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:
icecream : P -> FSo 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
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
f(n) := nLet'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
g : N -> Nwhere g(n) := 2n
So this function
g doubles the input.
0, 1, 1, 2, 3, 5, 8, 13, 21, ...
Define
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/CardinalityThe 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!
card : {Sets} -> {Cardinalities}Huh. This is kind of weird, isn't it?
So, the first part of the mapping (the Domain) is
{Sets}That is the Set of all sets. Now this is indeed a nebulous concept; sets containing sets??? Whatever next?!
1And as for the output we have
{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:
S1 = {13, 652, 7632411}What is the cardinality of this set? Three, right? Simple.
What about this one:
S2 = {Chocolate, Vanilla, Mint}Also three. Easy enough.
This one?
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:
g : N -> N, g(n) := 2nand
h : Z -> N, h(n) := n*nNote, 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:
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:
k : N -> N, k(n) := n + 50The 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,
{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
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_argumentCantor'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.
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