The simplest way of proving that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = c$

Question

What is the simplest way of proving (to a non-mathematician) that the power set of the set of natural numbers has the same cardinality as the set of the real numbers, i.e. how to construct a bijection from $\mathcal{P}(\mathbb{N})$ to $\mathbb{R}$?

Answer 1

Here's my favourite (most intuitive one I've seen): Define the bijection in three parts

First is easy: take your favourite bijection between $\mathbb{R}$ and $(0,1)$. (Even to a non-mathematician it is intuitively clear that one exists by just "contracting" $\mathbb{R}$, and one example can be give explicitly by $\frac{1}{\pi}arctan(x) +\frac{1}{2}$).

Then by considering the binary expansion of real numbers we almost get a bijection between $(0,1)$ and the set $S$ of infinite binary strings, where the number $0.1001110110010\dots$ corresponds to the string $0.1001110110010\dots$ etc. I say "almost" because the binary strings $100000\dots$ and $01111\dots$ correspond to the same real number $0.10000 \dots = 0.01111\dots$ To a non-mathematician you could say "we can fix these problems because there aren't many stings with repeated $1$'s, compared to the set of ALL binary strings" and move on.
(To a mathematician, this is because a string ending in all $1$'s is determined by its prefix, which has finite length, and mathematicians know that the set of finite binary strings is countable, and furthermore by the diagonalization argument $\mathbb{R}$ is UNcountable. This allows you to modify the natural map $S\rightarrow (0,1)$ into one which is a bijection using the method described at the bottom of this link: https://nrich.maths.org/discus/messages/67613/67678.html?1133563921 But like I said, to a non-mathematician these issues aren't important.)

Finally, another easy one: a binary string $s\in S$ is by definition a sequence of $0$'s and $1$'s, which can technically be defined as simply a function $s:\mathbb{N}\rightarrow\{0,1\}$. Such a function determines a subset $A_s\subset\mathbb{N}$ by "$a\in A_s$ iff $s(a)=1$" and similarly a subset determines a function (we call $s$ the characteristic function of $A_s$ in $\mathbb{N}$). An intuitive way to think of this phenomenon as that a binary string $s$ goes through the elements of $\mathbb{N}$ one-by-one and decides if it is in the set $A_s$ (with a $1$) or not (with a $0$). This correspondance gives the bijection between $S$ and the powerset $\mathcal{P}(\mathbb{N})$.

Combining it all together, we get $\left|\mathbb{R}\right|=\left|(0,1)\right|=\left|S\right|=\left|\mathcal{P}(\mathbb{N})\right|$

P.S. This is the sort of discourse in which the vast majority of people first come to understand this theorem, because this is the context in which it is phrased: sets and functions and clever bijections. If you would prefer a more elementary description then I will paraphrase what Euclid told a general under Alexander the Great: there is no royal road to mathematics. This is as elementary as I can make it without writing an introduction on set theory :)

Answer 2

If it has to be a bijection I would go for continued fractions (as pointed out by J.D.):

1. State that for every real number there is only one shortest representation of it as a continued fraction ($a_0 \in \mathbb{Z}$, $a_i \in \mathbb{N}-\{0\}$ and if the fraction is finite, $a_{\text{last}} \in \mathbb{N}-\{0,1\}$ ).
2. Then explain that $P(\mathbb{N})$ it the same as infinite sequence of 0s and 1s.
3. Show that infinite sequence of 0s and 1s is the same as sequence of natural numbers of any length (coded in base 1 with ones interleaved by zeros, i.e. 2,3,0,1,0,0,0... = 11 0 111 0 0 1 00000..., finite sequence have finite number of zeros, empty sequence = 11111...).
4. Sequence of natural numbers is a continued fraction:
   • empty sequence is 0;
   • sequence of length 1 should be transformed by any $\mathbb{N} \to \mathbb{Z} -\{0\}$ bijection;
   • if the sequence is of length 2 or greater, transform first term by any $\mathbb{N}\to\mathbb{Z}$ bijection and add $1$ to the rest;
   • if the sequence is finite, increase the last term by additional 1.

This is the simplest bijection I could think of. But probably it doesn't have to be a bijection, in which case the argument is much simpler:

1. For a given set $X$, first sort it.
2. Take first smallest number, if it is odd, the result is negative.
3. Take the second smallest, this will be the integer part of the real.
4. Take the rest (in ascending order) all modulo $10$--this will be the tail of the real.

Going back:

1. Take only reals from $[0,1]$, where $1$ is represented by $0.(9)$,
2. $n \in Y$ if and only if the digit on the $n$-th place is odd.

Edit 1: to simplify a bit the step 4 in the bijection, you could interpret the sequence of natural numbers in a way that the first one tell how many 1s are there on the end of stream.

Edit 2: I missed the case of finite number of zeros in (3), thanks to Marc van Leeuwen for pointing that out. I think that the result after the fix is even nicer than before.

Hope that helps ;-)

Answer 3

For this answer, use $\{1,2,\ldots\}=\mathbb{N}$.

A. For each positive real $x\in (0,1]$ there is a unique infinite subset $A$ of $\mathbb{N}$ such that $x = \sum_A 2^{-p}$.

(To find numbers in $A$ one could start with some $p$ so that $2^{-p}$ is at least half the distance to $x$, but not $x$ itself. Then continue with that idea for each remaining distance.)

B. Every infinite subset of $\mathbb{N}$ corresponds to an element of $(0,1]$ in this way.

(Have them imagine adding up all those $2^{-p}$'s and appeal to their intuitive sense of convergence of an increasing bounded series.)

C. Now just use $f(x) = \tan(\pi x - \frac\pi2)$ which takes $(0,1)$ onto $\mathbb{R}$ (and don't worry about the endpoint $1$). This shows that the infinite subsets of $\mathbb{N}$ are equinumerous with $\mathbb{R}$.

(One could explain it by stretching $(0,1)$ out until it "becomes" $\mathbb{R}$.)

D. As the collection of finite subsets of $\mathbb{N}$ is countable, this completes the proof.

(Wave hands and say something like "finite sets are smaller than infinite sets" ;)

Answer 4

Here is a bijection inspired by the one given by dtldarek, but I've tried to reformulate as non-technical as possible. Instead of continued fractions, I make use of the related Stern-Brocot tree (or better, one with root $0$ and completed with a reflected image added to its left so as to cover the negative numbers as well as the positive numbers).

But I won't need a technical description of it. In fact I can do with any tree labelled by rational numbers (or one could even allow arbitrary real numbers), with the following properties:

• For any node labelled by a number $y$, the labels $x$ of all nodes in the left subtree below it have $x<y$, and the labels $z$ of all nodes in the right subtree below it have $z>y$.
• If at this node $y$ node one goes to the left child, and then keeps going from a node to its right child indefinitely, one gets a sequence of (increasing) labels $x$ that converges to $y$.
• If at this node $y$ node one goes to the right child, and then keep going from a node to its left child indefinitely, one gets a sequence of (decreasing) labels $z$ that converges to $y$.
• If from the root node one descends to the left (resp. right) child indefinitely, the decreasing (resp. increasing) sequence of labels obtained diverges to $-\infty$ (resp. to $+\infty$).

These properties guarantee that the (countable) set of all labels forms a dense subset of the real numbers (the set of all rationals in the case of the symmetrized Stern-Brocot tree), and that for any real number $r$ that do not occur in the tree, one has a unique a infinite path whose labels converge to $r$, which can be found by using the tree as a binary search tree: when at a label $y$, descend to the left subtree if $r<y$ and to the right subtree if $r>y$. Note that since $r$ is not itself a label this is well defined, and the infinite path so obtained will not beyond some point continue to go in one same direction indefinitely, in other words, it keeps on (ultimately) changing direction, forever.

Now the basic idea is to use, given a subset $S$ of $\mathbb N$, the presence/absence of succesive numbers to determine the direction of descent in the tree; say go left at step $i$ if $i\notin S$ and right if $i\in S$. Except for the two extreme possibilities, the labels along the infinite path obtained converge, and taking their limit almost defines a bijection to the real numbers (with $\{-\infty,\infty\}$ added to it for the diverging extremes).

However the map is not injective: all numbers that occur as a label $y$ are the limit for $2$ different paths, both passing though the node labelled $y$: one that descends to the left child of $y$ and then to the right forever, and another that descends to the right child of $y$ and then left forever. This defect, as well as the diverging extremes, can almost be repaired by the following rule: for those paths that beyond some point continue indefinitely in the same direction (these correspond to subsets of $\mathbb N$ that are either finite or the complement of a finite set), stop after the last move in the opposite direction, and take as image the label of the node reached. This change removes the two infinite paths whose labels converged to $y$, and replaces it by the unique path that was of the form: descend to $y$, change direction there and after that never change direction any more.

The one tiny defect that remains is that for the two extreme paths, our rule stipulates to truncate after the last step in the opposite direction, which step does not exist: the natural thing to do here is to remove all steps, and map to the label of the root of the tree ($0$ for the symmetrized Stern-Brocot tree). The two subsets that "collide" here are the empty set and all of $\mathbb N$. To repair this I start by modifying $\mathcal{P}(\mathbb N)$ so as to "tuck away" (make disappear) the empty set. The following rule does this: for all finite initial segments, i.e. subset of the form $\{i\in\mathbb N\mid i<n\}$ for some $n\in\mathbb N$ (which includes the empty set for $n=0$), add to it the minimal element $n$ originally absent form it. This maps $\mathcal{P}(\mathbb N)$ bijectively to $\mathcal{P}(\mathbb N)\setminus\{\emptyset\}$, and after this transformation one can use the map described above, which now is a bijection to $\mathbb R$. Oof!

Answer 5

Here's my attempt to make the steps as mathematically non-threatening as possible, even at the expense of elegance:

We start from any real number, and wish to transform it into an unique set of natural numbers, such that every subset of $\mathbb N$ is hit exactly once.

Step 1: Squeeze the real line into the half-open interval between $0$ and $1$:

• If the number was $0$, then it stays unchanged.
• If the number was a positive integer $n$, then replace it with $\frac 1{2n}$.
• If the number was a positive non-integer, say $+x$, then replace it with $\frac1{2(x+1)}$.
• If the number was negative, say $-y$, then replace it with $\frac12 + \frac1{2(y+1)}$.

(This is not a particularly nice way to do this, mathematically speaking, but it is easier to understand than many slicker alternatives).

Pause. Satisfy yourself that every number between 0 (inclusive) and 1 (exclusive) is hit exactly once.

Step 2: Write down the decimal fraction for the number. It will start by "0." followed by a countably infinite sequence of decimals. If you can represent the number exactly with finitely many digits, top it up with infinite repeating 0s that the back end. Don't use representations that end in infinitely many '9's.

(There's a considerable amount of mathematical sophistication being swept under the rug in this step, but I'm assuming you're intuitively familiar with decimal fractions, and know that $0.999... = 1$).

Step 3: Remove the initial 0..

Pause. Satisfy yourself that every infinite sequence of digits is hit exactly once, except that sequences that end in infinite repeated 9s are not hit.

Step 4: Encode the digits as ones and zeroes, using a table such as

0 becomes 0000    5 becomes 0101
1 becomes 0001    6 becomes 0110
2 becomes 0010    7 becomes 0111
3 becomes 0011    8 becomes 10
4 becomes 0100    9 becomes 11

The precise details of the encoding are not important, except to note that if we encode any sequence of digits in this way, we don't need to keep around any separators between successive digits. You can reconstruct the digits without them, because the first bit of an encoded digit determines whether it's a 4-bit or a 2-bit encoding. Furthermore, a sequence that ends in infinite repeating 9s (but only such sequences) would map to a sequence ending in infinite repeating 1s.

Pause. Satisfy yourself that every infinite sequence of digits is hit exactly once, except that sequences that end in infinite repeated 1s are not hit.

Step 5: Get the sequences that end in infinite repeated 1s included, by this rule:

• If the sequence already has infinitely many 1s (which must be interspersed with infinitely many 0s in some pattern or non-pattern), then leave it as it is.
• If the sequence only has finitely many 1s, and it starts with a 0, then chop off the initial 0 and leave it otherwise as it is.
• If the sequence only has finitely many 1s and starts with an 1, then chop off the initial 1, and flip every remaining element in the sequence from 1 to 0 or vice versa.

Pause. Satisfy yourself that every infinite sequence of zeroes or ones is hit exactly once, no exceptions.

Step 6: Take the subset of $\mathbb N$ consisting of the positions in the sequence where an 1 is found.

Satisfy yourself that every subset of $\mathbb N$ is hit exactly once.

Answer 6

I will prove every reasonable "equinumerosity" involved. There is a key lemma (Schröder–Bernstein theorem) that states that $|A|=|B|$ iff there is an injective map from $A$ to $B$ and an injective map from $B$ to $A$.

1. $|\mathbb{R}|=|(0,1)|$ is trivial: $f(x)=\frac{1}{\pi}\left(\frac{\pi}{2}+\arctan(x)\right)$ is an increasing function that maps $\mathbb{R}$ to $(0,1)$;
2. $|(0,1)|=|[0,1]|$ is less trivial: take an enumeration of the rational numbers in $[0,1]$ such that $q_0=0,q_1=1$, then map $q_n$ into $q_{n+2}$ (Hilbert's hotel);
3. $|(0,1)|=|2^{\mathbb{N}}|$ is tricky: every number in $(0,1)$ has a unique canonical binary representation, where canonical means that no tails of $11111\ldots$ are allowed: that gives an injective map from $(0,1)$ to $2^\mathbb{N}$. On the other hand, if $\{a_n\}_{n\geq 0}$ is a sequence such that $a_i\in\{0,1\}$, the map $$ \{a_n\}_{n\geq 0} \mapsto \sum_{n\geq 0}\frac{2a_n+1}{5^{n+1}} $$ sends $2^{\mathbb{N}}$ into a Cantor subset of $(0,1)$ in a injective way.
4. $|\mathcal{P}(\mathbb{N})|=|2^{\mathbb{N}}|\,$ is trivial again: every non-empty subset $A\subseteq \mathbb{N}$ can be associated with the sequence $\{a_n\}_{n\geq 0}$ in which $a_n=1$ if $n\in A$ and $a_n=0$ otherwise.

By $(4)+(3)+(2)+(1)$, $$\left|\mathcal{P}(\mathbb{N})\right|=\left|\mathbb{R}\right|$$ as wanted. The trickiest part comes from noticing that this approach shows that a bijection exists without an explicit construction: the "i.e." in OP's question is not really an "i.e.", unless we are constructivists.