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This might be more philosophy than math, but it’s been bothering me for a while.
Question: If there’s an infinite amount of real numbers between $ 0 $ and $ 1 $, shouldn’t there be twice the amount of real numbers between $ 0 $ and $ 2 $? Wouldn’t that be $ 2 \times \infty $? What about the amount of real numbers between $ 0 $ and $ \infty $? Wouldn’t that be $ \infty \times \infty $?
I guess the whole concept of infinity just kind of evades me. Any help would be appreciated!
I answered a similar question on May 13th:
There are many different things that could be called "the infinite" in mathematics. None of them is a real number or a complex number, but some are used in discussing functions or real or complex numbers.
There are things called $+\infty$ and $-\infty$. Those appear in such expressions as $$ \lim_{x\to-\infty}\frac{1}{1+2^x} = 1 \text{ and }\lim_{x\to+\infty}\frac{1}{1+2^x}=0. $$
There is also an $\infty$ that is approached as $x$ goes in either direction along the line. That occurs in $$ \lim_{x\to\infty} \frac{x}{1-x} = -1\text{ and }\lim_{x\to1} \frac{x}{1-x}=\infty. $$ The second limit above could be said to approach $+\infty$ as $x$ approaches $1$ from one direction and $-\infty$ if from the other direction, but if one has just one $\infty$ at both ends of the line, then one makes the rational function a continuous function at the point where it has a vertical asymptote. This may be regarded as the same $\infty$ that appears in the theory of complex variables.
There are "points at infinity" in projective geometry. This is similar to the "infinity" in the bullet point immediately preceding this one. Two parallel lines meet at infinity, and it's the same point at infinity regardless of which of the two directions you take along the lines. But two non-parallel lines pass through different points at infinity rather than the same point at infinity. Thus any two lines in the projective plane intersect exactly once.
There are cardinalities of infinite sets such as $\{1,2,3,\ldots\}$ (which is countably infinite) or $\mathbb R$ (which is uncountably infinite). When it is said that Euclid proved there are infinitely many prime numbers, this sort of "infinity" is referred to.
One regards an integral $\int_a^b f(x)\,dx$ as a sum of infinitely many infinitely small quantities, and a derivative $dy/dx$ as a quotient of two infinitely small quantities. This is a different idea from all of the above.
Consider the step function $x\mapsto\begin{cases} 0 & \text{if }x<0, \\ 1 & \text{if } x\ge 0. \end{cases}$ One can say that its rate of change is infinite at $x=0$. This "infinity" admits multiplication by real numbers, so that for example, the rate of change at $0$ of the function that is $3.2$ times this function, is just $3.2$ times the "infinity" that is the rate of change of the original step function at $0$. This is made precise is the very useful theory of Dirac's delta function.
There is the "infinite" of Robinson's nonstandard analysis. In that theory, we learn that if $n$ is an infinite positive integer, then evern "internal" one-to-one function that maps $\{1,2,3,\ldots,n-3\}$ into $\{1,2,3,\ldots,n\}$ omits exactly three elements of the latter set from its image. Nothing like that holds for cardinalities of infinite sets discussed above.
I'm sure there are other examples that I'm missing here.
User wrote:
If there is an infinite amount of numbers between $ 0 $ and $ 1 $, shouldn’t there be twice that amount between $ 0 $ and $ 2 $?
No. Every real number in the interval $[0,1]$ can be mapped one-to-one to every real number in $[0,2]$ using the function $f(x)=2x$. So, there are no more numbers in $[0,2]$ than there are in $[0,1]$.
I have a rule of thumb: anything that says "infinity" (as opposed to $\aleph_0$, $\beth_2$, $\varepsilon_0$, $+\infty$, (projective) point at infinity, infinite, $\omega$, etc.) is probably not mathematics. It's not that mathematicians are afraid of the infinite -- hardly! -- but that it's an imprecise collection of heterogeneous concepts. Math is more precise than that.
In this case you're talking about the number of elements in a set (of real numbers), and so the particular concept you're talking about is called cardinality. The set [0, 1] has the same cardinality or 'size' as [0, 2]: precisely $2^{\aleph_0}=\beth_1.$
If you look at the fractions in [0, 1] and [0, 2] you'd find that, again, they have the same cardinality as each other, but a different cardinality from the above. This time they're 'just' $\aleph_0=\beth_0.$
If you drop all the way down to the integers the cardinality of the integers in [0, 1] is 2 while thee cardinality of the integers in [0, 2] is 3 and so these are no longer the same.
Intuitively, you are treating the "quantity" called infinity just like an ordinary real number. This is the defect.
You have a definition of $\infty$ and prove whether it satisfies the properties of real numbers, if yes, then you may say that,
If there is an infinite amount of numbers between $ 0 $ and $ 1 $, shouldn’t there be twice that amount between $ 0 $ and $ 2 $?
Note that you have assumed that the quantity $2$ $.$ $\infty$ is defined similarly as we define $x . y$ in $\mathbb {R}$.
Basically you may have assumed $\infty$ to be a very very big real number. This is the basic flaw, since $\infty$ is not a member of $\mathbb {R}$.
The ideas of being "of the same size", here , referring to your question we are dealing with cardinality, that we understand in the world of the finite, break down when we have infinite sets. There are formal definitions of what it means for two infinite sets to have the same size/cardinality or, informally, having the same amount of terms. There is a whole precisely-defined arithmetic of infinite cardinalities, with precisely-stated meanings, e.g.: $X^Y$ is the cardinality of all the functions from a set of cardinality represented by $X$ to a set with cardinality represented by $Y$.
Specifically, two infinite sets $X,Y$ have the same size if there is a one-one , onto map between them, i.e., if there is a bijection between $X,Y$ (there are related results like the Cantor-Schroder-Bernstein theorem that tell you that if there is an injection between $X,Y$ and an injection between $Y,X$ then there is a bijection between the two). In our case, since we can find a bijection between the two intervals $(0,1)$ and $(0,2)$, by our definition, the two have the same size (cardinality).
To answer your question, $2 \infty$ is decided by whther we can find a bijection between a set of that same infinite cardinality and the disjoint union of the set with itself.
Like Andre Nicolas stated in the comments, you can consider different measures of size, such as length, or measure in general. So one can have sets of the same cardinality but with different measure.
Infinity can be as big as you want it to be. It can be in the interval between 0 and 1, or it can be in the interval between 0 and 2, or in the interval between $(-3)^7$ to $10^{10^{10^{34}}}$, etc. I take it you've heard the term "infinitesimally small."
Try to count how many rational numbers there are between 0 and 1. Better yet, try to count how many rational numbers there are between 0 and 1 which can be expressed with a denominator that is a power of 2. There is $\frac{1}{2}$. There is $\frac{1}{4}$ and $\frac{3}{4}$. And $\frac{1}{8}$, $\frac{3}{8}$, $\frac{5}{8}$, $\frac{7}{8}$, ... you get the idea. Now try the same thing between 0 and 2. There's 1. There's $\frac{1}{2}$ and $\frac{3}{2}$. And $\frac{1}{4}$, $\frac{3}{4}$, $\frac{5}{4}$, and well, you run into the same problem.
In your mind, imagine a typical 12-inch ruler. But with this ruler, you can make the tickmarks as small as you want, whereas with a real ruler, you would get to a point at which you have to make the tickmarks so small you can't even see them.
The concept of infinity can easily lead to a lot of paradoxes, as well as statements that look nonsensical, like $\infty^2 = \infty^\infty = \infty$. I suggest you read http://www.suitcaseofdreams.net/infinity_paradox.htm
I think you should not try to tackle understanding "infinity" head on. It's literally too much for human mind. So I'll try to give an "intuitively" understandable answer, thus avoiding "infinity" from now on:
All the numbers in [0, 2] can easily fit into [0, 1]! We can play a game to show this: You provide a number from [0, 2] and I'll return a number from [0, 1]. For any different numbers, I'll lose, if I fail to provide different numbers. I can only win, if I convince you that I won't lose.
You may start with 2.
I'll go with 1.
Maybe 1?
0.5
square root of 2! (square root of 2) divided by 2.
Zero, got ya. Nope, zero, too.
Any number. Half that number.
See, what I'm getting at? There are as many numbers in [0, 1] as there are in [0, 2], because I can define a method to get one for every one you provide. I'll assume this is convincing enough and I won the game.
So to measure something that is too huge to measure, you just compare it, piece by piece, to something about the same size. Well, actually you don't compare, but tell how you would compare. If you can find a way to compare, piece by piece, it's not bigger.
The problem is to find a way to compare piece by piece. The only help I can give you in this, is the advice to avoid confusing yourself by the-bad-word. Instead ask yourself: "Where does the huge number of pieces come from?"
[0, 2] looks twice as big as [0, 1]. But it's not, if you half it!
[0, x] looks x times as big as [0, 1]. So as long as you can divide by x, everything is fine.
We all know that dividing by zero isn't allowed. So we found something that's "smaller": [0, 0] has only one element, so it's clearly smaller.
I disallowed myself from using the-bad-word. So I need to put it like this: If x keeps growing. What will happen to say 1/x?
At some point 1 divided by ever-growing-x will be indistinguishable close to 0. And if you can't distinguish between 0 and 1/ever-growing-x the piece to piece comparison fails. This makes [0, ever-growing-x] "bigger" than [0, 1] or [0, not-growing-x].
The next step would be to have a look at y-growing-way-faster-than-x. But I guess that'd overdo things.
This is a great anecdotal way to learn about different sizes of infinity.
Wikipedia Article - Hilbert's Hotel
So there are different sizes of infinity. Suppose you have two sets A and B and both of those sets are full of infinite things, that is, each of those two sets has infinite items. Not to say that they are they same size, one of their infinities may be bigger than the other.
If for every single item in A I can map it to exactly one item in B and I have all items in B covered, then the two sets are the same size of infinity.
If in all 1-1 mapping from A to B in which all items in A are used there are always items in B leftover, then the size of B is a bigger infinity than that of A.
You need a little set theory to understand this concept well.
If a set s has infinite amount of elements then there exists a function f from S to a subset of S ( say T, note that T != S ).
Just that easy !
Now, Why does N (1, 2, ,3 , 4, ...) have infinite elements?
Consider f(x) = x + 1 it maps whole N to "N - { 1 }" therefore N so .
Why does R have infinite members!? Because we can find functions like f(x) = 1/x which map whole r between a subset of R ( here between 0 & 1 ).
That's all I known ... I hope it helps. If you are interested in set theory I recommend "Lectures in logic & set theory vol2" ( hey I am not a mathematics expert if you ask a more experienced one you shall get better resources )
Its been mentioned before. But this TED-Ed video is totally cool ..
Infinity is so large, it can contain multiple copies of itself, within itself. It's not at all like any of the other numbers, so many of the "common sense" approaches to infinity don't make a lot of sense.
That answer that pointed you to Cantor's Diagonalization Argument... yes, that's the way to go...
If you want to be able to treat $\infty$ more or less as a real number, look into hyperreal numbers. A hyperreal number is essentially a number defined by a sequence, which may or may not be limited. Therefore it is possible to say that the sequence $\{2,4,6,...\}$ is twice as large as the sequence $\{1,2,3,...\}$, even though both approach $\infty$. However, good luck applying hyperreals to the set of numbers between $0$ and $1$. That particular type of infinity is very difficult to define or measure, and almost completely useless.
