Can x=x+1?

Posted on April 13, 2023 • 2 minutes • 408 words

A few of days ago I saw a post on Hacker News about extending numbers to allow “impossible” things. The discussions in the original MathOverflow thread and in the Hacker News comments made a lot of sense: extending the numbers should keep the existing properties consistent.

Can we create such an extension that satisfies $$x = x + 1$$ without creating contradictions? And without resorting to some esoteric maths concepts?

Actually, we can. But first, let’s draw a simple picture:

Coordinate grid with points A(2, 3.5) and B(4, 1)

And now we define an operation of projecting the point onto a selected axis:

$$A=3.5 \enspace (\text{proj } y)$$

$$B=1 \enspace (\text{proj } y)$$

This operation returns the value of the $y$ coordinate: the second number in the $(m, n)$ pair. The dotted line shows a physical meaning of the that value.

The syntax of this operation is similar to the modular arithmetic; the main difference, beside the way it’s calculated, is the usage of the equal sign = instead of the equivalence sign ≡.

And now we can extend any real number into the realm of points: any number $r$ can be represented as a point on the $x$-axis:

$$R(r,0)$$

In this case, its projection onto the $y$-axis is 0:

$$R=0 \enspace (\text{proj } y)$$

And this leads us to an equation ($B$ is defined above):

$$B=B+1 \enspace (\text{proj } y)$$

Obviously, you can put $A$ instead of $B$ and any other real number instead of 1 in the equation above, and it will hold.

Here we defined the addition operation as member-wise addition:

$$B+1= (4,1)+(1,0)=(5,1)$$

This way the projection of $B+1$ onto $y$-axis does not change:

$$B=1 \enspace (\text{proj } y)$$ $$B+1=1 \enspace (\text{proj } y)$$

It’s easy to see that other properties of addition remain consistent:

$$B+1=1+B \enspace (\text{proj } y)$$ $$B-1=B+(-1)=-(1-B) \enspace (\text{proj } y)$$

Then, what about subtracting $B$ from both sides of $B+1=B$ ?

$$B+1=B \enspace (\text{proj } y)$$ $$B+1-B=B-B \enspace (\text{proj } y)$$ $$1=0 \enspace (\text{proj } y)$$

And the last statement is correct as both $(1, 0)$ and $(0, 0)$ project onto 0 on the $y$-axis!

You can say using the projections here is cheating. Of course, this post is here for pure mathematical entertainment. And surely, there are many other ways of extending real numbers to satisfy $x = x + 1$ consistently. Using projections looked a quite visual one, though. And it is a good basic test for KaTex - a math typesetting library in JavaScript , too.