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Real Numbers Continued

Silly Druid

Entry posted by Silly Druid in Mathematics

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The title of this entry means we're still talking about the real numbers, but it also means we're talking about something called "continuum". But what is it? The answer is it's a kind of infinity, and it's different from the "countable infinity". (There are more kinds of infinity, but let's focus on these two.)

"Countable infinity" is, for example, the number of elements in the set of all integers. "Countable" means we can arrange the elements in a sequence, so for any integer there is a well-defined previous one and next one.

Rational numbers are an interesting case, because they can't be ordered in a sequence where every element is bigger than the previous one. Why? Because between any two different rational numbers we can find a third one, for example the arithmetic mean of them. But there are ways to order them in a different kind of sequence. We can put them in a big table, where the horizontal position is the numerator, and the vertical position is the denominator. Then we can go through the diagonals in this table, and add the unique fractions we find there to the list. Unique means we can omit the ones that are already there, for example if we already have 1/2, then we don't have to put 2/4 on the list. This way we can create a proper sequence, which means the set of rational numbers has "countable infinity" elements.

In the set of real numbers, it's different. They are not a sequence, and there is no way we can make it a sequence. Which seems strange because in the previous entry I proposed an algorithm that generates them all, and at any finite step they form a nice sequence. But the exponential growth of the number of elements in every step causes something strange when we go to infinity: a sequence stops being a sequence and becomes a total mess. And that's continuum for you.

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Splashee

@PawelS Just checking, when you say "integers", you mean "whole numbers which include the negative side of the number line"?

As soon as we go negative, there is the negative infinity as well. So I am having a little bit of a problem to visualize "a different kind of infinity", when there are already two on the number line to being with, both on opposite ends.

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Silly Druid

@Splashee Yes, that's what I mean by integers. It's a set of all positive and negative whole numbers (and zero), and it doesn't actually contain positive or negative infinity.

Maybe I was a little unclear about that, but in this entry I'm talking about the number of elements in a set, which obviously can't be negative. I don't mean the infinities that you can find at both "ends" of the number line.

So the number of elements in the set of integers is countable infinity by definition, it just means you can assign an integer to every element in the set. With some tricks, you can also do that with rational numbers, and even algebraic numbers. But with real numbers you can't, no matter how hard you try. That's why I'm calling it a "different kind of infinity", which is called "continuum" and is bigger than countable infinity.

But is there an infinity that falls between these two in terms of size? That's another story...

Edited by PawelS
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Splashee
1 minute ago, PawelS said:

Maybe I was a little unclear about that, but in this entry I'm talking about the number of elements in a set, which obviously can't be negative.

Kinda like how a length of a vector can't be negative because multiplication by itself (two negatives) becomes positive.

 

It will take some time before I fully understand this blog entry. I'll get there eventually ^_^

So "counts" (number of elements in a set) can't be negative, and they must be whole numbers, right?

 

Just thinking, Real Numbers could be used as an index (count to a specific index), in between indices, like an interpolation thing. But then we have different interpolation methods as well, such a bilinear and bicubic, so no way of specifying what it means. I don't like Real Numbers :worry: They are too "lossy".

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Silly Druid

@Splashee I admit it can be hard to understand. I'm trying to explain some elements of Cantor's set theory, without using too many technical terms like "cardinality" or "bijection", but it's still pretty abstract stuff.

I like how you think about this as an indexing problem. So countable sets are the ones that can be indexed by integers, while uncountable ones can't.

(Btw I edited my previous comment, so you might want to read it again.)

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