Topics Numbers and logic

What is infinity?

Infinity is the idea of something that never ends. In everyday speech it means "bigger than anything you can count," but in mathematics it is not an ordinary number you can add to a bank balance. It is a way of describing sets, processes and limits that have no last step, like the counting numbers 1, 2, 3, and so on forever.

What makes it interesting is that infinity comes in different sizes. In the late 1800s, Georg Cantor showed that the whole numbers and the real numbers on a number line are both endless, yet the second kind cannot be matched up one to one with the first. Some infinities are bigger than others. Meanwhile, infinity also shows up in calculus, in geometry, and in physics, where it often signals that a theory has reached its limit rather than that nature really contains something infinite. Whether the universe itself is infinite is still an open question.

An episode on bre would start from the plain picture of counting without stopping, then walk through the strange parts one at a time. bre's hosts are AI, so they can get things wrong, and you can press Talk to ask a question whenever something does not sit right.

What a bre episode would cover

An outline of the episode bre would make for this question. Every episode is written fresh when you ask, so yours will differ.

  1. Counting without a last numberWhy there is no biggest whole number, and why infinity is better understood as a process or a size than as a place you can arrive at.
  2. The hotel with endless roomsA famous thought experiment, Hilbert's hotel, shows how a full hotel with infinitely many rooms can still fit new guests.
  3. Some infinities are biggerCantor's diagonal argument, in plain words: why the real numbers cannot be listed, even by an endless list.
  4. Infinity in calculusHow limits let mathematicians handle things that grow without bound or shrink toward nothing, without ever treating infinity as a normal number.
  5. Infinite sums that add upWhy adding endlessly many pieces can still land on a finite total, and why other endless sums really do blow up.
  6. Is the universe infinite?What physics can and cannot say. Whether space goes on forever is unresolved, and infinities in equations often mark where a theory breaks.
  7. Why infinity feels wrongWhy our intuition, built on finite things, keeps tripping over endless ones, and how mathematicians learned to trust careful rules instead.

How the episode might open

A sample exchange between two of bre’s AI hosts, bre and Cal. Both are AI; this is written by AI, as every bre episode is.

  1. breAI host

    Okay, let's start with the question everyone asks quietly. Is infinity a number? Like, could I write it on a check?

  2. CalAI host

    Please don't. Short answer: no. Longer answer, and I'm going to wander a bit: it behaves so differently from numbers like seven or a million that mathematicians treat it as a different kind of thing, closer to an idea about size or about something that never stops.

  3. breAI host

    So what's wrong with treating it like a really big number?

  4. CalAI host

    Because the usual rules break. Take any number you like and add one, you get a new number. But if infinity were a number, infinity plus one would just be... infinity again. Nothing changed. That's a red flag.

  5. breAI host

    That feels like cheating. Where does it come from, then?

  6. CalAI host

    From counting. You count 1, 2, 3, and you notice you never get to a last one. Infinity is the name for that never-ending-ness, and once you take it seriously, it gets weirder, because it turns out there's more than one kind.

  7. breAI host

    Okay, hang on. More than one kind of endless? I don't know about you, but I'd like that explained slowly.

  8. CalAI host

    Slowly it is. We'll start with a hotel that has no last room.

Questions people also ask

Is infinity a number?
Not in the ordinary sense. It does not follow the usual rules of arithmetic, so mathematicians treat it as a concept describing endlessness or size. In some systems, symbols for infinity are used carefully with special rules, but it is not a regular number you can count to.
Are there different sizes of infinity?
Yes. Georg Cantor showed that the counting numbers and the real numbers are both infinite, yet the real numbers form a larger infinity because they cannot be paired off one to one with the counting numbers. This is a standard, well-established result in set theory.
What is infinity plus one?
For the usual infinity of counting numbers, adding one gives a set of the same size, so the size is unchanged. In other systems, like ordinal numbers, infinity plus one is a different, larger position in an order. The answer depends on which kind of infinity you mean.
Is the universe infinite?
Nobody knows. Observations are consistent with a universe that is very large, but whether it goes on forever or eventually closes up or ends is an open question. We can only observe a limited part of it directly.

Related topics

More: all 300 topics, numbers and logic, or the longer reads on /learn.

bre’s hosts are AI, and every episode is generated, so they can be wrong: check anything that matters. This page outlines what an episode would cover. It is for interest and learning, not medical, financial or legal advice.