Why can't we define sequential numbers on the real number line?

[Mates]

Forgive me if the answer to the title is "we can".

I have a logical issue if the answer to the title is "no". Put quite simply, my logical issue is that the real number line is made up of only real numbers. Therefor, geometrically speaking, there can only be a real number next to, say, the number 4. Let's call it x. Then x = 4.0000...(infinite 0's)...00001. How can this be disputed?

 

[MarkFL]

Pick any two numbers on the real number line, no matter how close to one another they are. There is still an uncountably infinite numbers in between them.

I think the flaw you have is an infinite number of zeroes followed by a one. This is meaningless.

 

[Mates]

Pick any two numbers on the real number line, no matter how close to one another they are. There is still an uncountably infinite numbers in between them.

I think the flaw you have is an infinite number of zeroes followed by a one. This is meaningless.

Then what if we just left it at x? Or x = 4.000... like pi

 

[MarkFL]

I think the issue here is you are trying to discretize the real number line, when it is a continuum. x can be exactly 4, and you cannot say any other number sits right next to it.

 

[Mates]

x can be exactly 4, and you cannot say any other number sits right next to it.

But going back to my original issue, we know that there are only real numbers next to it. I mean what happens when we ask geometrically what is next to the number 4. All I can understand there being is another real number. And can't we call that real number x?

 

[MarkFL]

What is next to 4 is a neighborhood, or range of numbers. We cannot identify a real number closer to 4 than any other. We can speak of infinitesimals, but I lack the formal training to be rigorous.

 

[Otis]

Hi Mates. On the Real number line, there is no number "next to" any other. As Mark said, between any two points on the Real number line there exists an infinite quantity of other points. We say the Real number line is 'infinitely dense'.

Using your visualization for the "next" number after 4, watch the progression:

You say 4.000000001
So I say 4.0000000001
Then you 4.00000000001
Then me 4.000000000001
Then you 4.0000000000001
Then me 4.00000000000001
…

It never ends.
[imath]\;[/imath]

 

[Mates]

Hi Mates. On the Real number line, there is no number "next to" any other. As Mark said, between any two points on the Real number line there exists an infinite quantity of other points. We say the Real number line is 'infinitely dense'.

Using your visualization for the "next" number after 4, watch the progression:

You say 4.000000001
So I say 4.0000000001
Then you 4.00000000001
Then me 4.000000000001
Then you 4.0000000000001
Then me 4.00000000000001
…

It never ends.
[imath]\;[/imath]

Yeah, that makes sense, but it is not how I would argue my case. In a response above, I put 4.0000 ... infinite 0's ... 00001 as just how it would seem to exist intuitively. I don't suppose there is a formula that can express this ... I didn't think so.

Another way to show my issue, is when I think about a real number line segment, say, from 2 to 7. The line is made up of only points, and any point along that line would seem to have to be a real number. And we seem to be able to even remove these points, like the number 5 or pi for example.

Now imagine removing the number two from a segment [2,7] so that it is (2,7] (I hope my notation is correct). Now we have a different line segment. Taking limits out of this if we can, does this segment still end on both sides just like it did before? If the answer is yes, why can't we remove another number from the end just like we did with the number 2? If the answer is no, what makes the point that 2 occupies so special that we can't do the same thing to the segment (2,7]

 

[Otis]

I put 4.0000 ... infinite 0's ... 00001

Hi Mates! That construction does not represent a Real number. It describes an idea, not an actual value. So, there is no point on the Real number line corresponding to it.

I think you agree with that now.

I think about a real number line segment…made up of only points [that] seem to [represent] real number[s].

That is correct, but why did you use the phrase, "seem to"?

imagine removing the number two from a segment [2,7] so that it is (2,7]…does this [new] segment still end on both sides just like it did before?

No. The left endpoint of interval [2,7] is the number 2. The interval (2,7] has no left endpoint.

We call (2,7) an 'open' interval because it has no endpoints.

We call (2,7] and [2,7) 'half-open' intervals because each has only one endpoint.

We call [2,7] a 'closed' interval because it has two endpoints.

why can't we remove another number from the end just like we did with the number 2?

Don't say "another number". Be specific! Tell us which number you have in mind.

PS: Most Real numbers are useless to humans. They don't match any physical quantities or measurements. We have no names for them. We could express some in decimal form, but why would we want to?

EG: 4.000…0001, where the ellipsis dots represent 44 trillion googols of zeros
[imath]\;[/imath]

 

[Zermelo]

Yeah, that makes sense, but it is not how I would argue my case. In a response above, I put 4.0000 ... infinite 0's ... 00001 as just how it would seem to exist intuitively. I don't suppose there is a formula that can express this ... I didn't think so.

Another way to show my issue, is when I think about a real number line segment, say, from 2 to 7. The line is made up of only points, and any point along that line would seem to have to be a real number. And we seem to be able to even remove these points, like the number 5 or pi for example.

Now imagine removing the number two from a segment [2,7] so that it is (2,7] (I hope my notation is correct). Now we have a different line segment. Taking limits out of this if we can, does this segment still end on both sides just like it did before? If the answer is yes, why can't we remove another number from the end just like we did with the number 2? If the answer is no, what makes the point that 2 occupies so special that we can't do the same thing to the segment (2,7]

Hello, here's my take on all of this.

As mentioned, 4.0000...01, where there is "an infinite amount of zeros", is a very complicated idea. But this seems to be the same idea behind dividing numbers with 0, for example, 1/0. In mathematics, dividing by 0 is undefined, but it's interesting to think what happens with 1 / 0.000....01, aka "an infinite amount of zeros", or as you stated, the point to the right of 0.

Firstly, let's show that "the first point right to 0" doesn't exist.
Let's assume that it does exist, and we'll arrive at a contradiction.
Let's denote that point as x. So x has the property that x > 0, and that there aren't any numbers in between 0 and x, aka there doesn't exist any number y such that 0 < y < x. But, let's look at x / 2. It's still positive because both x and 2 are positive. But because it's half of x, it's smaller than x. So we have 0 < x/2 < x, thus x can't be "the first point right to 0".
We assumed that such a point exists, and then proved that we can squeeze in another one in there.
Thus, we proved that THERE CAN'T EXIST A POINT RIGHT NEXT TO 0.

This argument is generalizable to any point, not just 0, and as some helpers said, R is infinitely dense in this sense.

But still, 0.0000...001 is an interesting idea to think about, and gave birth to a whole field of mathematics called calculus, or mathematical analysis. Calculus asks the question of what happens when a number approaches 0 from the right, aka what happens in an arbitrarily small neighborhood of points right to 0. How does the function 1/x behave when x can take values from (0, 001)? What about (0, 00000001)? How does it behave when this interval is arbitrarily small?

You asked the question what is 4.0000...0001. As we proved, this number doesn't exist. But you can ask the question, what happens to the number 4 in an arbitrarily small neighborhood of points right to it? In calculus, we would call this a limit, and denote it with [imath]\lim_{x \downarrow4}x[/imath], and it turns out that [imath]\lim_{x \downarrow4}x = 4[/imath]. But still, this number 4.0000...01 doesn't exist, and this limit represent the idea of what happens when we look at arbitrarily small neighborhoods of points. This gave rise to differential calculus, (when you think of instant speed, number of meters traversed in 0.000....01 seconds), integral calculus (how to approximate a circle with arbitrarily small rectangles, aka rectangles of width 0.000...01) etc etc. Depending on the amount of your math education, you may or may not have heard some calculus, but the point is, the problems you stated are researched using calculus, and the point you tried to construct doesn't exist.

EDIT:
I just read the original question, and you ask something like "why isn't the real line a sequence of numbers?", if I understood correctly. This is also a very complicated idea that gave rise to the study of cardinal numbers. We can bypass the whole 4.000...01 idea, because it's only 1 idea of turning the reals into a sequence, someone might have a better idea than 4.000..001 that would have worked.
But this indeed isn't true, you can't put real numbers in a sequence r1, r2, r3, .....
The proof for this is really cool and is called Cantor's diagonal argument, you can find many videos on youtube that explain why this isn't possible, by searching for "cantor's diagonal argument", for example: This YouTube video.
[imath]\;[/imath]

 

[Mates]

Hi Mates! That construction does not represent a Real number. It describes an idea, not an actual value. So, there is no point on the Real number line corresponding to it.

I think you agree with that now.

No, not really, that is what I am trying figure out.

That is correct, but why did you use the phrase, "seem to"?

I am just trying to be careful as I am not a mathematician, clearly.

No. The left endpoint of interval [2,7] is the number 2. The interval (2,7] has no left endpoint.

Ok, then I will try to just think about this geometrically.

Can we think about a line of real numbers, say, 5 meters long, and then just "superimpose" (for lack of the correct term) onto the 5 meter long line. We would assume that this line has the exact same properties as the reals number line.

Don't say "another number". Be specific! Tell us which number you have in mind.

Let's call it x for now.

 

[Mates]

Hello, here's my take on all of this.

As mentioned, 4.0000...01, where there is "an infinite amount of zeros", is a very complicated idea. But this seems to be the same idea behind dividing numbers with 0, for example, 1/0. In mathematics, dividing by 0 is undefined, but it's interesting to think what happens with 1 / 0.000....01, aka "an infinite amount of zeros", or as you stated, the point to the right of 0.

Firstly, let's show that "the first point right to 0" doesn't exist.
Let's assume that it does exist, and we'll arrive at a contradiction.
Let's denote that point as x. So x has the property that x > 0, and that there aren't any numbers in between 0 and x, aka there doesn't exist any number y such that 0 < y < x. But, let's look at x / 2. It's still positive because both x and 2 are positive. But because it's half of x, it's smaller than x. So we have 0 < x/2 < x, thus x can't be "the first point right to 0".
We assumed that such a point exists, and then proved that we can squeeze in another one in there.
Thus, we proved that THERE CAN'T EXIST A POINT RIGHT NEXT TO 0.

No, we would define it as the first number, sort of like how the naturals work.

This argument is generalizable to any point, not just 0, and as some helpers said, R is infinitely dense in this sense.

But still, 0.0000...001 is an interesting idea to think about, and gave birth to a whole field of mathematics called calculus, or mathematical analysis. Calculus asks the question of what happens when a number approaches 0 from the right, aka what happens in an arbitrarily small neighborhood of points right to 0. How does the function 1/x behave when x can take values from (0, 001)? What about (0, 00000001)? How does it behave when this interval is arbitrarily small?

You asked the question what is 4.0000...0001. As we proved, this number doesn't exist. But you can ask the question, what happens to the number 4 in an arbitrarily small neighborhood of points right to it? In calculus, we would call this a limit, and denote it with [imath]\lim_{x \downarrow4}x[/imath], and it turns out that [imath]\lim_{x \downarrow4}x = 4[/imath]. But still, this number 4.0000...01 doesn't exist, and this limit represent the idea of what happens when we look at arbitrarily small neighborhoods of points. This gave rise to differential calculus, (when you think of instant speed, number of meters traversed in 0.000....01 seconds), integral calculus (how to approximate a circle with arbitrarily small rectangles, aka rectangles of width 0.000...01) etc etc. Depending on the amount of your math education, you may or may not have heard some calculus, but the point is, the problems you stated are researched using calculus, and the point you tried to construct doesn't exist.

I have first-year university calculus and linear algebra.

EDIT:
I just read the original question, and you ask something like "why isn't the real line a sequence of numbers?", if I understood correctly. This is also a very complicated idea that gave rise to the study of cardinal numbers. We can bypass the whole 4.000...01 idea, because it's only 1 idea of turning the reals into a sequence, someone might have a better idea than 4.000..001 that would have worked.
But this indeed isn't true, you can't put real numbers in a sequence r1, r2, r3, .....
The proof for this is really cool and is called Cantor's diagonal argument, you can find many videos on youtube that explain why this isn't possible, by searching for "cantor's diagonal argument", for example: This YouTube video.
[imath]\;[/imath]

I know about Cantor's diagonal argument. It is not about disproving my idea. It's about proving that countable sets do not have the same cardinality as the reals. Afterall, the rationals have the same cardinality as the naturals, yet there is no "next number" in the case of the rationals.

 

[lookagain]

Afterall, the rationals have the same cardinality as the naturals, yet there is no "next number" in the case of the rationals.

The rational numbers can still be listed, not from smallest to largest. By using the
diagonalization approach, and skipping over equivalent fractions, the rationals can
be listed. And, for any rational number in this method of listing, there is a given next
number, as in a follow-up number.

 

[Zermelo]

Hello, here's my take on all of this.

As mentioned, 4.0000...01, where there is "an infinite amount of zeros", is a very complicated idea. But this seems to be the same idea behind dividing numbers with 0, for example, 1/0. In mathematics, dividing by 0 is undefined, but it's interesting to think what happens with 1 / 0.000....01, aka "an infinite amount of zeros", or as you stated, the point to the right of 0.

Firstly, let's show that "the first point right to 0" doesn't exist.
Let's assume that it does exist, and we'll arrive at a contradiction.
Let's denote that point as x. So x has the property that x > 0, and that there aren't any numbers in between 0 and x, aka there doesn't exist any number y such that 0 < y < x. But, let's look at x / 2. It's still positive because both x and 2 are positive. But because it's half of x, it's smaller than x. So we have 0 < x/2 < x, thus x can't be "the first point right to 0".
We assumed that such a point exists, and then proved that we can squeeze in another one in there.
Thus, we proved that THERE CAN'T EXIST A POINT RIGHT NEXT TO 0.

This argument is generalizable to any point, not just 0, and as some helpers said, R is infinitely dense in this sense.

But still, 0.0000...001 is an interesting idea to think about, and gave birth to a whole field of mathematics called calculus, or mathematical analysis. Calculus asks the question of what happens when a number approaches 0 from the right, aka what happens in an arbitrarily small neighborhood of points right to 0. How does the function 1/x behave when x can take values from (0, 001)? What about (0, 00000001)? How does it behave when this interval is arbitrarily small?

You asked the question what is 4.0000...0001. As we proved, this number doesn't exist. But you can ask the question, what happens to the number 4 in an arbitrarily small neighborhood of points right to it? In calculus, we would call this a limit, and denote it with [imath]\lim_{x \downarrow4}x[/imath], and it turns out that [imath]\lim_{x \downarrow4}x = 4[/imath]. But still, this number 4.0000...01 doesn't exist, and this limit represent the idea of what happens when we look at arbitrarily small neighborhoods of points. This gave rise to differential calculus, (when you think of instant speed, number of meters traversed in 0.000....01 seconds), integral calculus (how to approximate a circle with arbitrarily small rectangles, aka rectangles of width 0.000...01) etc etc. Depending on the amount of your math education, you may or may not have heard some calculus, but the point is, the problems you stated are researched using calculus, and the point you tried to construct doesn't exist.

EDIT:
I just read the original question, and you ask something like "why isn't the real line a sequence of numbers?", if I understood correctly. This is also a very complicated idea that gave rise to the study of cardinal numbers. We can bypass the whole 4.000...01 idea, because it's only 1 idea of turning the reals into a sequence, someone might have a better idea than 4.000..001 that would have worked.
But this indeed isn't true, you can't put real numbers in a sequence r1, r2, r3, .....
The proof for this is really cool and is called Cantor's diagonal argument, you can find many videos on youtube that explain why this isn't possible, for example:

No, we would define it as the first number, sort of like how the naturals work.



I have first-year university calculus and linear algebra.



I know about Cantor's diagonal argument. It is not about disproving my idea. It's about proving that countable sets do not have the same cardinality as the reals. Afterall, the rationals have the same cardinality as the naturals, yet there is no "next number" in the case of the rationals.

"No, we would define it as the first number, sort of like how the naturals work."
- Define what? 4.000..00001? As I proved in my answer, it doesn't exist, so you can't define it to be anything That would be like saying "Let's define the smallest even number not divisible by 2, as X". You can't define an even number not divisible by 2 to be anything, because it doesn't exist. In the same way, because of the 0 < x/2 < x argument, we proved that 4.000..001 doesn't exist.

"I know about Cantor's diagonal argument. It is not about disproving my idea. It's about proving that countable sets do not have the same cardinality as the reals. Afterall, the rationals have the same cardinality as the naturals, yet there is no "next number" in the case of the rationals."
- Oh but no my friend, this is exactly what we mean by cardinality! Saying that a set A has the same cardinality as the set N of natural numbers, is saying that there is a 1-1 correspondence between them! That means, for every element a in N = {1, 2, 3, 4, ...}, we can map it to a unique element in A.
So we map 1 -> a2, 2 -> a2, 3->a3... Thus, A can be listed! There is a first element, a1, there is a next number a2, and the next a3...

As you have stated, the rationals Q also have the same cardinality as N.
Thus, we can construct a map
1 -> q1
2 -> q2
3 -> q3 ...
Which means there is a first rational q1, then a second rational q2, etc...

One important thing to note is that we're not necessarily listing them in a way they're sorted. This would be impossible because there is no first rational, rationals aren't bounded from below. But we can take
1 -> 0/1
2 -> 1/1
3 -> -1/1
4 -> 1/2
5 -> -1/2 ...

You can find a way they can be listed somewhere online, and one helper also mentioned it, it's similar to the diagonal argument, this is just an example of how we could map the first 5

As you have mentioned, you knew already that R and N have different cardinalities. But in maths, you really need to think about what this means. If any set has the same cardinality as N, then it can be listed, where the list is basically the function f: N -> A.

This is how it would look for the integers Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

1 -> 0
2 -> 1
3 -> -1
4 -> 2
5 -> -2
6 -> 3
7 -> -3
...

We are able to "list" every element in Z. Pick an arbitrary z in Z. If it's positive, we can find a corresponding n in N, as n = 2z. If it's negative, we can find n = -2z + 1. It can easily be proven that this map is 1-1, there are no repeating mappings, 1 is only mapped to 0, 2 is only mapped to 1 etc. From this we conclude, N and Z have the same cardinality.

But Cantor's diagonal argument says "if we try to list all the elements in R, we can prove that we missed some, aka that we didn't manage to list them all", which is a contradiction, and means that R has a higher cardinality than N, or there can't exist a bijective function f: N -> R, which would uniquely map 1 -> r1, 2->r2 ...

Once you understand what cardinality really means, you see that R can't be put in a sequence, because R has bigger cardinality than N. There's really nothing more to say than this
You tried to put the reals in a sequence, with a broken idea 4.000...0001, and are trying to argue about that idea. But it makes no sense arguing about the idea if you already know it's impossible. Hope that now everything is clear

 

[Mates]

"No, we would define it as the first number, sort of like how the naturals work."
- Define what? 4.000..00001? As I proved in my answer, it doesn't exist, so you can't define it to be anything That would be like saying "Let's define the smallest even number not divisible by 2, as X". You can't define an even number not divisible by 2 to be anything, because it doesn't exist. In the same way, because of the 0 < x/2 < x argument, we proved that 4.000..001 doesn't exist.

I will try to explain what I mean again. The first real number would be defined that way just like 1 is for the naturals (leaving the N definition with 0 out of this). You can divide naturals, but it does not mean that 5/6 has to be a natural. Just like you can divide reals, it would not mean that (4.0000...00001)/2 has to be a real or anything at all.

As you have stated, the rationals Q also have the same cardinality as N.
Thus, we can construct a map
1 -> q1
2 -> q2
3 -> q3 ...
Which means there is a first rational q1, then a second rational q2, etc...

Wait, it is not saying that there is a next rational.

You tried to put the reals in a sequence, with a broken idea 4.000...0001, and are trying to argue about that idea.

Maybe to make the "jump" to 4.000...0001, one would have to divide a real number by a function that goes to infinity or something like that.

But it makes no sense arguing about the idea if you already know it's impossible.

Ok, I think your mind is set in a state that assumes everything has to be exactly as it is therefore this new idea can't exist. This probably hopeless argument might have to make amendments to the reals in one way or another. Why this idea? Maybe it would add to the variety of tools that math could use to solve other solutions.

 

[Zermelo]

I will try to explain what I mean again. The first real number would be defined that way just like 1 is for the naturals (leaving the N definition with 0 out of this). You can divide naturals, but it does not mean that 5/6 has to be a natural. Just like you can divide reals, it would not mean that (4.0000...00001)/2 has to be a real or anything at all.



Wait, it is not saying that there is a next rational.



Maybe to make the "jump" to 4.000...0001, one would have to divide a real number by a function that goes to infinity or something like that.



Ok, I think your mind is set in a state that assumes everything has to be exactly as it is therefore this new idea can't exist. This probably hopeless argument might have to make amendments to the reals in one way or another. Why this idea? Maybe it would add to the variety of tools that math could use to solve other solutions.

I will try to explain what I mean again. The first real number would be defined that way just like 1 is for the naturals (leaving the N definition with 0 out of this). You can divide naturals, but it does not mean that 5/6 has to be a natural. Just like you can divide reals, it would not mean that (4.0000...00001)/2 has to be a real or anything at all.

As one of the helpers stated, there is a theory if infinitesimals, the infinitesimal is "the positive number closer to 0 than any other number". As we proved, this number isn't real, so yeah, as you stated, we can construct a set of hyperreal numbers, that contain reals and infinitesimals. These quantities were used as an alternative to calculus, but calculus was shown to be a much more elegant alternative, that's why we teach calculus instead of infinitesimals in universities. But nonetheless, there exists a field of non-standard analysis that expresses the same ideas as calculus, but without using limits, and instead using infinitesimals. When you write y' = dy/dx, it can be thought of as a ratio of infinitesimals, but in standard calculus, we define the derivative as a limit.

Wait, it is not saying that there is a next rational.
What it's saying is that rationals can be listed. The natural numbers N are interising because of the property that there always exists the next natural number, n + 1. That's the most elementary infinity we can think of, there is always one more, without an end.
But you have to be rigorous in what do you mean by "next rational", in mathematics, every concept has to be clearly defined in order to discuss it. If we don't define it, we will reach these types of situations, where I say that it does mean next, and you say that it doesn't, because we didn't agree what "next" means.

In the question title, you said "why can't we put reals in a sequence", the Cantor's diagonal argument tells you why. A sequence of objects is well defined, it is a map f: N -> X, where X is some set of objects. You can have sequences of numbers, a1, a2, a3..., sequences of functions f1(x), f2(x), ..., sequences of triangles T1, T2, T3...
In that sense, your question was answered.
Now let's get back to "next number". Please define what do you mean by next number.

Definition
Let X be a subset of real numbers (X can be rationals, reals, anything), and let x0 in X be defined as the first element of X. For any x in X, we define the next number x' as ________________. You need to fill the blank.

The first idea would be what I used, if X has the same cardinality as N, we can define the order using a function N -> X (which must exist because of the same cardinality), and get a sequence x1, x2, x3, ..., xn, xn+1, xn+2,,, For xn, the next number x' can be defined as xn+1.

The second idea would be x' is the number such that x' > x, and it is the closest number to x, |x - x'| is minimal. Both in Q (rationals) and R (reals), this kind of x' doesn't exist, as I proved.

Then you're saying, x' doesn't have to be real! But then, you're not putting R in a sequence, are you? You're putting some other set R U {x'}, which is a superset of the reals. You can try to pursue this idea, or you can check out infinitesimals online, but still, you're not putting R in a sequence, you're constructing a bigger set with extra elements in there.

Maybe to make the "jump" to 4.000...0001, one would have to divide a real number by a function that goes to infinity or something like that.
This is exactly the idea behind calculus. This would be described as [imath]\lim_{x\downarrow0}( 4+x)[/imath], or [math]\lim_{x \rightarrow \infty} (4 + \frac{1}{x})[/math]. But still, I'm not sure what you're trying to achieve

Ok, I think your mind is set in a state that assumes everything has to be exactly as it is therefore this new idea can't exist. This probably hopeless argument might have to make amendments to the reals in one way or another. Why this idea? Maybe it would add to the variety of tools that math could use to solve other solutions.
Hey man, I'm only answering the question of why reals can't be put in a sequence. They just can't, because of Cantor's proof. In the same way, my mind is in a state that there can't exist an even number that isn't divisible by 2, and I don't think that's a bad thing We have to take some things as true (axioms), and respect the results that are derived from them using some rules of logic. If you use the Zermelo-Frankel set theory axioms, along with the axioms of real analysis that give us the properties of real numbers, and the standard mathematical logic, you will eventually reach a theorem that says there can't exist a bijective function f: N -> R. You are free to question the Zermelo-Frankel set theory axioms and construct a different set theory. You are also free to question the axioms of real analysis and come up with a much cooler set called MatesReals. You can even experiment with different rules of logic. But, with ZF set theory, standard real axioms and mathematical logic, you can't put reals in a sequence.

But all in all, I think you just need to read up a bit on infinitesimals, they seem to be exactly what you're describing.
But even when you see what infinitesimals are, it's still clear that we can't list the new expanded reals R*, because that set is LARGER than the reals, R is a subset of R*. And R has higher cardinality than N (aka can't be listed), and R is a subset of the new set R*, which means R* also has a higher cardinality than N. The idea of putting reals in a sequence is literally impossible, if you use ZF set theory, real axioms, and standard logic. That's just the way math works, from what I said above, I proved the theorem:

Theorem
There can't exist any set R* which is a superset of R, such that R* can be listed.

Proof
R is a subset of R*, thus R* has equal or higher cardinality than R. By Cantor's theorem, we know that R has higher cardinality than N. Because R* has the same or higher cardinality than R, it must have higher cardinality than N, which means it can't be listed, qed.

The 4.000...01 idea is fine if you want a system to describe very small quantities, but the notion of listing reals, or anything bigger than reals is broken from the start, Cantor proved it can't be done

 

[Mates]

I will try to explain what I mean again. The first real number would be defined that way just like 1 is for the naturals (leaving the N definition with 0 out of this). You can divide naturals, but it does not mean that 5/6 has to be a natural. Just like you can divide reals, it would not mean that (4.0000...00001)/2 has to be a real or anything at all.

As one of the helpers stated, there is a theory if infinitesimals, the infinitesimal is "the positive number closer to 0 than any other number". As we proved, this number isn't real, so yeah, as you stated, we can construct a set of hyperreal numbers, that contain reals and infinitesimals. These quantities were used as an alternative to calculus, but calculus was shown to be a much more elegant alternative, that's why we teach calculus instead of infinitesimals in universities. But nonetheless, there exists a field of non-standard analysis that expresses the same ideas as calculus, but without using limits, and instead using infinitesimals. When you write y' = dy/dx, it can be thought of as a ratio of infinitesimals, but in standard calculus, we define the derivative as a limit.

Wait, it is not saying that there is a next rational.
What it's saying is that rationals can be listed. The natural numbers N are interising because of the property that there always exists the next natural number, n + 1. That's the most elementary infinity we can think of, there is always one more, without an end.
But you have to be rigorous in what do you mean by "next rational", in mathematics, every concept has to be clearly defined in order to discuss it. If we don't define it, we will reach these types of situations, where I say that it does mean next, and you say that it doesn't, because we didn't agree what "next" means.

In the question title, you said "why can't we put reals in a sequence", the Cantor's diagonal argument tells you why. A sequence of objects is well defined, it is a map f: N -> X, where X is some set of objects. You can have sequences of numbers, a1, a2, a3..., sequences of functions f1(x), f2(x), ..., sequences of triangles T1, T2, T3...
In that sense, your question was answered.
Now let's get back to "next number". Please define what do you mean by next number.

Definition
Let X be a subset of real numbers (X can be rationals, reals, anything), and let x0 in X be defined as the first element of X. For any x in X, we define the next number x' as ________________. You need to fill the blank.

The first idea would be what I used, if X has the same cardinality as N, we can define the order using a function N -> X (which must exist because of the same cardinality), and get a sequence x1, x2, x3, ..., xn, xn+1, xn+2,,, For xn, the next number x' can be defined as xn+1.

The second idea would be x' is the number such that x' > x, and it is the closest number to x, |x - x'| is minimal. Both in Q (rationals) and R (reals), this kind of x' doesn't exist, as I proved.

Then you're saying, x' doesn't have to be real! But then, you're not putting R in a sequence, are you? You're putting some other set R U {x'}, which is a superset of the reals. You can try to pursue this idea, or you can check out infinitesimals online, but still, you're not putting R in a sequence, you're constructing a bigger set with extra elements in there.

Maybe to make the "jump" to 4.000...0001, one would have to divide a real number by a function that goes to infinity or something like that.
This is exactly the idea behind calculus. This would be described as [imath]\lim_{x\downarrow0}( 4+x)[/imath], or [math]\lim_{x \rightarrow \infty} (4 + \frac{1}{x})[/math]. But still, I'm not sure what you're trying to achieve

Ok, I think your mind is set in a state that assumes everything has to be exactly as it is therefore this new idea can't exist. This probably hopeless argument might have to make amendments to the reals in one way or another. Why this idea? Maybe it would add to the variety of tools that math could use to solve other solutions.
Hey man, I'm only answering the question of why reals can't be put in a sequence. They just can't, because of Cantor's proof. In the same way, my mind is in a state that there can't exist an even number that isn't divisible by 2, and I don't think that's a bad thing We have to take some things as true (axioms), and respect the results that are derived from them using some rules of logic. If you use the Zermelo-Frankel set theory axioms, along with the axioms of real analysis that give us the properties of real numbers, and the standard mathematical logic, you will eventually reach a theorem that says there can't exist a bijective function f: N -> R. You are free to question the Zermelo-Frankel set theory axioms and construct a different set theory. You are also free to question the axioms of real analysis and come up with a much cooler set called MatesReals. You can even experiment with different rules of logic. But, with ZF set theory, standard real axioms and mathematical logic, you can't put reals in a sequence.

But all in all, I think you just need to read up a bit on infinitesimals, they seem to be exactly what you're describing.

Okay, I think my question has been answered in that there is already hyperreals. I did not know this, or something similar to this, was in the hyperreal set. Thanks a lot for your input!

 

[Otis]

The 4.000...01 idea is fine

Hi. Yes, it's fine as an idea. But it's not fine as a Real number. Part of Mate's issue is a misconception in their definition of Real numbers.
[imath]\;[/imath]

 

[Otis]

I think my question has been answered

Hi. If that statement means that you're now willing to accept the answer given to you in post#2, then I think that's great progress!
[imath]\;[/imath]

 

[Mates]

Hi. If that statement means that you're now willing to accept the answer given to you in post#2, then I think that's great progress!
[imath]\;[/imath]

Yes, I realize now that I was trying to redefine the reals to something that already exists, namely the hyperreals as I have been told.