Why does this go into an infinite loop?

Note: Originally I posted C# code in this answer for purposes of illustration, since C# allows you to pass int parameters by reference with the ref keyword. I’ve decided to update it with actual legal Java code using the first MutableInt class I found on Google to sort of approximate what ref does in C#. I can’t really tell if that helps or hurts the answer. I will say that I personally haven’t done all that much Java development; so for all I know there could be much more idiomatic ways to illustrate this point.

---

Perhaps if we write out a method to do the equivalent of what x++ does it will make this clearer.

public MutableInt postIncrement(MutableInt x) {
    int valueBeforeIncrement = x.intValue();
    x.add(1);
    return new MutableInt(valueBeforeIncrement);
}

Right? Increment the value passed and return the original value: that’s the definition of the postincrement operator.

Now, let’s see how this behavior plays out in your example code:

MutableInt x = new MutableInt();
x = postIncrement(x);

postIncrement(x) does what? Increments x, yes. And then returns what x was before the increment. This return value then gets assigned to x.

So the order of values assigned to x is 0, then 1, then 0.

This might be clearer still if we re-write the above:

MutableInt x = new MutableInt();    // x is 0.
MutableInt temp = postIncrement(x); // Now x is 1, and temp is 0.
x = temp;                           // Now x is 0 again.

Your fixation on the fact that when you replace x on the left side of the above assignment with y, “you can see that it first increments x, and later attributes it to y” strikes me as confused. It is not x that is being assigned to y; it is the value formerly assigned to x. Really, injecting y makes things no different from the scenario above; we’ve simply got:

MutableInt x = new MutableInt();    // x is 0.
MutableInt y = new MutableInt();    // y is 0.
MutableInt temp = postIncrement(x); // Now x is 1, and temp is 0.
y = temp;                           // y is still 0.

So it’s clear: x = x++ effectively does not change the value of x. It always causes x to have the values x₀, then x₀ + 1, and then x₀ again.

---

Update: Incidentally, lest you doubt that x ever gets assigned to 1 “between” the increment operation and the assignment in the example above, I’ve thrown together a quick demo to illustrate that this intermediate value does indeed “exist,” though it will never be “seen” on the executing thread.

The demo calls x = x++; in a loop while a separate thread continuously prints the value of x to the console.

public class Main {
    public static volatile int x = 0;

    public static void main(String[] args) {
        LoopingThread t = new LoopingThread();
        System.out.println("Starting background thread...");
        t.start();

        while (true) {
            x = x++;
        }
    }
}

class LoopingThread extends Thread {
    public @Override void run() {
        while (true) {
            System.out.println(Main.x);
        }
    }
}

Below is an excerpt of the above program’s output. Notice the irregular occurrence of both 1s and 0s.

Starting background thread...
0
0
1
1
0
0
0
0
0
0
0
0
0
0
1
0
1