What Is Infinity Plus One?

At first glance, infinity seems like the largest possible quantity. So what happens if we add one to it? How is it possible to add anything to something that, by definition, never ends?

A seemingly simple question reveals a deeper one:
Is infinity really a number?
And, for that matter, are numbers really quantities?

A Hotel with Infinite Rooms

To explore this question, let’s start with a famous thought experiment by mathematician David Hilbert: the Hilbert Hotel paradox.

Imagine a hotel with rooms numbered 1 to ∞, and all of them are occupied.
Now a new guest arrives. It seems there’s no room… but the receptionist has a clever idea: each guest moves to the next room. Room 1 to 2, room 2 to 3, and so on.

Room 1 is now free — and the new guest can check in.

When Infinitely Many New Guests Arrive: The Even Room Trick

Now suppose an infinite number of new guests arrive — each assigned a number: 1, 2, 3, …

Again, the hotel is full. What now?

Here’s the trick:

  • Each current guest moves to the room twice their current number: 1 to 2, 2 to 4, 3 to 6, and so on.
  • This frees up all the odd-numbered rooms: 1, 3, 5, …
  • The new guests can now move into those odd-numbered rooms in order.

Result:

  • Old guests occupy the even rooms
  • New guests occupy the odd rooms

The hotel is still full — but it has welcomed infinitely many new guests.
In the world of infinity, there’s always room for more.

Infinity Plus One… Still Infinity?

From a formal standpoint, mathematics gives us a crisp answer:

\[ \infty + 1 = \infty \]

But then — if we subtract \( \infty \) from both sides, do we get:

\[ \infty + 1 - \infty = \infty - \infty \Rightarrow 1 = 0? \]

Of course not.
This only highlights a key point: we can’t treat infinity like an ordinary number.
Some operations — like \( \infty - \infty \) — aren’t defined, because infinity isn’t a concrete value.

So what is infinity, really?

Infinity as a Concept

The idea of infinity is ancient — from Zeno’s paradoxes in Greek philosophy to theological and metaphysical debates in the Middle Ages.

But the symbol ∞ is quite modern.
It was introduced by John Wallis in 1655, possibly inspired by the shape of a sideways “8” — a loop that can be traced forever.

With Newton and Leibniz, infinity became central to calculus, as a way of expressing endlessly growing quantities or infinitesimal values approaching zero.

Over time, mathematics developed multiple meanings of infinity, each within its own structured context.

Numbers Are Not Quantities

To truly grasp what’s going on, we must distinguish two fundamental ideas:

  • Quantity: something measurable (3 apples, 5 liters)
  • Number: an abstract symbol that can describe quantity, position, order, or identity

For instance, the number 3 might mean:

  • Three oranges → a quantity
  • Page 3 → a position
  • Jersey number 3 → an identifier

Numbers are not quantities — but they can be used to describe quantities.

They’re abstract tools, created to model aspects of the world.
Zero, negative numbers, imaginary and complex numbers, even infinite ones — all are conceptual extensions introduced to meet the evolving needs of mathematics.

Infinity Means Different Things in Different Contexts

Infinity — like numbers — does not have a single meaning. It takes on different forms depending on the domain in which it’s used.

Let’s look at a few examples:

🧮 Extended Arithmetic

\[ \infty + 1 = \infty \]

This is treated as a symbolic statement, expressing the idea that “adding one to infinity changes nothing.”
But beware: expressions like \( \infty - \infty \) are undefined.

📐 Calculus and Limits

\[ \lim_{x \to \infty}(x + 1) = \infty \]

Here, infinity represents asymptotic behavior, not a fixed value.
Adding 1 doesn’t change the trend toward infinity.

🔢 Set Theory and Cardinality

The set of natural numbers \( \mathbb{N} \) has cardinality \( \aleph_0 \).
Adding a single element doesn’t change its size:

\[ \aleph_0 + 1 = \aleph_0 \]

Even adding another countably infinite set still gives \( \aleph_0 \).

🧠 Philosophy of Mathematics

In this broader view, “infinity plus one” becomes a conceptual statement:

Infinity cannot be completed, counted, or exceeded by one.
It is a mental construct to express the absence of limit.

Conclusion: Infinity Plus One Is Not a Calculation — It’s an Idea

When we say \( \infty + 1 = \infty \), we’re not solving an equation.
We’re recognizing the conceptual nature of infinity.

Infinity isn’t measured, summed, or exhausted.
It is thought.

And from that perspective, our original question is answered not by arithmetic, but by insight.
Not because there’s a clever trick — but because that’s how the idea of infinity works.

As so often happens in mathematics, a simple formula can carry deep philosophical weight.
And sometimes, that depth asks us not just to recalculate, but to rethink the very way we understand ideas.