Is the "space" between one and two infinite?

  • Context: Graduate 
  • Thread starter Thread starter Toppy123
  • Start date Start date
  • Tags Tags
    Infinite Space
Click For Summary

Discussion Overview

The discussion explores the concept of whether the "space" between the numbers one and two can be divided infinitely, and how this relates to the nature of discrete and continuous quantities. It also touches on the implications of infinite divisibility in the context of time and parallel universes, as well as the philosophical and scientific ramifications regarding the beginning of the universe.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the difference between one and two can be infinitely divided, suggesting that discrete numbers may be separated by an infinite "space."
  • Others argue that while there are infinitely many numbers between two distinct numbers, this does not imply an infinite "space" exists between them.
  • A participant questions whether the infinite divisibility of time means we can never reach the absolute beginning of the universe, suggesting a paradox in the context of the Big Bang.
  • Another participant mentions Cantor's theory of infinity, noting that any interval of real numbers can be put in one-to-one correspondence with any other interval, which has implications for calculus.
  • Some participants express skepticism about the existence of parallel universes and their relationship to the concept of infinite divisibility.
  • There is a discussion about the nature of sets, with distinctions made between integers, rationals, and reals regarding their infinite properties.
  • A later reply questions the proof of infinite divisibility of time, suggesting that it may not be relevant to the discussion at hand.

Areas of Agreement / Disagreement

Participants express a range of views on the nature of infinite divisibility and its implications, with no consensus reached on the relationship between discrete and continuous quantities or the existence of parallel universes. The discussion remains unresolved regarding the philosophical implications of these concepts.

Contextual Notes

Limitations include varying definitions of "space" and "distance," as well as the dependence on the mathematical sets being discussed (integers, rationals, reals). The discussion also highlights the unresolved nature of the relationship between infinite divisibility and physical reality.

Toppy123
Messages
6
Reaction score
0
Can the difference between the two numbers (one) be divided infinitely?

If so, are discrete counted numbers actually separated by an infinite "space"?

Does this relationship between the discrete and the continuous (infinite) lie at the heart of reality?

* I am not a scientist, as you may have already figure out!
 
Mathematics news on Phys.org
And if parallel universes are a reality, are they parallel in time as well as space? (I assume so).

If so, and if time can be divided infinitely, does that explain the idea that a parallel universe can exist "next to" our own but be infinitely far away and therefore unreachable because time (as thought of as the "space" between one thing and another) can be divided infinitely? In other words, we can never inhabit the same time as the parallel universe.

* I may be talking nonsense, but I am curious and willing to learn!
 
Toppy123 said:
Can the difference between the two numbers (one) be divided infinitely?

If so, are discrete counted numbers actually separated by an infinite "space"?

Does this relationship between the discrete and the continuous (infinite) lie at the heart of reality?

* I am not a scientist, as you may have already figure out!

In mathematics, the space between one and two is indeed infinitely divisible. In fact, between any two distinct numbers, you can find infinitely many other numbers.
 
Can the difference between the two numbers (one) be divided infinitely

If so, are discrete counted numbers actually separated by an infinite "space"?

Does this relationship between the discrete and the continuous (infinite) lie at the heart of reality?*

And if parallel universes are a reality, are they parallel in time as well as space? (I assume so).

If so, and if time can be divided infinitely, does that explain the idea that a parallel universe can exist "next to" our own but be infinitely far away and therefore unreachable because time (as thought of as the "space" between one thing and another) can be divided infinitely? In other words, we can never inhabit the same time as the parallel universe.**

* I am not a scientist, as you may have already figure out!
** I may be talking nonsense, but I am curious and willing to learn!
 
Does this have implications for studying the big bang?

We are looking at the big bang from a backwards perspective. So, if we look at the first second of the start of our universe in the big bang model, we try to chip away from 1 second to try to get to the beginning, eg half a second from the big bang, 0.1 seconds from the big bang, 1 millionth of a second from the big bang. However, if time is infinitely divisible, can we ever get to 0 (the start) or are we destined to chip away without getting to the beginning.

Does this attribute of time (its infinite divisibility) mean that there is a paradox - there can be a start but that the start is infinitely far away (in time) and can never actually be located by reference to discrete numbers?
 
Toppy123 said:
If so, are discrete counted numbers actually separated by an infinite "space"?

You need to study Cantors theory of the infinite. Any interval of real numbers - and the natural numbers are real numbers - can be put in 1-1 correspondence with any other interval.

This has important implications for the rigorous theory of the calculus, called analysis, with things like the least upper bound axiom and what not.

However physically its simply a model - whether whatever its modelling has all those properties is anyone's guess - all you can say is as far as has been checked its a good model.

Thanks
Bill
 
There are an infinite number of numbers between (for example) zero and one.
That doesn't mean that there's an infinite "space" between zero and one, just that some of the numbers between zero and one are very close to one another.

None of this has anything to do with parallel universes, which do not exist.
 
Last edited:
That's a lot of questions. I'll answer the first and comment on a couple of others.

Suppose we divide 1 by 2, then divide the result by 2, and then divide again, and so on. Suppose we get to a smallest unit that cannot be divided into smaller units. Let's call this unit n.

It follows that n/2 is greater than or equal to n, because it cannot be divided again.

Let's look at, 2*n/2 which reduces to n because the 2s cancel. If n/2 >= n, then 2n/2 >= 2n. It follows that n >= 2n, which is only true for n=0. 0 can never be the result of the division of two positive numbers, so 1 is infinitely divisible.

(The above is not a rigorous proof because I make some "common sense" assumptions.)

A comment: your other answers cannot be answered meaningfully if "space" and "distance" are undefined.
 
I may be able to infinitely divide the time between the moment that I started this post and the moment I submitted it, but the period of time is still finite and I have lived through several such periods.

You are falling for Zeno's supposed paradox.

That said, a lot of things in science are limits that are unattainable, such as absolute zero temperature and the speed of light (attainable only by light, but it is still the limit for matter).

Also, I'm not sure that there is any proof of the infinite divisibility of time. Either way, it doesn't matter for this question.
 
  • #10
Toppy123 said:
Can the difference between the two numbers (one) be divided infinitely?

If so, are discrete counted numbers actually separated by an infinite "space"?

Does this relationship between the discrete and the continuous (infinite) lie at the heart of reality?

* I am not a scientist, as you may have already figure out!

If by the space between two numbers you mean the open interval (a, b), then yes. It is an infinite uncountable set.
 
  • #11
Of course, it depends on what set you're working in. For integers, it is finite. For rationals, it is countably infinite. For reals, it's uncountably infinite.
 
  • #12
Nick O said:
Of course, it depends on what set you're working in. For integers, it is finite. For rationals, it is countably infinite. For reals, it's uncountably infinite.

Of course. I meant reals, assuming the person who posted meant the same thing.
 
  • #13
I know. Your post just made me think of that distinction, which may or may not interest the OP.
 
  • #14
Nick O said:
I know. Your post just made me think of that distinction, which may or may not interest the OP.


No problem mate. :)
 
  • #15
Thanks for the replies.

I won't pretend that I understand the details of your answers but I think I grasp the gist.

Regarding our ability ever to peer back to the moment of creation, do you think that is possible? I understand what you said here: "I may be able to infinitely divide the time between the moment that I started this post and the moment I submitted it, but the period of time is still finite and I have lived through several such periods." However, does that change when you are looking at the very beginning (an "absolute beginning" if you like) instead of just the beginning of any interval of time (eg the period of time you start a post and the moment you submit it)? Is looking at the absolute beginning as unattainable as absolute zero for example? If so, does it matter? Does it matter practically - will there be a gap in our knowledge if we can never get back to the very beginning?
 
  • #16
Toppy123 said:
Thanks for the replies.

I won't pretend that I understand the details of your answers but I think I grasp the gist.

Regarding our ability ever to peer back to the moment of creation, do you think that is possible? I understand what you said here: "I may be able to infinitely divide the time between the moment that I started this post and the moment I submitted it, but the period of time is still finite and I have lived through several such periods." However, does that change when you are looking at the very beginning (an "absolute beginning" if you like) instead of just the beginning of any interval of time (eg the period of time you start a post and the moment you submit it)? Is looking at the absolute beginning as unattainable as absolute zero for example? If so, does it matter? Does it matter practically - will there be a gap in our knowledge if we can never get back to the very beginning?

This is a very complicated, perhaps impossible question to answer. All I am saying in my response is that, as far as numbers, even an open interval such as (0,1) (that is, all the real numbers between 0 and 1), is uncountably infinite. look up Cantor's work.

But I attempt to make no connection between the abstract nature of mathematics and the real nature of the universe.
 
  • #17
Toppy123 said:
Regarding our ability ever to peer back to the moment of creation, do you think that is possible?

There is no reason why not - who knows what future progress may bring.

But maybe not:
http://en.wikipedia.org/wiki/Eternal_inflation

Basically - watch this space.

Thanks
Bill
 

Similar threads

High School Cartesian Space vs. Euclidean Space
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Expressing any given point on plane with one unique number
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Shauder basis for Hilbert spaces
  • · Replies 0 ·
Replies
0
Views
628
High School Reprise: Calculate the distance between two points without using a coordinate system
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad What is the role of geometry and dimensional operations?
  • · Replies 3 ·
Replies
3
Views
2K
High School The infinite and the infinitesimal
  • · Replies 10 ·
Replies
10
Views
3K
Graduate Infinite scaling limit of an elliptic charged surface
  • · Replies 5 ·
Replies
5
Views
414
High School Is one out of infinity different from zero?
  • · Replies 31 ·
2
Replies
31
Views
3K
Undergrad Expansion of Space: Questions on How the Universe Works
  • · Replies 1 ·
Replies
1
Views
2K
MHB Is There a Mathematical Basis for an Infinite Number of Twin Primes?
  • · Replies 7 ·
Replies
7
Views
4K