this doesn't make sense to me, smart people? (infinity)

[kiwiMark]

Okay so I'm dumb and he's smart and probably right and all the rest of it but I learn best by assuming that I am king of the world and arguing. As such:

He seems to be saying:

let's take allllllll the real numbers between 0 and 1. (0.4, 0.7235762355 etc.)
let's take allllllll the natural numbers (1, 2, 3 etc.)

So far these two sets are equal. If you like, imagine that you're counting the first set, this automatically generates the second. He's saying if we were to list these out to infinity (also seems troubling that he keeps saying that like it's a logical possibility) then we get all of them.

Then he says, now imagine a real number X that isn't equal to any of the real numbers "already listed".

So now we're rocking it like this:

1. We've already "listed" "all" the natural numbers.
2. Before this whole X thing, we have a "list" of real numbers that is exactly as long as the "list" of "all" the natural numbers.
3. So once we add X to that second list, the second list becomes longer than the first.

This is all well and good but he fails to prove that X can exist.

To my mind, any possible value that X could be, we've already covered since we kept listing real numbers out to infinity.

If you want to say that there's a magical real number X that's not equal to any of the "other" real numbers, why not a magical natural number Y, that's not equal to any of the other natural numbers and restores the 1:1 equilibrium?

[oskar]

Okay so I'm dumb and he's smart and probably right and all the rest of it but I learn best by assuming that I am king of the world and arguing. As such:

He seems to be saying:

let's take allllllll the real numbers between 0 and 1. (0.4, 0.7235762355 etc.)
let's take allllllll the natural numbers (1, 2, 3 etc.)

So far these two sets are equal. If you like, imagine that you're counting the first set, this automatically generates the second. He's saying if we were to list these out to infinity (also seems troubling that he keeps saying that like it's a logical possibility) then we get all of them.

Then he says, now imagine a real number X that isn't equal to any of the real numbers "already listed".

So now we're rocking it like this:

1. We've already "listed" "all" the natural numbers.
2. Before this whole X thing, we have a "list" of real numbers that is exactly as long as the "list" of "all" the natural numbers.
3. So once we add X to that second list, the second list becomes longer than the first.

This is all well and good but he fails to prove that X can exist.

To my mind, any possible value that X could be, we've already covered since we kept listing real numbers out to infinity.

If you want to say that there's a magical real number X that's not equal to any of the "other" real numbers, why not a magical natural number Y, that's not equal to any of the other natural numbers and restores the 1:1 equilibrium?

I'll just come back to that because it's super easy, and I can feel smart by answering it.
If you want to start representing natural numbers by real numbers, you can just take a natural number that is

2164984351684684684....
where each digit is
1a2a3a4a5a6a
And represent it by a real number that is
0.1a2a3a4a5a6a...
or
0.0001a2a3a4a5a6a...
or
99.123bacon1a2a3a4a5a6a...

You can already see that intuitively you can fit infinite infinites of natural numbers in a real number any which way you like, so proving that there are infiniely more numbers in any arbitrary interval of real numbers than there are in all of N is just a formality.

[rong]

I'll just come back to that because it's super easy, and I can feel smart by answering it.
If you want to start representing natural numbers by real numbers, you can just take a natural number that is

2164984351684684684....
where each digit is
1a2a3a4a5a6a
And represent it by a real number that is
0.1a2a3a4a5a6a...
or
0.0001a2a3a4a5a6a...
or
99.123bacon1a2a3a4a5a6a...

You can already see that intuitively you can fit infinite infinites of natural numbers in a real number any which way you like, so proving that there are infiniely more numbers in any arbitrary interval of real numbers than there are in all of N is just a formality.

I get this, but I don't think it has any value whatsoever, and maybe isn't even right, or perhaps the problem is that the mathematical language we have created is finite. It's basically a chicken and an egg thing. Because for any real number that you can come up with that is between two natural numbers I can just add another natural number.

I think it becomes a problem with a definition of infinity. I don't think any infinity can be greater than any other infinty, because doesn't infinity mean no limit? In which case there is no difference between the two. Trying to compare two infinitys is stupid because they are exactly the same thing, quite simply unlimited, so one can't possibly be larger than the other because they aren't a size, they are an absence of one.

[daviddem]

I get this, but I don't think it has any value whatsoever, and maybe isn't even right, or perhaps the problem is that the mathematical language we have created is finite. It's basically a chicken and an egg thing. Because for any real number that you can come up with that is between two natural numbers I can just add another natural number.

I think it becomes a problem with a definition of infinity. I don't think any infinity can be greater than any other infinty, because doesn't infinity mean no limit? In which case there is no difference between the two. Trying to compare two infinitys is stupid because they are exactly the same thing, quite simply unlimited, so one can't possibly be larger than the other because they aren't a size, they are an absence of one.

No they are not the "same" infinity. Let's start with an example with finite sets.

The cardinality of a finite set is simply the number of elements in the set. It is pretty trivial to see that when two finite sets have the same cardinality, you can always create some relation between the two sets so that one and only one element of one set corresponds to one and only one element of the other. Example:

set A
a
b
c

set B
d
e
f

and you can always create a relationship simply by pairing one element of one set with one element of the other, for example a-e, b-d and c-f. Each element in set A has one and only one buddy in set B and each element of set B has one and only one buddy in set A.

But if the sets have different cardinality:
set A
a
b
c

set B
d
e
f
g

Then whatever you do there is always one element of set B which won't have a buddy in set A...

Well now for infinite sets it's exactly the same story: you can't find a natural number buddy for every single real number...

Consider the set of all natural numbers:
0, 1, 2, 3, ...
and the set of all even natural numbers:
0, 2, 4, 6, ...

Both these sets are infinite, and we can say that they are the same kind of inifinity (in mathematical language, you say that both sets have the same cardinality) because you can find a function that puts in correspondence one element of one of these sets with one and only one element of the other set. In this example, the function would be
f(x) = 2*x where x belongs to the set of natural numbers. With this function:
f(0) = 0
f(1) = 2
f(2) = 4
f(3) = 6
...
So via this function, to each element of the set of natural numbers corresponds one and only one element of the set of natural even numbers. The inverse of the function is of course invf(y) = y/2 where y belongs to the set of natural even numbers:
invf(0) = 0
invf(2) = 1
invf(4) = 2
invf(6) = 3

So via these two functions:
- if you give me any natural number, I can tell you which is the corresponding even natural number
- if you give me any even natural number, I can tell you which is the corresponding natural number

And that is why we say that these two sets have the same cardinality.

Another way of saying it is that you can unequivocally and non-ambiguously "map" one set onto the other and vice-versa.

You can try for your entire life to find such a function (and its inverse) that would unequivocally put in correspondence one and only one real number with one and only one natural number: you won't find one because it does not exist.

So while it is not really correct to say that there are "more" real numbers than natural numbers because "more" is an adjective that applies only in the world of finite sets, mathematicians needed a way to express the existence or non-existence of such an unequivocal relation between two sets, and the way they did that was to say that the cardinality of two sets is the same if such a function can be found and the cardinality of two sets is different if such a function cannot be found.

Additionally, it can be demonstrated that the cardinality of the set of real numbers is strictly greater than the cardinality of the set of natural numbers, because there exist injective functions that map N on R but no bijective functions (which means that you can create a function that finds a real number corresponding to each and every single natural number, but its inverse won't cover all the real numbers: for example f(x) = x with x being a natural number finds a real number for every natural number, but its inverse, which is invf(y) = y with y being a real number does not work for example for 0.5 because to the real number 0.5 would supposedly correspond the natural number 0.5 but since 0.5 is not a natural number, that doesn't work. In other words, via these two functions:
- if you give me any natural number, I can tell you which is the corresponding real number
- but for some real numbers (actually most of them), I wouldn't be able to tell you which is the corresponding natural number, because there isn't one (they don't have a natural number buddy).

er... clear?