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!
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!
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?
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.
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.
If by the space between two numbers you mean the open interval (a, b), then yes. It is an infinite uncountable set.
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.
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.
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