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I remember from school that the number of points on a section of a line is infinite. On the other hand, when you reach the number two in a number sequence, that is a number and how big the number is, it will never become infinite. But on the line between one and two you have passed an infinite numbers of fractions. So, my question is, is the infinite between the two numbers a limited infinite or a real infinite, and if so, how can the next number be reached, just as it is impossible to reach the infinite?

Thanks, Simon.

Simon G
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  • What's limited infinite and what's a real infinite? – Asaf Karagila Feb 05 '14 at 20:14
  • Also, run a Google search about Zeno's Paradoxes. – Asaf Karagila Feb 05 '14 at 20:15
  • You will probably not have great luck finding someone to answer your question satisfactorily, because you're branching into the philosophy of mathematics. You might be interested in some of the Paradoxes of Zeno, which have a similar spirit to yours. These are still somewhat debated, although learning calculus tends to make these "paradoxes" less paradoxical. – pjs36 Feb 05 '14 at 20:16
  • @pjs36: Where are these paradoxes debated? (Unless you subscribe to the theory that time is an artifact of our brains to deal with the amount of information, and everything is happening at once, in which case Zeno is alive an well, and debates these paradoxes this very "moment".) – Asaf Karagila Feb 05 '14 at 20:23
  • A limited infinite was something I made up myself as a description of what is between two points. If it could be described as such, was my question. I know about Zeno's Paradoxes but that is what this is about. – Simon G Feb 05 '14 at 20:38
  • @AsafKaragila Perhaps "debated" was a little too strong of a word (I doubt, but wouldn't be surprised to find, that these topics are formally debated). What I mean to say is that the paradox resolution via calculus doesn't sit well with everyone. I would consider Bertrand Russel a pretty serious guy, and even he took the time to weigh in on the subject fairly recently. I don't think he would have wasted the time if there remained nothing to be said. – pjs36 Feb 05 '14 at 20:39
  • @pjs36: I find this to be more of a critique of philosophy than a commentary about Zeno's paradoxes. – Asaf Karagila Feb 05 '14 at 20:56

1 Answers1

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Between every rational number (fraction) there is an irrational (think root 2 and stuff), between every two irrationals there is a rational.

A line in the real numbers does have infinitely many points, but the line that contains $x$ if $0<x<1$ has a mapping ($f$ say) with $-\infty<f(x)<\infty$ - this is bijective so they have the same cardinality, number of elements.

It can be shown that there's a bijection from the power set of the natural numbers (1,2,3....) to this interval $0<x<1$, thus this has the same number of elements.

the power set is the set of every set that can be made, so {1,2} has {}, {1}, {2} and {1,2} as its power set.

If a set has $n$ elements its power set has $2^n$

if we call the number of elements in the set of natural numbers $\infty_0$ Then the number of points in the interval $(0,1)$ is $2^{\infty_0}$ and the number of real numbers is also $2^{\infty_0}$

This shows the real numbers are "uncountable", there's no way you can have the first real number, there's no process you could iterate over where you could write down them all.

I have chosen infinity 0 as this is what I called it when I independently discovered this concept, I believe but I am not confident that there's a "funky $N_0$" used to denote it, if someone can confirm this please tell me how to enter this symbol in latex and I'll make the edit, or accept it if you make the edit.

Alec Teal
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