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

[Dr.Peterson]

To be honest, I actually do not see how it is logical for there to be no next real number.

If we can think about the real number line geometrically, say, from 1 to 2, we know that there are only real numbers, no spaces. These numbers increase making them unique.

Now, if everything that I said is correct, then we can ask what is the closest thing to the number 1 in the interval? The answer should be another point/real since that is all there is. This point is larger than 1.

It's not a knock-out proof, but it is at least deductive reasoning.

No, deductive reasoning has to be based on valid premises. You are only assuming that there is a particular closest point to 1. What you're doing is circular reasoning!

Any point you choose other than 1 itself will be some non-zero distance from 1; so there will be another point at half that distance, which will be closer than the point you picked. So no point can be the closest. Your assumption is false.

What you are really doing is expecting infinite things (the points in an interval) to behave like the finite things you are accustomed to. I fully understand; it is hard to get used to the fact that they don't! Only logical reasoning can convince you; feelings or assumptions will fail you.

 

[Otis]

I suppose the question becomes: can we "zoom in on" the reals like we can with the naturals?

No, we can't, and (for me) the question becomes: Is there a smallest, positive Real number? No, there is not.

At this point, I'm not sure why you're still talking about the set of Real numbers. On February 12th, you'd posted that you're in agreement with the answer in post#2. You'd also said that you considered your question answered and you'd been unwittingly trying to redefine the set of Real numbers.

It seems as though you might have changed your mind since then — or have nagging doubts that won't leave your mind. If so, maybe try talking back to your subconscious. Tell it over and over, "There is no smallest, positive Real number".
[imath]\;[/imath]

 

[Mates]

No, deductive reasoning has to be based on valid premises. You are only assuming that there is a particular closest point to 1. What you're doing is circular reasoning!

Any point you choose other than 1 itself will be some non-zero distance from 1; so there will be another point at half that distance, which will be closer than the point you picked. So no point can be the closest. Your assumption is false.

I stated, "if everything that I said is correct ..." That is a crucial part of my reasoning. So I hope you will critique my premises.

What you are really doing is expecting infinite things (the points in an interval) to behave like the finite things you are accustomed to.

And what you say here would ease my mind except that I am haunted by the Reimann integral. The naturals behave as a continuum but also granular. It seems to depend on what point of view is taken, very close, or very far.

 

[Mates]

No, we can't, and (for me) the question becomes: Is there a smallest, positive Real number? No, there is not.

At this point, I'm not sure why you're still talking about the set of Real numbers. On February 12th, you'd posted that you're in agreement with the answer in post#2. You'd also said that you considered your question answered and you'd been unwittingly trying to redefine the set of Real numbers.

Yes, but now it doesn't even make sense to me that the reals can be defined this way. It seems contradictory geometrically speaking.

 

[Dr.Peterson]

And what you say here would ease my mind except that I am haunted by the Reimann integral. The naturals behave as a continuum but also granular. It seems to depend on what point of view is taken, very close, or very far.

Please explain what you mean by this. The natural numbers are definitely not a continuum.

And then explain why the Riemann integral (however you are envisioning that) says anything about real numbers. But take these one at a time.

What you really need to do is to answer the arguments you have been given, such as mine:

Any point you choose other than 1 itself will be some non-zero distance from 1; so there will be another point at half that distance, which will be closer than the point you picked. So no point can be the closest.

Keeping in mind that there is no such thing (in the real numbers) as 1.000...01 with infinitely many digits, how do you find a "next real number" that gets past this issue?

 

[Mates]

Please explain what you mean by this. The natural numbers are definitely not a continuum.

The Reimann integral takes an infinite number of natural numbers, and puts them in such a way that they behave like a continuum.

And then explain why the Riemann integral (however you are envisioning that) says anything about real numbers. But take these one at a time.

Just like the reals, the integral never lets you get to a next number.

What you really need to do is to answer the arguments you have been given, such as mine:

Okay here is my response/question for your argument. What happens if you keep dividing the distance in half an infinite number of times?

 

[Dr.Peterson]

The Reimann integral takes an infinite number of natural numbers, and puts them in such a way that they behave like a continuum.

No, the Riemann integral is one number, obtained as a limit. You are still totally misunderstanding it.

The process that defines the Riemann integral involves the continuum of real numbers, but adds nothing to the concept. As I said, don't mix this question with the main question; take them one at a time.

What happens if you keep dividing the distance in half an infinite number of times?

You can't do anything an infinite number of times. You can never get there.

That's why we use limits: to avoid needing to contemplate an actual infinity. As I think you've been told, working with infinity is always counterintuitive, so you need to let go of your expectations.

 

[Mates]

No, the Riemann integral is one number, obtained as a limit. You are still totally misunderstanding it.

I am sorry. I have a habit of not being specific.

I meant that the process of the Reimann integral takes n sub-intervals of the reals. n goes to infinity until there is zero length of the sub-intervals. The result is an area dense enough to have the same property as the reals that you are using as an argument for why there is no next number on the real number line.

The process that defines the Riemann integral involves the continuum of real numbers, but adds nothing to the concept. As I said, don't mix this question with the main question; take them one at a time.

I am not sure what exactly you want. I don't understand the questions.

You can't do anything an infinite number of times. You can never get there.

When I take a step, I pass an infinite number of halved distances.

 

[NotJeffM]

I can hope to eliminate your perplexity two different ways.

We can say that real numbers are defined as a set of numbers such that there is no smallest positive real number. In that case, the premise that x is the smallest positive real number is nonsense from the get-go.

We can define the real numbers as a set of numbers that satisfy the axioms of an ordered field. From that we can derive some theorems such as

[math]a, \ b \in \mathbb R \text { and } a > 1 \text { and } b > 0 \implies \\ \dfrac{b}{a} \in \mathbb R \text { and } b > \dfrac{b}{a} > 0.[/math]
If you are not OK with that definition or theorem, then you are perfectly free to develop those thoughts wherever they may take you, but you are not free to say that you are talking about real numbers because those are properties that are considered true for real numbers by mathematicians. If you say you you are talking about things with different properties but call them “real numbers,” you simply are misusing language, which is super-individual. It is like saying “United States“ when you mean Saudi Arabia. Using the theorem above, we can PROVE that there is no smallest positive positive real number.

[math] \text {ASSUME for purposes of contradiction that } x \text { is the smallest positive real number.}\\ \therefore \ x > \dfrac{x}{2} > 0 \text { and } \dfrac{x}{2} \text { is a real positive number} \implies\\ x \text { is not the smallest positive number, a contradiction.}\\ \therefore \text {There is no smallest positive real number.}[/math]
You are allowing your intuition about physical points observed in the actual world of experience, which have positive breadth, to distort your thinking about the analogy between the real numbers and the idealized points of the “real number line,” which have zero breadth.

 

[stapel]

Thread locked as (apparently) pointless.

Eliz.