Larger sets of infinity

This seems to be a widely accepted theory on larger sets of infinite numbers, originally shown by Cantor.

After watching the video, I am trying to grasp in layman's terms why this is true. My understanding is the following, and I wondered if someone could confirm if this is correct;

The reason there are more real numbers between 0 and 1, than all the natural numbers, is because in this example, each real number can have a length of infinity.

Initially I thought that there is an infinite number of real numbers between 0 and 1, and an infinite number of natural numbers. This would allow for (to use the videos metaphore) a line to be drawn between every real and natural number. But if the real numbers are also infinite in length, there are "infinity to the power of infinity" real numbers, and just infinity natural numbers.

Have I understood this correctly? If not, could someone spell it out for me please?

• I have not watched the video, but there as as many integers as there are fractions (i.e. the number of fractions are countable). However, there are uncountably many real numbers. – picakhu May 30 '12 at 17:23
• There are "countably infinite" natural numbers. There are "countably infinite" rational numbers. What does "countably infinite mean? A set A is said to be "countably infinite" if there is a bijection from N to A. The set of real numbers R is not countably infinite (we say that R is uncountably infinite). When we try to show that R is countably infinite, we fail due to a contradiction. You can use google and/or wiki for proofs of these facts. They are standard and accepted by most mathematicians. – Adam Rubinson May 30 '12 at 17:33
• Also, what do you mean by length of infinity ? Do you mean, infinite decimal places after the decimal point? If so, then yes, each real number can be written as an infinite string of numbers after the decimal point (but not uniquely, e.g. 0.99999... = 1) – Adam Rubinson May 30 '12 at 17:36
• If you look at all infinite sequences $x_1,x_2, x_3,\dots$ of real numbers, then it turns out that there are no more of these (in the sense of cardinality) than there are real numbers. So even though the explanation given is not wrong, there is detail to fill in before one can come up with a tight argument. – André Nicolas May 30 '12 at 18:08

You seem to get the gist of the idea by stating that there are "infinity to the power of infinity" real numbers, and "just infinity" natural numbers. The problem is that you haven't clarified "which" infinity you are talking about, nor have you shown that raising "infinity to the power of infinity" (whatever that means) gets you a "bigger" infinity. Speaking loosely again, you know that there are infinitely many even numbers, and infinitely many integers, and that they are the same "size" of infinity, even though one is "infinity" and the other is "two times infinity". So, it is important that we are precise about defining what we mean by "infinity".

The cardinality (size of infinity) of the natural numbers is $\omega$. This is also the cardinality of the rational numbers, algebraic numbers, etc. In set theory, we can think of the real number line $\mathbb{R}$ as $\omega^\omega$, which is just the set of functions from $\omega$ to itself (this is the same thing as $\omega$ to the power of $\omega$). Each function from $\omega$ can be thought of as a function on the natual numbers, and hence is just a sequence. Thus each real number is just a sequence of natural numbers.

If you look at Cantor's Diagonalization Argument, he uses the fact that each real number is a sequence of $\omega$-many natural numbers to inductively construct a real number that could not have been enumerated. The crux of the argument relies on being able to demonstrate the existence of a real number that "defies" the enumeration up until the $n$th step, where $n$ is made arbitrarily large. This, in turn, requires that the real numbers can contain an "infinite amount of information," which I'm guessing is what you intuitively mean by stating that real numbers are "infinite in length."

The reason there are more real numbers between 0 and 1, than all the natural numbers, is because in this example, each real number can have a length of infinity.

That's pretty much right, but you must be a little bit careful about what you mean by length. If you just look at the real numbers whose decimal expansions terminate, like 23.5043958495, or 0.000000000004 or whatever, there are only as many of those as there are natural numbers. The "lines" thing will work, though it would be a bit tricky to say exactly which line goes where. But when you allow decimal expansions to go on forever to the right, that is when you look at the set of all real numbers, there are strictly more of these than there are naturals.

there are "infinity to the power of infinity" real numbers, and just infinity natural numbers.

Isaac talked about this, so I won't. Have a look at http://en.wikipedia.org/wiki/Cardinal_number to find out more about how people measure the relative sizes of infinite sets.

• In saying "when you allow decimal expansions to go on forever to the right" you have oversimplified. Nonterminating repeating decimal expansions "go on forever to the right", but the set of those is merely countable. Austin Mohr's answer makes this distinction correctly. – MJD May 30 '12 at 19:12

To supplement the answers already given, I would add that one should take care to distinguish between sets in which decimal representations use a finite, but unbounded number of digits (such as the naturals) and those that use an infinite number of digits (such as the irrationals).

The former means that the decimal representation of any particular natural number has finite length, but I may choose as large a finite length as I want. Considering numbers like this will only get you up to a countable set.

On the other hand, the decimal representation of any particular irrational number truly uses a (countably) infinite number of digits, and so may be said to be of (countably) infinite length. It is these kinds of numbers that give the "infinity to the power infinity" that you need to construct an uncountable set.