Is this conception of countable vs. uncountable infinity adequate?

Question

I am not a mathematician, so please point out any mistakes I am making here - I am trying to grasp the concept of countable vs. uncountable infinity in a somewhat informal way and would like to know whether that conception makes sense.

We can imagine the set of natural numbers as an axis that goes from some fixed point ($0$, or $1$ if you want) to infinity:

Clearly, the points on this axis are countable, because we know exactly how the axis goes on and can therefore make a precise calculation about how many points the rest of the axis will contain.

For the integers, we no longer have a fixed starting point, but the axis grows infinitely in two directions:

However, we still have only one axis of fixed points and can make a precise calculation of how many points the axis as a whole will contain.
The first thing that bugs me: Since the axis is calculably exactly twice as long, this should be a "larger infinity" than the for the natural numbers, right? But still, we would say that $\mathbb{Z}$ has the same cardinality as $\mathbb{N}$?

For the rational numbers, things get a little more difficult, but we can still handle it: Any rational number can be displayed as the fraction between two integers - if I understood it correctly, this is what the Cantor pairing function does? - so we can just add a second axis to account for the combinatoric possibilities yielding $\mathbb{Q}$:

The amount of points now doesn't simply add up, i.e. the axis doesn't just get longer (as in the step from $\mathbb{N}$ to $\mathbb{Z}$), but it multiplies, i.e. there are more axes now, so that's even a "larger increase of infinity". Is this correct?
But we still have finitely many axes with countably many points, so the whole amount of points is countable too.

Now for the case of real numbers, things look a bit differently.
Clearly, a one-dimensional system doesn't suffice because we need to account for the digits behind the comma, so we need at least two axes, in order to create $0.0, 0.1, 0.2, ..., 1.0, 1.1, ...$:
Now that doesn't suffice either, because from $1.1$, we can decide to either stay at $1.1$, which would be $1.10$ (Is it true that $1.1$ is in fact $1.10$ which is in fact $1.1000000...$, so that rational numbers are actually never really finite, or is this idea false and $1.1$ is really just $1.1$?) or go further to $1.11$, so we need another axis:

We are now three-dimensional and can thus account for all the numbers with two digits behind the comma, but that still doesn't suffice, because between $1.10$ and $1.11$, we also have the numbers $1.101, 1.102, 1.103, ... $, and from any point we are, we are recursively stepping one dimension deeper, because for any digit we add, we again have all the possibilities to go on from that point, so we never reach a point where we can stop adding axes:

(At this point I'm running out of imagination on how to draw a 7D diagram, sorry)

Now we are at the point where we run into an infinite number of axes - and this corresponds to the set of real numbers $\mathbb{R}$ no longer being countably infinite.

My question is:
Is it adequate to say that countably infinite corresponds to finitely many dimensions, while uncountably infinite corresponds to infinitely many dimensions, or did I go anywhere wrong in my conception?

Answer 1

The notion of "countably infinite" is well named. Another word you can use is "enumerable," which is even more descriptive in my opinion.

I understand your intuition on the subject and I see where you're coming from. Let me try to give you some insight into the agreements about infinity that have been reached over the years by the mathematicians who've worked on this problem. (This is what I wish someone had explained to me.)

"Countably infinite" just means that you can define a sequence (an order) in which the elements can be listed. (Such that every element is listed exactly once.)

The natural numbers are the most naturally "countable"—they're even called "counting numbers"—because they are the most basic, natural sequence. The word "sequence" itself is wrapped up in what we mean by natural numbers, which is just one thing following another, and the next one coming after that, and the next one after that, and so on in sequence.

But the integers are countable as well. In other words, they are enumerable. You can specify an order which lists every integer exactly once and doesn't miss any of them. (Actually it doesn't matter, for the meaning of "countable," if a particular element gets listed more than once, because you could always just skip it any time it appears after the first time.)

Here is an example of how you can enumerate (count, list out) every single integer without missing any:

$0, 1, -1, 2, -2, 3, -3...$

The rational numbers are also countable, again because you can define a sequence which lists every rational number. The fact that they are listed means (at the same time) that they are listed in a sequence, which means that you can assign a counting number to each one.

The real numbers are not countable. This is because it is impossible to define a list or method or sequence that will list every single real number. It's not just difficult; it's actually impossible. See "Cantor's diagonal argument."

This will hopefully give you a solid starting point to understanding anything else about infinite sets which you care to examine. :)

Answer 2

You are somewhat right, in the sense that it is true that $\mathbb N^n$ is countable for positive $n$, and $\mathbb N^\mathbb N$ is uncountable (which is the set of functions from $\mathbb N$ to $\mathbb N$). Even $\{1,2\ldots n\}^\mathbb N$ is uncountable for $n>1$ (think about writing the numbers in the interval $(0,1]$ in an arbitrary base).

So a reliable way to prove that a set $A$ is countable is to build an injective function $f:A\rightarrow \mathbb N^n$, and a reliable way to prove that a set $B$ is uncountable, is to build an injective function $g:\{1,2\ldots n\}^\mathbb N\rightarrow B$

Answer 3

You are using non-standard imagery but, as I read it, the point where the "intuition" you describe fails is this:

Now we are at the point where we run into an infinite number of axes - and this corresponds to the set of real numbers $\mathbb{R}$ no longer being countably infinite.

Consider for example the set of algebraic numbers which contains all numbers that are roots of some polynomial with rational coefficients. This set obviously includes $\mathbb{Q}\,$, and it is a strict superset of $\mathbb{Q}$ since e.g. $\sqrt{2}$ is an algebraic number (as being a root of $x^2-2=0$) but $\sqrt{2} \not \in \mathbb{Q}\,$.

If you tried a construction similar to those "axes" you'd likely end up with an infinite number of them to cover all algebraic numbers. However, the set of algebraic numbers is provably countable.

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[ EDIT ]  To try and address @Yakk's point raised in a comment...

You should be more concrete about your construction of a counter example to the intuitive understanding of the OP. As an example, talk about the ring of polynomials directly instead of the algebraic numbers in your answer, then illustrate how it has the infinite axis property, and point out it remains countable.

I chose algebraic numbers as an example since it seemed to better fit the pattern of "towering" extensions of number sets in the original post. That said, OP's construction of "axes" is not clear enough to me to even attempt to duplicate it for algebraic numbers. I got the idea from this (quoting) "uncountably infinite corresponds to infinitely many dimensions" that the "axes" may be related to "dimensions" or loosely speaking some other measure of "degrees of freedom". My "counterexample" about algebraic numbers was meant to give an example of such a case where an "infinitely dimensional" super-set (again, loosely speaking) happened to still be countable.

The relation between algebraic numbers and polynomials in $\mathbb{Q}[X]$ is obvious by definition, so I did not elaborate much on that. One problem with open-ended questions like this one is that it's hard to second-guess the right language that "connects" best.

Answer 4

While certainly the "intuition" you have does help, if you want to begin to tackle cardinality questions, it will not suffice. It is a bit like saying that:

A countably infinite set is one where you can count the next number, while uncountably infinite is you can't even begin to go to the next number. For example, the next natural number after 1 is 2. What's the next real number after 1?

(In actuality, I think the above is actually more useful). To begin tackling a few of your comments:

The first thing that bugs me: Since the axis is calculably exactly twice as long, this should be a "larger infinity" than the for the natural numbers, right? But still, we would say that $\mathbb{Z}$ has the same cardinality as $\mathbb{N}$?

Yes, it should rightfully bother you. It is true since there is a bijection from $\mathbb{Z} \rightarrow \mathbb{N}$. If you want to use your "axes" argument, you can view it as the $x-y$ axis of just the natural numbers, consisting of only the positive $x$ axis and negative $y$ axis.

The amount of points now doesn't simply add up, i.e. the axis doesn't just get longer (as in the step from $\mathbb{N}$ to $\mathbb{Z}$), but it multiplies, i.e. there are more axes now, so that's even a "larger increase of infinity"

Ok..you can possibly extend your argument to the entire $x-y$ plane of just the natural numbers. The main point you are actually using is the fact that whenever you have trouble counting the set you have, you just add another axis of the integers/natural numbers. This is of course...true, if you can explicilty count out a set using this method, then for sure it is countable since it is proven that the cross product of countably infinite sets is countable. However, it is quite nonsensical to say that "uncountably infinite corresponds to infinitely many dimensions" because a countably infinite cross product of countably infinite sets is also countably infinite...and your argument denies this.

Not to mention, it is far from useful to prove more complicated cardinalities and ones of actual mathematical interest. If you want to actually understand "cardinality" and countable vs. uncountable, you can only start by using the actual definition that is that a set is of the same cardinality if you can form a bijection between them.

Answer 5

On different sizes of infinity

There is really no controversy in mathematics over the claim that there are an infinitude of different sizes of infinity, but there ought to be.

To convey the mathematical view very briefly: Is the infinite series of integers (1, 2, 3, and so forth) the same size (cardinality) as the infinite series of even integers? (2, 4, 6, and so forth). While at first, it might appear that there are twice as many integers as even integers, because you can map every other integer onto each of the even integers, and this leaves unmapped the infinite series of odd integers, (which itself is presumably the same size as the set of even integers), so one should conclude the integers is a larger set, in fact, exactly twice as large:

Integers: .......1..2..3..4..5..6..7..8..9..10…

Even Integers: ..2......4......6......8.....10…

(In mathematical parlance, integers are surjective but not injective with respect to even integers)

However, the above is just one of many ways to map between the two sets. One could as easily map the integers onto powers of two (2n), thus:

Integers:...........1..2.......3..............................4 …

Even Integers:..2..4..6..8..10.11.12.13.14.15.16.17 …

According to this mapping, for every integer there is a power of two, but there are plenty of even numbers that are not a power of two, so the even integers must by this mapping far out-number the integers. (In mathematical parlance, the integers are injective but not surjective with respect to even integers.) This contradicts the conclusion based on the earlier mapping.

There is a particularly valued mapping, called a one-to-one correspondence, valued because it is the core to the mathematical definition of numerosity (or cardinality). And, interestingly, these two sets can be mapped in a one-to-one correspondence by doubling each integer:

Integers:............1...2...3...4...5...6 …

Even Integers:..2...4...6...8...10..12…

Mathematicians take the amenability of these two sets to be mapped onto each other in a one-to-one correspondence as a demonstration that they have the same numerosity. (In mathematical parlance: with respect to the even integers, the set of integers is both injective and surjective, hence bijective, and thus both sets have the same cardinality.)
Infinite sets of this size are said to be countable.
Of course, counting and infinity do not belong in the same sentence. Counting is a process whose value is only known once the process has ended. Infinity never comes to an end, so it is logically impossible to count it. But the field of mathematics has cleverly circumvented this inconvenience using the core feature of numerosity: one-to-one correspondence. Using this principle, you cannot actually claim to have counted the elements of an infinite set, but you can compare the numerosity of one infinite set to another infinite set by establishing a one-to-one correspondence that includes all of the members of the first set exactly once, and likewise all the members of the comparison set exactly once. The correspondence itself will be infinite, but we can avoid having to enumerate the entire correspondence using logic: you can establish whether the method of enumeration will result in the entire set of elements being visited exactly once.
There are many other infinite sets that likewise can be mapped one-to-one with the integers, such as the set of all positive rational numbers (numbers that can be represented as fractions). So, all of these sets which are amenable to being mapped one-to-one to each other are considered to be the same (countable) size of infinity.

However, there are some sets that cannot be mapped one-to-one with the set of integers. They are “uncountable”. Evidently, any effort to map integers onto these sets leaves an infinitude of unmapped entities. This is demonstrated through Cantor’s diagonal method, or cousin methods. Accordingly, mathematicians have concluded that these sets are larger, that is, they constitute a larger infinity. Following similar logic, mathematicians have “mapped out” a dizzying list of ever larger infinities.

There is a serious flaw to all of this enumeration. While the mathematics of enumeration works quite well with finite sets, it falls apart when applied to infinite sets. Mapping one infinite set to another is problematic. As we have already seen, the two infinite sets, integers and even integers can, by mapping, be shown to be both the same and different in size; furthermore, mapping “proves” that both the former and the latter set are the larger of the two. (In mathematical parlance: whether a set interjects or surjects onto a second set is not a characteristic of the sets themselves, nor of their relationship, but a characteristic of the particular mapping process selected.) So, wouldn’t the proper conclusion be that when it comes to infinite sets, mapping is nonsensical and should not be trusted? Isn’t this precisely the type of contradictory outcomes that has led to the unanimous rejection of division by zero by the community of mathematicians? Mapping of infinite sets is the set-theoretic counterpart to algebra’s division by zero.

So, when a mathematician demonstrates by a diagonal method that there is at least one element of a target set (codomain) not mapable to (not in the image of) the set of integers, they may wish to conclude that consequently no such one-to-one (bijective) mapping is possible, yet all they have shown is that the diagonal method is a method that does not happen to result in a one-to-one mapping. They have in no way shown that there is no mapping method that could result in a one-to-one mapping.

It gets worse. The diagonal method and its ilk are specifically biased to result in the conclusion that the target set (codomain) is larger than the set of integers. It is not possible for the method to show the opposite, even in cases where the opposite may be true. I suspect it is not possible for the method to show that two sets of equal size are indeed equal. So, the use of the diagonal method as proof that a set is larger than the set of integers is a rigged test. Or, to be less inflammatory, it is fallacious.

There were warning signs that this was the case. The determination of the size (cardinality) of infinite sets is an application of set theory, and set theory, while it works perfectly for sets of finite size, is well known for creating self-contradictory paradoxes when dealing with sets of infinite size. While mathematicians have been scrambling to repair set theory and come up with a version not susceptible to such paradoxes, even when a repaired theory is proposed, it is difficult to know whether the revised theory is not also susceptible, after all. I don’t believe that there is any version of set theory that has been proven to be immune to self-contradiction when dealing with infinite sets, and even if it turns out there has been, such a very special version of set theory is not the version that has been used to establish the relative sizes of infinity. Proof that the practice of mapping infinite sets to each other results in contradictory conclusions has already been demonstrated in the simple case of even numbers.

Let us revisit diagonal mapping ala Cantor. (We will deviate from Cantor’s process in certain key particulars.) Let us begin the enumeration of the set of all numbers that can be expressed as a binary decimal beginning with 0.0 and adding the digit 1 to the right of the decimal point thereafter a finite number of times. The enumeration of this infinite set could be represented thus:

S1 = 0 0 0 0 0 0 0 0 0 …

S2 = 1 0 0 0 0 0 0 0 0 …

S3 = 1 1 0 0 0 0 0 0 0 …

S4 = 1 1 1 0 0 0 0 0 0 …

S5 = 1 1 1 1 0 0 0 0 0 …

S6 = 1 1 1 1 1 0 0 0 0 …

S7 = 1 1 1 1 1 1 0 0 0 …

S8 = 1 1 1 1 1 1 1 0 0 …

S9 = 1 1 1 1 1 1 1 1 0 …

. . . Using the digits falling along the diagonal, as Cantor did, if we consider the first n enumerations, we will get a string of n zeros. If we then invert the zeros into ones, as Cantor did, we get a string of n ones. This string of n ones appears in the n+1 item in the series. There is no finite integer value of n that can be selected for which this fails to be true. Thus, the inversion of Cantor’s diagonal does not result in an item that does not belong to the set thus enumerated. This set is countable.

Before considering the so what of all this, let us first note that even though the set is itself infinite, none of the items within the set are infinite—the position of the item is always finite, and the length of the corresponding string of ones is likewise finite. As a corollary point, note that (unlike Cantor) we never attempt to take the entirety of the diagonal as we generate inversions; rather, we take the diagonal up to a finite number of digits, n. Attempting to take the entire (infinite) diagonal would not make sense, inasmuch as we began by defining the set under question as consisting only of finitely long strings. Next, let us note that the set being enumerated does not even include all the rational numbers, let alone all the real numbers on the interval from 0 to 1. So, clearly we are not surprised to find this Cantoresque method leading to the conclusion that the set is enumerable.

Now, I have already admitted that I did not exactly use Cantor’s method. I distorted it by examining the diagonal at finite lengths rather than at one infinite length; also, by not paying any attention to the digits to the right of the diagonal. So, with this same distortion, let us revisit Cantor’s original proof using real numbers instead of a series of ones. Recall that in his proof the real numbers are enumerated in any random order, in binary (using just ones and zeros). As we consider the diagonal up to the nth item, and then invert ones and zeros, we get a finite string that, so far has not (could not have) appeared at the onset of any of the n enumerated rows. But for every inverted diagonal at n, there is the non-contradictory possibility of the n+1 row beginning with that exact string. As for what follows after that exact string (on this n+1 item) there are no restrictions, so absolutely anything could follow. Thus, there is no real number starting with this string which could not be listed as the n+1 item. This is true for every finite integer value of n. There is never a point reached where the next row cannot contain the inverse of the diagonal up to that point. Thus, the set being enumerated is enumerable.

Wait. There must be something wrong with this perversion of Cantor’s proof. Cantor’s proof does not check any nth item, but checks one specific infinitely long item. So, how is it that the proof shows countability through every element of the series, but not when taken as an infinity? Because when at infinity, we are (nonsensically) claiming that we have all the items, and that there can be no next item. Which is false. There is no last item in an infinite set, and so there is always an n+1 after any item in the set, and that n+1 item could easily be the missing number we thought could not be enumerated. So, the fallacy of Cantor’s proof and all the cousin proofs that attempt to show uncountability is simply that there is an assumption that you can have all of infinity, and that with that entirety there can be no next item in the set.

Wait, when we consider a one inch long line, do we not have, in effect, an interval with an infinite number of subdivisions, and all of those subdivisions present? So, doesn’t this contradict my claim that we can never have all of an infinite set? Not really. First of all, as far as physics is concerned, there does in fact come a point where we can no longer subdivide reality. But math is not beholden to quantum physics, and we can by fiat presume a line has infinite subdivisions. But the fallacy in this is that while we do indeed have the line, we do not thereby have all its subdivisions. To have those, we must subdivide: We must enumerate all the places where subdivisions could come. And that is an unending process that always allows for some next subdivision to be made.

Wait, common irrational numbers such as pi, e, and the square root of 2 “have” all infinity of their digits, do they not? Well, in some sense that is true, but not in the sense necessary here. These irrational numbers have no last digit, no digit such that there can be no next digit, so mathematical proofs that presume the possession of a listing out of all the digits of pi, or e, or the square root of two, excluding the possibility of there being any further digits to list out, are fallacious, as already described.

The upshot of all this is that (1) the enumeration of infinite sets is nonsensical, and (2) claims that one infinite set is larger than another infinite set are likewise nonsensical. At best, one could discuss the relative sizes of sets under a given mapping.

Let us explore this possibility. If numerosity conclusions change under different mappings, then in order to coherently discuss the sizes of infinity, we must first specify which mapping we are using. We would then refrain from generalizing any conclusions we might reach to any other mappings.

One could ask which mapping(s) would be preferable to use? Mathematicians have answered: always use only the one-to-one mapping (achieved by any means possible) unless no such mapping exists, otherwise, use the diagonal mapping. While this choice can lead to some interesting conclusions about countability and the relative size of uncountable infinities, I am hard pressed to imagine any physical-world implications under this mapping. More importantly, the diagonal mapping depends on the fallacy of asserting there can be no next item after the inverted diagonal, so it is not even a legitimate mapping.

Another, perhaps less versatile, yet more easily interpreted mapping is the simplest (most parsimonious) mapping, which consists of lining up identical elements. If all the elements of one set happen to be identical to a subset of elements in the other set, then the first set is a proper subset of the latter. If such subset mapping is used (with a little tweaking to allow the odd numbers to be numerically mapped onto the even numbers*), then the even integers are indeed half as numerous as the integers. Such an approach would (finally) make sense of the statement that “the set of integers is the union of the set of even integers and the set of odd integers.” This approach opens up the possibility of quantifiable differences in size among the so-called countable infinities: Twice infinity plus triple infinity equals quintuple infinity. The arithmetic of algebra works just fine (i.e., without creating internal inconsistencies) when the object is an infinite rather than a finite variable (so long as the infinities are “countable”). Tellingly, when inverted to the infinitesimal, the properties of subset-reckoned infinities constitutes precisely the algebraic relationships among dx’s that is the basis of integral calculus.
For more than a century mathematicians have enjoyed decrying the subset approach to numerosity as not only false, but naïve, so I suspect there will be a cultural resistance to adopting this path.

(To get an intuitive sense of the relationship of algebraically related countable infinities to the dx’s of calculus, consider the following thought experiment.) Let’s imagine a string of ten LED’s that can be turned on and off at will. Let us collect an infinite number of these ten-diode strings. Let’s shrink these strings down to infinitesimal size, and line all infinity of them up, and arrange them evenly over the surface of a square meter. Now, let’s turn on just one of the ten diodes in each string. There will be a faint glow along the entire square-meter surface so constructed (with a total brightness the same as the brightness of however many full-sized diodes would cover a tenth of a square meter). Now, let’s turn on a second diode in each. The light will glow twice as brightly. Turn on all ten, the light will glow ten times as brightly. The classical one-to-one correspondence view insists that the same number of diodes is lit up in all three cases, so there should be no difference in brightness, but the subset method correctly predicts the exact difference in brightness. This thought experiment suggests intuitively that the subset approach to infinities has a sensible relation to the physical world, the same as dx does.

*One tweak that could allow the odd integers to be mapped onto the even integers: For sets with no common (identical elements), if a one-to-one correspondence can be established wherein both sets are identical through simple addition of a constant to each element of one set, then the sets can be said to be the same size.

Answer 6

Your understanding of the whole thing is a little off the road of regular Mathematical sense. I am not a mathematician or even a math major, but I know some math and let me tell you some of my understanding. The major difference btw set R and N is that N is listable but R is not, and listable is just another saying of countable. N is somewhat discrete while R is "continous". Btw every 2 reals, there will always be a real, no matter rational or irrational while between 2 consecutive integer, no integer exists. There is no "next" for a real. As to the dimension thing, you have to know linear combination and dimension is defined as the minimal number of elements such that each element of the space can be represented as a linear combination of those elements. Those elements must also be linear independent. In finite dimensional space, the situation is always simpler than infinite dim space. In finite dim, people tend to find element whose span is dense instead of being the whole space. For R, it is one dimensional because a single real can represent all others by linear combination.