The book I am working from (Introduction to Set Theory, Hrbacek & Jech) gives a proof of this result, which I can follow as a chain of implications, but which does not make natural, intuitive sense to me. At the end of the proof, I found myself quite confused, and had to look carefully at the build-up to see how the conclusion followed. I get the steps, now - but not the intuition.

The authors took sets $X$ and $Y$ and assumed injections $f: X \rightarrow Y$, $g: Y \rightarrow X$. Since $X \sim g[f[X]]$ and $X \supseteq g[Y] \supseteq g[f[X]]$, and $Y \sim g[Y]$, the authors went to prove the lemma that $A \sim A_1, A \supseteq B \supseteq A_1 \implies A \sim B$.

For the lemma, they defined recursive sequences $\{A_n\}_{n \in \omega}, \{B_n\}_{n \in \omega}$ by $A_0 = A, A_{n+1} = f[A],$ $ B_0 = B, B_{n+1} = f[B].$ Since $A_0 \supseteq B_0 \supseteq A_1 \implies f[A_0] \supseteq f[B_0] \supseteq f[A_1]$, we get from induction that $A_{n+1} \subseteq A_n$. Putting $\{C_n\}_{n \in \omega} = \{A_n - B_n\}_{n \in \omega},$ they noted that $C_{n+1} = f[C_n]$ (since $A_n \supseteq B_n$ inductively, $f[A_n - B_n] = f[A_n] - f[B_n]$). Putting

$$ C = \bigcup_{n=0}^\infty C_n, \text{ } D = A - C,$$

they noted that $f[C] = \bigcup_{n=1}^\infty C_n$, and that $h(x): A \rightarrow f[C] \cup D$ can be defined by sending $x \in C \rightarrow f(x),$ and $x \in D \rightarrow x$. It is clear that $h$ is one-to-one and onto.

Inductively, for $n > 0$, $C_0 \cap C_n = \varnothing$; it follows that $C_0 \cap \bigcup_{n=1}^\infty C_n = C_0 \cap f[C] = \varnothing$. We know that $C_0 \cup f[C] \cup D$ = A, and since all three sets are disjoint, we may conclude that $f[C] \cup D = A - C_0 = A - (A - B) = B$. Thus, our bijection $h$ maps $A$ to $B$, as we wanted.

What is the intuition here? What were H&J, or Cantor, Bernstein, etc. thinking when they went to prove this - the "high-level" idea? Is there a clearer proof, in the sense of thought process (not necessarily shorter)?

  • $\begingroup$ If I can count all the $Y$s using $X$s and all the $X$s using $Y$s, then there must be the same number of $X$s as $Y$s. $\endgroup$ – Kevin Carlson Oct 30 '12 at 22:33
  • $\begingroup$ Little remark: If you just care about the result, there's a (in my opinion very beautiful) proof described on the wiki page on the theorem. (I think it's clearer. Obviously it's also shorter.) $\endgroup$ – k.stm Oct 30 '12 at 23:29

$\newcommand{\ran}{\operatorname{ran}}$Here’s one way to think about it. Suppose that $y\in\ran f$; then we can pull $y$ back to $f^{-1}(y)\in X$. If $f^{-1}(y)\in\ran g$, we can pull it back to $g^{-1}(f^{-1}(y))\in Y$. If we continue this pulling back, one of two things must happen: either we reach a dead end at a point of $X$ or $Y$ that can’t be pulled back (because it’s in $Y\setminus\ran f$ or $X\setminus\ran g$), or we don’t.

Let $X_0=X\setminus\ran g$, the set of points of $X$ that cannot be pulled back at all, and let $Y_0=Y\setminus\ran f$. More generally, for each $n\in\omega$ let $X_n$ be the set of points of $X$ that can be pulled back exactly $n$ times, and let $Y_n$ be the set of points of $Y$ that can be pulled back exactly $n$ times. Finally, let $X_\omega$ and $Y_\omega$ by the subsets of $X$ and $Y$, respectively whose points can be pulled back infinitely many times.

At this point a sketch helps; it should show the partitions $\{X_n:n\le\omega\}$ of $X$ and $\{Y_n:n\le\omega\}$ of $Y$, and it should include arrows indicating what parts of $X$ get mapped to what parts of $Y$ and vice versa. To avoid having arrows crossing, I’ve taken $X$ and $Y$ apart in the following diagram.

$$\begin{array}{} X_0&\overset{f}\longrightarrow&Y_1&\overset{g}\longrightarrow&X_2&\overset{f}\longrightarrow& Y_3&\overset{g}\longrightarrow&X_4&\dots&X_\omega\\ Y_0&\overset{g}\longrightarrow&X_1&\overset{f}\longrightarrow&Y_2&\overset{g}\longrightarrow&X_3&\overset{f}\longrightarrow&Y_4&\dots&Y_\omega \end{array}$$

Each of the arrows is a bijection, so I can break up the diagram into $\omega$ self-contained parts. The first two parts are:

$$\begin{array}{} X_0&\overset{f}\longrightarrow&Y_1\\ Y_0&\overset{g}\longrightarrow&X_1 \end{array}\qquad \begin{array}{} X_2&\overset{f}\longrightarrow&Y_3\\ Y_2&\overset{g}\longrightarrow&X_3 \end{array}$$

Ignoring $X_\omega$ and $Y_\omega$ for the moment, I can rearrange the rest of the diagram to give my a bijection from $X\setminus X_\omega$ to $Y\setminus Y_\omega$:

$$\begin{array}{ccc} X_0&\overset{f}\longrightarrow&Y_1\\ X_1&\overset{g^{-1}}\longrightarrow&Y_0\\ X_2&\overset{f}\longrightarrow&Y_3\\ X_3&\overset{g^{-1}}\longrightarrow&Y_2\\ \vdots&\vdots&\vdots\\ X_{2k}&\overset{f}\longrightarrow&Y_{2k+1}\\ X_{2k+1}&\overset{g^{-1}}\longrightarrow&Y_{2k}\\ \vdots&\vdots&\vdots \end{array}$$

Finally, I claim that $f[X_\omega]=Y_\omega$: everything in $X_\omega$ can be pulled back infinitely often, so everything in $f[X_\omega]$ can be pulled back infinitely often, and therefore $f[X_\omega]\subseteq Y_\omega$. On the other hand, if $y\in Y_\omega$, then $y$ can be pulled back infinitely often, so it must be possible to pull $f^{-1}(y)$ back infinitely often, and therefore $f^{-1}(y)\in X_\omega$. Thus, $Y_\omega\subseteq f[X_\omega]$ as well. The diagram above can now be completed to show a bijection from $X$ onto $Y$:

$$\begin{array}{ccc} X_0&\overset{f}\longrightarrow&Y_1\\ X_1&\overset{g^{-1}}\longrightarrow&Y_0\\ X_2&\overset{f}\longrightarrow&Y_3\\ X_3&\overset{g^{-1}}\longrightarrow&Y_2\\ \vdots&\vdots&\vdots\\ X_{2k}&\overset{f}\longrightarrow&Y_{2k+1}\\ X_{2k+1}&\overset{g^{-1}}\longrightarrow&Y_{2k}\\ \vdots&\vdots&\vdots\\ X_\omega&\overset{f}\longrightarrow&Y_\omega \end{array}$$

The bijection is defined piecewise, but that’s no problem.

There are a few details to be filled in to make this a fully rigorous proof, but I think that it does give a reasonable idea of one possible intuition.

Added: Here’s a very rough sketch. Arrows from left to right are (parts of) $f$, and arrows from right to left are (parts of) $g$.

enter image description here

  • $\begingroup$ I'm having a hard time following this. Where you note that "at this point, a sketch helps" - what does this sketch look like? How have you taken the sets X and Y apart into their partitions? $\endgroup$ – Chris Oct 30 '12 at 23:38
  • $\begingroup$ @user1296727: I thought that the diagrams made that pretty clear; I don’t have very good facilities for doing so, but I’ll try to add a picture in a bit. $\endgroup$ – Brian M. Scott Oct 31 '12 at 0:00
  • $\begingroup$ Thank you for the diagram; it helps. When you say something is in $X_n$, do you mean that, by $n$ iterations of alternately pulling it back into Y, and then X, and then Y... you can get to something which can't be pulled back? (Otherwise, I don't see why the functions are bijections.) $\endgroup$ – Chris Oct 31 '12 at 0:26
  • $\begingroup$ @user1296727: Yes, that’s exactly right. Each pull-back is from one set to the other, so they must alternate. $\endgroup$ – Brian M. Scott Oct 31 '12 at 0:28
  • $\begingroup$ This looks like Eilenberg's Swindle. See en.wikipedia.org/wiki/… $\endgroup$ – Baby Dragon Sep 20 '13 at 21:42

It is a reasonably long and seemingly complicated proof, but actually the idea is very simple (or at least it is in the proof I've seen). We have injections $f:X\to Y$ and $g:Y\to X$. To build a bijection $h:X\to Y$ we need to send each point in $X$ to a unique point in $Y$, making sure we hit everything.

We can start by saying $h(x) = f(x)$. But of course, this doesn't hit everything if f is not surjective. But , for each element $y$ we don't hit in $Y$ (i.e. the set ${Y\setminus{f(X)}}$) there is a unique element $g(y)$ in $X$ which we can make map to $y$. So we edit $h$ such that now $$h(x) = \begin{cases} g^{-1}(x) & \mbox{if } g^{-1}(x) \notin f(X) \\ f(x) & \mbox{otherwise} \end{cases}$$

So this time we hit everything we didn't get first time from the $g^{-1}(x) $ part. But now we're missing some other parts! (namely the values $f(x)$ where $x \in g(Y\setminus f(X))$ but $f(x) \notin Y\setminus f(X)$.) We can define the sets that we "miss" in each iteration of improving $h$ as $C_n$ and say $C = \displaystyle \bigcup_1^{\infty}C_n$. Then defining $$h(x) = \begin{cases} g^{-1}(x) &\mbox{if } g^{-1}(x) \in C \\ f(x) &\mbox{otherwise} \end{cases}$$

And by thinking of how we built it up, it kind of makes a bit more sense as to why it is a bijection.

Please note that this isn't the most efficient way to carry out the proof, but I explained it in the way that I would come up with it whilst trying to see why it works, rather than what I'd write down formally. Also please note it's been a long time since I've seen the proof and I may have gotten some of the specifics wrong (please anyone correct me if this is the case!), but this is the right general idea (I hope!)


My favorite proof is due to R. H. Cox, and I learned it from the wonderful Handbook of Analysis and its Foundations by Schechter, from which I've taken the illustration below. The basic idea is that an injection from $Y$ to $X$ identifies $Y$ with a subset of $X$. So we can reduce the problem to showing that if we have $Y\subseteq X$ and $f:X\to Y$ is an injection, then $|X|=|Y|$ and this is less confusing. let $C=X\backslash Y$. The sequence of sets $\big(f^n(C))$ is disjoint since $f(X)\subseteq Y$ and $f$ is injective. Let $S=\bigcup_{n=0}^\infty f^n(C)$. Define $g:X\to Y$ by

$$g(x) = \begin{cases} f(x)&\mbox{if } x\in S \\ x & \mbox{if }x\notin S. \end{cases}$$

It is easily verified that $g$ is a bijection.

Remark: According to Kunen's book on the foundations of mathematics, Dedekind already proved this version of the Cantor-Bernstein theorem in 1887.

enter image description here

  • $\begingroup$ Yes, this theorem has quite the history. It's buried in a link [maybe in the comments] in one of my answers about the topic. $\endgroup$ – Asaf Karagila Oct 30 '12 at 23:40
  • $\begingroup$ @Asaf: I actually found the original reference online. I guess you refer to this comment by Theo. $\endgroup$ – Michael Greinecker Oct 30 '12 at 23:52
  • $\begingroup$ This proof is more succinct, but I don't know about clear. Why the heck would I expect your function $g$ to be a bijection? Why would I expect that this continued iteration of $f$ from the image of $X/Y$, over and over again, coupled with $X/Y$, would give me something in bijection with $X \cap Y$?! $\endgroup$ – Chris Oct 30 '12 at 23:54
  • $\begingroup$ @Michael: Yes, that's the one. $\endgroup$ – Asaf Karagila Oct 30 '12 at 23:57
  • 1
    $\begingroup$ I did...? (Please do be explicit, I'm juggling two proofs at once.) $\endgroup$ – Chris Oct 31 '12 at 0:04

When Y is a subset of X, Y can be assumed a proper subset of X otherwise the result is trivial. f is then called a reflection. If Z is a subset of X that contains Y then X is partitioned into Z and the rest of X which can be imagined as a frame of Z. This partitioning is carried over by f to Y, and then again to the image of Y under f and so on. Thus we obtain a sequence of disjoint frames and a sequence of nested subsets of Z. Picture the frames as an infinite stack. f then pushes the stack one level down. So if you take f on the stack of frames and the identity on the subsets you get a 1-1 mapping from X onto Z.

This is the essence of Dedekind's proof mentioned above. cf. my book http://link.springer.com/book/10.1007/978-3-0348-0224-6/page/1


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.