back to my first post. I explained that size (or cardinality) is defined in terms of being able to put a set in one-to-one correspondence with another. Any two sets with cardinality 3, for example, can be put in one-to-one correspondence with each other. No set with cardinality 3 can be put in one-to-one correspondence with cardinality 4; if you try to pair the elements, you will always have one left out.

By the same sword, I cannot abide when people hold onto "established results" so rigidly and stubbornly that they refuse to open their mind up to any possible alternative perspective, however reasonable/absurd it may be. So, at least we feel the same way about that, if only from different perspectives!

Well, this is one of the advantages of mathematics. Any result we can rigorously prove to be true is established for all time.

[∫tan x dx]

First off, I want to apologise again for any insulting statements I made in my previous post. My intention was not to belittle you for not having the same kind of formal training that I or others my have.

It is certainly true that mathematics can be an especially opaque subject, especially regarding topics that have little or no basis in the everyday world around us - such as infinity. I would therefore like to take this opportunity to teach you about some of the concepts involved.

I think what is best here is to start with some explanations of some of the terminologies mentioned in this thread. That way we are all at least talking about the same things.

... what I mean is the distance between any given number and the next number in a pattern or series of numbers

A sequence of numbers is generally given by what is known as a formula, or perhaps a function.

A quick aside, functions are themselves mathematical objects. They are sometimes referred to as mappings, since in a technical sense they map between sets. That is, a function is fed elements from one set, and spits out an element from another set. (Some functions can actually have multiple inputs, but for the sake of argument, we'll restrict ourselves to single input -> single output.)

Now, consider a function called "icecream". This function takes a single input of a person, and outputs a flavour of ice cream which that person thinks is best. So the input set for this function (called the Domain in mathematical terms) is the set of all People (Denoted here as P). Likewise, the output set (the Range in mathematical terms) is the set of all ice cream flavours (Denoted here as F).

Thus we may define:

Code Select Expand

icecream : P -> F
So icecream maps People to Flavours.

Let's plug in a few values:
icecream(tan x dx) = Chocolate.
icecream(Proxima) = Vanilla.
icecream(WillLem) = Mint.
(I apologise in advance for any inaccuracies in the output of this function).

Now this is all very well and good, but it is by no means exhaustive. In order to fully define this icecream function, we would have to poll every single person on earth - or to put it another way, each element of the input set has to be special cased. Which is why we generally prefer true mathematical functions that are given by a formula - we don't have to do a lot of silly work for every possible number, simply plug into the formula and there is the answer.

Let's get back to maths.

For example, the sequence
0, 1, 2, 3, 4, 5, ...
may be defined as

Code Select Expand

f : N -> N
(where in this case, N stands for the Natural numbers: N = {0, 1, 2, 3, ...})
So f is a function (or mapping) from the Naturals to the Naturals. The function can be evaluated using the formula

Code Select Expand

f(n) := n

Let's pick apart this notation. The n in the parenthesis is the parameter - the input to the function. The ':=' bit says 'this is how to calculate the result', and the n at the end is what the output is. In this case, we simply return the input. Thus the zeroth term is 0, the first is 1, and so on.

We may do this since the set of Natural numbers N is in fact a well defined mathematical object. It is an infinite set which contains every single non-negative integer by definition. There are no qualms about compiling an infinite list in a finite universe, because we are working with abstract mathematical objects and not any kind of actual physical object. That's one of the things you can do in mathematics; define something rigorously and you can use it as you please, even if it is an infinite set.

As for your other examples:

0, 2, 4, 6, 8, ...
Define

Code Select Expand

g : N -> N
where g(n) := 2n
So this function g doubles the input.

0, 1, 1, 2, 3, 5, 8, 13, 21, ...
Define

Code Select Expand

fib : N -> N
           fib(0) := 1
where{ fib(1) := 1
           fib(n) := fib(n-1) + fib(n-2), for n >=2

Now this one is obviously trickier to define, largely because of its recursive nature. But it is a valid definition; it results in a single unambiguous answer for each of the inputs. Like with the icecream function, the inputs for 0 and 1 are special cased - both are defined to specifically yield a value of 1. But for all other input values, we recursively call the fib function over and over again until it eventually terminates with a fixed answer.

There is obviously more nuance to functions than just this, but as an introduction I think this will do.

Now, let's move onto cardinality.
https://en.wikipedia.org/wiki/Cardinality

The cardinality (sometimes called "Size") of a set S is defined as the "number of elements" of S.

Wait a minute, we can make a function out of this!

Code Select Expand

card : {Sets} -> {Cardinalities}
Huh. This is kind of weird, isn't it?
So, the first part of the mapping (the Domain) is

Code Select Expand

{Sets}
That is the Set of all sets. Now this is indeed a nebulous concept; sets containing sets??? Whatever next?!¹
And as for the output we have

Code Select Expand

{Cardinalities}
Which seems even more esoteric.

So... What is a cardinality then?

It's like a natural number. But different. And there are infinite cardinalities.

Well that's not really helpful, is it?

We'll start somewhere simpler. Consider the following set:

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S1 = {13, 652, 7632411}
What is the cardinality of this set? Three, right? Simple.
What about this one:

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S2 = {Chocolate, Vanilla, Mint}
Also three. Easy enough.
This one?

Code Select Expand

S3 = {Hurgle, Blorpo, Snrouse}
I have no idea what those things are, but I know there are three of them.

So... What's the point?
Well the point is that even though these sets are all utterly different, they still have something in common with each other - their cardinality. They are the same size.

This gives us a way to compare sets. If we have a set with 100 elements, then that set is larger than a set with 20 elements. We don't necessarily need to compare the sets directly, we can instead compare their cardinalities.

But how can we do this for arbitrary sets? As Proxima suggested, we look for bijections.

Think of an ordinary movie theatre, with some unknown number of seats. Now imagine that a crowd of cinema patrons enters the theatre. (I realise how unrealistic such a thing would be in the modern day COVID-19 world, but bear with me here). Even if we do not know the exact numbers involved, either seats or patrons, we can still make some true statements about whether or not the cinema is full.

Suppose an usher looks into the theatre and makes some observations about the relationships between seats and patrons:

• If every patron has a seat, but there are seats without patrons, then we know that the theatre is not at max capacity - there are more seats than patrons. That is, the cardinality of seats is greater than the cardinality of patrons.
  
• If every seat has a patron, but there are still patrons standing, then we know that the theatre is overloaded. The cardinality of patrons is greater than the cardinality of seats.
• If every seat has a patron and every patron has a seat, then we are at the exact capacity. The cardinality of patrons is equal to the cardinality of seats.

This is the power of bijections! A bijection between two sets is a pair of mappings (oh hey, these are functions!) which have certain properties.

• Both mappings must be injective.
• Both mappings must be surjective.

Uh... more mathematical terminology.

Injective:
Suppose we have a one-way mapping between two sets f : A -> B.
f is injective if no two elements of A are mapped to the same element in B.
For example, consider the following two functions:

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g : N -> N, g(n) := 2n
and

Code Select Expand

h : Z -> N, h(n) := n*n
Note, Z here means the set of all integers, positive and negative. Z = {... -3, -2, -1, 0, 1, 2, 3, ...}

Now, can you see why g is injective but h is not?
g is a "doubling" function. It simply doubles the input. It is easy to see that no two distinct inputs are mapped to the same output - if you take any two different natural numbers (like 5 and 6) and double them, you ain't going to get the same answer (10 and 12).

Whereas h is a "squaring" function. What's the problem here? Look at the elements +2 and -2:

Code Select Expand

h(-2) = (-2)*(-2) = 4
h(2) = 2*2 = 4

Ah. Two distinct elements are both mapped to the same result. The function h is not injective.

Surjective:
This one is more subtle. Our mapping f : A -> B is surjective if for each element b in B, there is at least one element a in A that is mapped onto it.

For example:

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k : N -> N, k(n) := n + 50
The function k simply adds 50 to each input. Now, both the Domain (the set of valid inputs) and the Range (the set of possible outputs) of k are defined as N. So let's ask a question:
Is there any input which is mapped onto the value 42?

The answer is no, since the input set only contains non-negative integers, and adding 50 to any of those cannot result in a value of 42. Thus k is not surjective.

If you look back to the movie theatre example, we tried to construct simple mappings between patrons and seats.
Remember that there are two mappings, one going each way
Patrons -> Seats (does each patron have a seat?)
Seats -> Patrons (is each seat occupied by a patron?)

It was only when the number of seats and patrons matched that we had a bijection. You can check this against the definitions of injective and surjective as above.


Okay. So... What are we doing again?


Quick recap:

• We want to rigorously talk about sizes of sets, including infinite sets.
• Cardinalities are ways of comparing sizes of sets without directly caring about the specific contents of each sets.
• If we can construct a pair of mappings (a bijection) between two sets, we know those sets have the same cardinality (same size).

As Proxima mentioned in their post, this is how one-to-one correspondences work. A bijection between two sets is a one-to-one correspondence!

Proxima also gave an excellent example of a bijection between two distinct infinite sets,

Code Select Expand

{0, 1, 2, 3, 4,  5,  6...}
|  |  |  |  |   |
v  v  v  v  v   v
{0, 1, 4, 9, 16, 25, 36...}


Here the input set is N, the natural numbers, and the output is the set of square numbers (denoted here as Sq). Both sets are infinite, but they are of the same cardinality.
The bijections involved are to do with squaring/taking square roots. You can check that the functions

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f1 : N -> Sq, f1(n) = n*n
f2 : Sq -> N, f2(n) = sqrt(n)

Are indeed both valid functions in this context.


So onto the main result - Cantor's diagonal argument.
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Cantor's diagonal argument that there is no bijection between the Natural numbers N, and the Real numbers between 0 and 1 - denoted (0,1).
It is certainly possible to create arbitrary mappings between these sets, but the point is that no matter how you do it, the functions involved will fail in at least one of injectivity or surjectivity.

And this is the beauty of the argument: by attempting to list every real number we are implicitly trying to create a mapping between the two sets! We say that the first element is such-and-such, and the second is blah-blah, and so on! But no matter how clever we try to be when listing the reals, we can never create a bijection, there are always an infinite number of Reals still not on the list! No function f : N -> (0,1) can be both injective and surjective! Thus we are forced to conclude that the set (0,1) is in fact, larger than N!


I hope you find this illuminating. Yes, the concepts involved can be tricky to wrap your brain around, but there really is no mystery about infinity or infinite sets. And of course, reality is under no obligation to coincide with your personal interpretations of it.
The fact that infinite sets can be larger than one another is a well established consequence of mathematical logic. Just because you do not fully understand the concepts involved, does not mean that the mathematicians are wrong. In fact, the reverse is almost certainly true. The fact that you made your original post with such blatant misunderstandings suggests to me a great deal of arrogance. This is what I take issue with; this is what irritates me.

Show me a picture of a corridor that seems to go on forever, or a video of an endlessly spinning globe, or even just a picture of a circle, and I'll have a far better, deeper comprehension of what "infinity" is.

I find it ironic that you admit your picture only "seems to go on forever". Even your video of the spinning globe is necessarily finite in length. Even then, with the endlessly spinning globe, what is infinite about this? The number of revolutions? Number. Hmm. That's mathematics, then. Same for the circle. Infinity is an inherently mathematical concept, and your suggestion that mathematics is not the best tool to understand it smacks of willful ignorance. What would be the best tool then?


*** Note 1: Actually it gets worse. The "set of all sets" isn't technically a set; such a collection is too big to be a set! It is instead an example of what is known as a Proper Class, which I'm not going to go into here because that would be opening up a whole other can of worms. That's the thing about mathematics: the more you dig, the worse (or better) it gets.

[kaywhyn]

How awesome there are other math enthusiasts like me here in Proxima and integral of tan x dx (love your user name, btw)! I have two math degrees,  so you can imagine the numerous amount of math courses I had to take as a math major in the last 11-12 years. In particular, my math degrees are specifically for the teaching field (i.e, not applied or pure math). However, set theory was something that I didn't really study in college/university, so I'm not as well-versed in it as tan x dx is in it, though I'm familiar with it enough to be able to follow his entire explanation completely. Also, even with two math degrees, I'm definitely not an extreme expert on the matter. I cannot do the Putnam problems, for example.

I think I can see where you're confused, WillLem. I, too, initially had trouble with the concept of how some infinite sets can have the same size when I first read about this on my own years ago, which I believe is the source of your confusion? However, with tan's explanation, I hope it's clear to you that the set of natural numbers and the set of even positive integers have the same cardinality (size), even though you would think the set of even numbers is smaller due to how it misses some numbers, eg, {1, 3, 5, ...} while the set of natural numbers includes the missing odds and the evens together. As he has said, the correct way is to look at it as a bijection, as we say infinite sets have the same cardinality if there exists a bijection between them by definition. Yes, both sets are infinite, and so it's impossible to say what the exact cardinality of each set is with a specific number, but the fact that a bijection exists means both sets are still countable, countable because a bijection exists (definition of countable set). Even more, they're countably infinite.

In contrast, it has been proven that the set of real numbers is uncountable because no matter how you do it, you cannot make a bijection with the set N = {0, 1} and R. In other words, it does not satisfy the definition of what it means to be countable, and so the reals is uncountable, uncountably infinite to be exact.

This is what I love about mathematics: the results are intriguing and have beauty to them. In contrast to other subjects, it's easy to check people's work, as most problems you either have the correct answer or you don't. The results aren't going to change on a daily basis thanks to mathematical properties.

[grams88]

Yes we are getting more people doing their own blogs, I like that. L) We get to see the person behind the lemmings player. I'm not a flat earth guy myself and have to say the discussion here seems quite in-depth.

[WillLem]

The term you want is first differences

...

"Cardinality" just means "size", as in the size of a set, and it's used because it's more specific, the word "size" having many possible shades of meaning.

Thanks! This makes sense.

So then, the question I still have is... how can one infinite set be bigger than another infinite set if both are infinite? Would this be one way of understanding it:

You could, in theory, have a Lemmings level that is infinitely long but still 160px high, whilst at the same time having a Lemmings level that is infinitely long and infinitely high. Whilst both have infinite properties, one is bigger than the other.

I say "and so on" because there is a clear pattern here, but it's important to note that if you carry out this process literally, it would never end. You can never express 1/3 exactly as a sum of tenths, hundredths, thousandths and so on; there will always be a little bit left over... That doesn't mean the number 1/3 is ill-defined, though! It just means it doesn't fit neatly into our decimal system.

I have a question about this that I can't quite put into words, but... basically, doesn't this prove that number systems are flawed?

(On a side note: you mean "analogue". "Analogous" is a separate word that means "behaving in a similar way to something, so that you could draw an analogy between them".)

Right you are, I've corrected the OP.

It proves that some infinite sets are larger (have greater cardinality) than others.

How can a set have a definable size if it is infinitely large?

First off, I want to apologise again for any insulting statements I made in my previous post. My intention was not to belittle you for not having the same kind of formal training that I or others my have.

That's OK, I understand it can be frustrating when people seem to be ignorant and you certainly seem passionate about this subject, which is great! I appreciate you taking the time to write such a substantial reply, as well.

Thus we may define:

Code Select Expand

icecream : P -> F
So icecream maps People to Flavours.

Let's plug in a few values:
icecream(tan x dx) = Chocolate.
icecream(Proxima) = Vanilla.
icecream(WillLem) = Mint.

I do love a bit of Mint ice cream.

I understand this very well, so thank for the explanation. One question that arises might be: how could this function be expanded to account for variations within the same ice cream flavour, or the fact that any given person might choose a different flavour as their favourite on a given day?

There are no qualms about compiling an infinite list in a finite universe, because we are working with abstract mathematical objects and not any kind of actual physical object. That's one of the things you can do in mathematics; define something rigorously and you can use it as you please, even if it is an infinite set.

I can certainly see why that's incredibly useful. The question here would be: if mathematics is a system of understanding and interacting with reality in the abstract, as useful and accurate as it tends to be, is it not still dependent on its creators (i.e. the human race, specifically mathematicians) for its development, definition and maintenance?; and, that being the case, is it not therefore limited by current existing limits of our understanding?

If so: then, surely, further investigation and development is needed - and this may involve consideration of radical or seemingly nonsensical ideas (although not necessarily) in order to expand.

If not: then how can we prove it?

There is obviously more nuance to functions than just this, but as an introduction I think this will do.

I understood your explanations perfectly, thanks for that!

This gives us a way to compare sets. If we have a set with 100 elements, then that set is larger than a set with 20 elements. We don't necessarily need to compare the sets directly, we can instead compare their cardinalities.

How can there be a cardinality of infinite size if infinity itself is not a number. I think this is where my main question is coming from: I understand that a set of 3 things, whether it's fruit, numbers, snorples or anything else, has a cardinality of 3. Easy enough.

But a set of infinite things suddenly doesn't match with the use of numbers to define its size, and is therefore incompatible. I may be wrong about this, of course, but this is just how it appears to me given the information I have.

A bijection between two sets is a one-to-one correspondence!

This is another thing I'm struggling to get my head around: my above question regarding Lemmings level sizes states it well enough, but to refer to bijections specifically:

Take an infinite set, A. It has infinity things in it (which is already a slightly flawed statement since infinity is not a number, but bear with me...)

We then get another infinite set, B. It also has infinity things in it.

The bijection rule would have me understand that both sets must be the same size, with each value in set A corresponding to a value in set B. Even in the case of the two Lemmings levels (one Infinity x 160 pixels, the other Infinity x Infinity pixels) has some bijection: each 160-pixel-tall vertical column of pixels in the first level corresponds to an infinite column of pixels in the second level, so in this sense, they have bijection, since we aren't concerned with what the value is, just whether it has correspondence.

Therefore, it seems that both levels have the same cardinality. If it works like this for an example of a rectangle 160 x infinity and a square infinity x infinity, then why doesn't it work for natural numbers and real numbers?

And this is the beauty of the argument: by attempting to list every real number we are implicitly trying to create a mapping between the two sets! We say that the first element is such-and-such, and the second is blah-blah, and so on! But no matter how clever we try to be when listing the reals, we can never create a bijection, there are always an infinite number of Reals still not on the list!

Again, totally understand what's happening here. However: if the list can always have an extra value added to it, then it is indeed infinite. But the same is true of the infinite list of natural numbers. So:

For example, let's just say that the 337th number on Cantor's list is:

0.947374092730918749703986518724977...

The 1,689,236,899,172th number is:

0.74928370942803642739287097209847098...

The 1,456,423,978,354,535,764,234,980th number is:

0.370927309487203947209387409286906902...

I could go on... both lists can always be added to, so how can they not have the same cardinality, as we understand it? And... if they can be different sized sets, then this is what leads me to question the integrity of the idea of "infinite sets", particularly with reference to their size/cardinality.

I hope you find this illuminating. Yes, the concepts involved can be tricky to wrap your brain around, but there really is no mystery about infinity or infinite sets. And of course, reality is under no obligation to coincide with your personal interpretations of it.

Haha - believe me, I know it doesn't!

However, I would disagree that there is no mystery about infinite sets: it's a mystery we're currently engaging in discussion about. If it wasn't a mystery, there would be nothing to explain from either side.

The fact that you made your original post with such blatant misunderstandings suggests to me a great deal of arrogance. This is what I take issue with; this is what irritates me.

I apologise for this, but there may have been a misunderstanding: I'm not saying that mathematics or mathematicians are wrong, I'm saying that I am investigating the concept of infinity from a mathematical point of view and finding only questions, rather than answers. I've updated the OP so it's worded more carefully and in a more open, question-asking tone so as not to cause further irritation or misunderstanding.

Even your video of the spinning globe is necessarily finite in length. Even then, with the endlessly spinning globe, what is infinite about this? The number of revolutions? Number. Hmm. That's mathematics, then.

If the globe were a uniform colour and moving at an inconstant speed, it would be impossible to visually measure the number of revolutions. The fact that it is spinning at all is what suggests the concept of infinity, not that the spinning can be measured.

Infinity is an inherently mathematical concept, and your suggestion that mathematics is not the best tool to understand it smacks of willful ignorance. What would be the best tool then?

Again, I'm not suggesting that maths isn't the best tool, I'm simply asking the question as to whether or not it is. Perhaps I worded the question as a somewhat ignorant-sounding statement. My bad, I apologise if that's how it came across.

Since you ask, I would say: I don't know what the best tool is, that's why I'm investigating the subject.

[Proxima]

I'll quote things a bit out of order, since it might help to clarify the basic concepts before using them to build more complex ones.

However: if the list can always have an extra value added to it, then it is indeed infinite. But the same is true of the infinite list of natural numbers. [...]

I could go on... both lists can always be added to, so how can they not have the same cardinality, as we understand it?

Not entirely sure what you mean by "can be added to". A set of any size can be added to, in the sense that you can add another element to it. But of course, for a set of size 3, adding an element makes it size 4. Indeed, for any finite size, adding an element increases the cardinality by 1. But for an infinite size (cardinality), adding an element does not change the cardinality, and in fact this is precisely the mathematical definition of "infinite".

In case this isn't clear, here's an example. Take the set of positive integers {1, 2, 3, 4...} and add the element 0. The two sets can be put into one-to-one correspondence:

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{1, 2, 3, 4, 5...}
|  |  |  |  |
v  v  v  v  v
{0, 1, 2, 3, 4...}

So, both sets have the same cardinality, and this cardinality is infinite. Remember that "infinity" is not the name of a number; it's a description of a cardinality. When we say a cardinality is infinite, we mean that adding a single element does not change the cardinality.

How can a set have a definable size if it is infinitely large?

The size (cardinality) of the set of positive integers {1, 2, 3, 4...}" is clearly larger than 1, 100, 1000000, or any other finite number. So we need a new name for it: aleph-0. We define aleph-0 to mean "the cardinality of the set of positive integers". Any set that can be put into one-to-one correspondence with the set of positive integers has this cardinality. There you go, I defined it

So then, the question I still have is... how can one infinite set be bigger than another infinite set if both are infinite? Would this be one way of understanding it:

You could, in theory, have a Lemmings level that is infinitely long but still 160px high, whilst at the same time having a Lemmings level that is infinitely long and infinitely high. Whilst both have infinite properties, one is bigger than the other.

That's a resonable guess, but no. Firstly, note that the even positive integers can be put into one-to-one correspondence with all positive integers:

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{1, 2, 3, 4, 5...}
|  |  |  |  |
v  v  v  v  v
{2, 4, 6, 8, 10...}

So, (2 x aleph-0) = aleph-0.

Similarly, 160 x aleph-0 = aleph-0, so your first level (160 by aleph-0) has aleph-0 pixels. To envisage this, suppose you try to count the pixels by starting from one corner, going down the first column, then moving to the next column and going up, and so on... this counting process would include every pixel on the level, so the number of pixels is countable (and remember that "countable" is the same as "has cardinality aleph-0").

Now, you might think the second level (aleph-0 by aleph-0) has greater cardinality, but in fact it doesn't. Suppose it has a top-left corner and continues right and down to infinity. We can count the pixels in a pattern like this:

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1--2  9--10
   |  |  |
4--3  8  11
|     |  |
5--6--7  12
         |
16-15-14-13

Since the pixels can be counted, i.e. can be put into one-to-one correspondence with the positive integers, the set of pixels on this level is countable (aleph-0). In other words: aleph-0 x aleph-0 = aleph-0.

I have a question about this that I can't quite put into words, but... basically, doesn't this prove that number systems are flawed?

It proves that one system can't do everything. Our decimal system is perfectly adequate for its intended purpose: expressing real numbers to any desired degree of accuracy and making it easy to compare the sizes of numbers and perform arithmetic with them. There are simple operations we sometimes want to do that can't be done within the decimal system, such as adding 1/3 and 1/6. To get around this, we can either add to the decimal system (by adding the recurring notation, together with rules for performing arithmetic on recurring decimals), or use a different system.

Similarly, our decimal system can't express all possible cardinalities of sets, which is why we add the notation "aleph-0". That doesn't mean aleph-0 is ill-defined, because I have given a precise definition: it's the cardinality of the set of positive integers. It also doesn't mean the decimal system is "flawed", because expressing the cardinalities of infinite sets is outside its intended purpose in the first place.

[Proxima]

This is another thing I'm struggling to get my head around: my above question regarding Lemmings level sizes states it well enough, but to refer to bijections specifically:

Take an infinite set, A. It has infinity things in it (which is already a slightly flawed statement since infinity is not a number, but bear with me...)

We then get another infinite set, B. It also has infinity things in it.

The bijection rule would have me understand that both sets must be the same size, with each value in set A corresponding to a value in set B. Even in the case of the two Lemmings levels (one Infinity x 160 pixels, the other Infinity x Infinity pixels) has some bijection: each 160-pixel-tall vertical column of pixels in the first level corresponds to an infinite column of pixels in the second level, so in this sense, they have bijection, since we aren't concerned with what the value is, just whether it has correspondence.

Therefore, it seems that both levels have the same cardinality. If it works like this for an example of a rectangle 160 x infinity and a square infinity x infinity, then why doesn't it work for natural numbers and real numbers?

You start by talking about two sets of pixels, but then you say that each column in one set corresponds to a column in the other. That's not a bijection. A bijection is specifically a correspondence whereby each element (here, each pixel) in one set is paired with a unique element of the other.

Now, it is possible to set up a bijection between a 160 x infinity level and an infinity x infinity level. I showed how in my post above, but to reiterate:

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1--2--3--4...159--160
                   |
320...--163--162--161
|
321--322--323...

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It doesn't work for natural and real numbers because there is no way to set up a bijection between the natural numbers and all real numbers. This is what the diagonal argument proves. If there were such a bijection, we could turn it into a list of all real numbers (by listing whichever real number is paired with 1, then the real number paired with 2, and so on), and then use the diagonal argument to construct a real number that is not on the list. That's a contradiction; therefore there is no such bijection.

[ccexplore]

Finally took a dip here.  So I guess this is the "WillLem is insulting all the mathematicians out there" topic.

So basically, just because you seem to personally have trouble grasping the mathematical treatment of infinity, therefore they must be "necessarily and conceptually flawed".  I can't even find a smiley for this.

Show me a picture of a corridor that seems to go on forever, or a video of an endlessly spinning globe, or even just a picture of a circle, and I'll have a far better, deeper comprehension of what "infinity" is.

That's completely equivalent to mathematicians giving the set of all natural numbers ({1, 2, 3, ...}) as an example of infinite set.  You seem to argue the set of all natural numbers is abstract and does not "exist in the real world", but does either of your examples actually exist in the real world either?  The picture of the corridor is almost certainly of a real corridor that is in fact not infinite, you are only imagining in your mind that it goes on forever.  The endlessly spinning globe in the real world will likely be rudely terminated at some point in time far in the future, be it due to our Sun going nova, the heat-death of the universe, etc.  So they are actually just as abstract as talking about a set of natural numbers created simply by endlessly adding 1 to a previous number to get more and more numbers into the set.  It is quite fair to argue that infinity is intrinsically an abstract concept.  You can try to ground it against "real-world examples" that are approximations, but they are still approximations that require some mental idealizations to be absolutely true to the properties of infinity.

And don't be so sure about the world being analog either.  Some physicists postulates that space and time themselves might be subjected to the laws of quantum mechanics and therefore may actually be quantized, such that there may actually be a minimum length/duration enforced by the laws of physics.  In effect, our physics may well turn out to be digital and not truly analog.  There's just so much we don't yet fully understand about the physics of our world.

You mentioned music and art, but I'm having trouble following what examples you have mentioned that are actually coming from those disciplines and relating to infinity.  I'm rather surprised you didn't mention philosophy.  The nature of infinity had actually been historically discussed as philosophy far, far longer than the modern mathematical set-theoretic treatment had existed.

If the globe were a uniform colour and moving at an inconstant speed, it would be impossible to visually measure the number of revolutions. The fact that it is spinning at all is what suggests the concept of infinity, not that the spinning can be measured.

It may not be visually measurable, but if there is truly no way at all (not even non-visually) to measure the spinning, then how do you prove that it is even spinning at all?

Anyway, the spinning globe is no better than, say, a light beam moving forward forever unimpeded.  The only difference is that the former only assumes time is infinite, while the latter would require both space and time to be both infinite (although it could also be that maybe the universe actually has a looping topology, and the light beam could potentially loop back to a formerly visited position in space after traveling forward a sufficient distance, in which case we don't actually need space to be infinite.  Again, so much we don't know about the universe.).

[Dullstar]

This topic is certainly producing some interesting discussion.

Now, it is possible to set up a bijection between a 160 x infinity level and an infinity x infinity level. I showed how in my post above, but to reiterate:

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I'm not really following this argument (that, of course, doesn't necessarily mean it's wrong). To give an idea of what my current understanding is: If I had to compare the area of the levels, my instinct would be to say that the infinity*infinity level is larger than the infinity*160 level (I'm intentionally avoiding terms like "cardinality" here, since I'm not quite sure I understand how it works with infinite sets). My reasoning would be that, suppose we compare an infinity*160 level with an infinity*320 level. In the x-direction, each level would be infinite. Pixels are countable; if we are at pixel n, there exists a pixel n+1 and there aren't any pixels in between (real numbers, by comparison, are uncountable; while you can compare which of two real numbers is larger, there is no "next" real number, since if you tried to define one, you'd always be able to insert another one in between). So there should be a one-to-one correspondence between pixels in each level in the x-direction. But the second level has twice as many pixels in the y direction for each pixel in the x-direction, thus the second has a greater area. An infinity*infinity level would have an infinitely greater area than either.

Other possibly wrong thoughts about infinity:
- Suppose we have an infinite set that represents real numbers between 0 and 1. We have a second infinite set representing the real numbers between 0 and 2. Both sets are, of course, infinite. But the first set is a subset of the second set - any number that belongs in the first set will also belong in the second set, but a number that is in the second set might not belong in the first set. Thus, the second set is larger than the first set.
- But maybe this argument breaks down if we look at natural numbers and then use the doubling function. If we have the set of all natural numbers, we could define a second set that is the numbers from the first set doubled. Each number in the first set has exactly one number in the second set that it corresponds to. But every number in the second set would also exist in the second set, so, arguably, the second set is therefore a subset of the first set and thus it is smaller...
- I'm not really sure what's a correct statement here.