Is there a number before infinity?

Question

Is there a number before infinity?

In: Mathematics

Answer 1

It depends on what kind of infinity you mean, because we’ve actually defined a whole bunch of different types of infinity, and they’re not all quite the same to each other. And it’s not really agreed-upon in all fields that infinity *is* a number. Some of the infinities in math have more number-like properties than others.

There is a concept called “cardinality”, which treats numbers as quantities or amounts. The first infinite cardinal number we know is called “aleph-null” and it expresses the idea of a ‘countable infinity’, it’s how many whole numbers there are. The next bigger one is “aleph-one” and it’s how many real (ie, rational and irrational) numbers there are. For both of these infinities, we would answer your question “no.” If we subtracted one or added one to aleph-null, we just still have aleph-null.

But there’s a different way of approaching infinity called “ordinality”, and with this approach, things like “infinity plus one” and “infinity minus one” may have a quite sensible meaning. We call the first named infinite ordinal, “omega.” You might imagine the ordinal numbers as denoting places in an infinitely long queue. If you’re standing in line for an infinite rock concert, and there’s an infinite number of ticketholders in line in front of you, we might say that you are the infinity-eth person in line.

Now, if we asked you “how many people are ahead of you in line?” the answer would be aleph-null. And if we asked the person 3 places ahead of you the same question, they’d give the same answer! But still, you are clearly in different positions in line.

In this situation it makes perfect sense to talk about the person three places ahead of you, and to say they’re at the “infinity minus three” position in line. So if you’re using ordinal numbers, then yes: every infinite ordinal has a (also infinite) number which comes immediately before it. [edit: see below, this isn’t true; we define some ordinals specifically as limits which don’t have one.]