Is there a number before infinity?
In: Mathematics
Infinity is not a number, per se. It’s a size. If you could cram every possible number, all of them, into an endless box, infinity would be how you’d describe the size of that box. A box with enough room to contain something that never ends.
No, infinity minus one is still infinity.
Subtracting any finite amount doesn’t change the fact that the infinite is infinite.
Technically it’s just infinity minus 1 since infinity is really just a concept not a value.
“Infinity” describes how long (in terms of time or length) numbers can go on. It’s not an actual number.
The problem with this question is that “infinity” is somewhat ill-defined.
In mathematics, “infinite” just means “not finite”. While that sounds like a very obvious and almost meaningless statement, it turns out that in fact we are often interested in distinguishing a large number of infinite objects, and it is possible in many areas for one infinite object to be “bigger” (in some sense) than another infinite object.
But under every definition I can think of, the answer to your question is “no”, even if we use the “smallest infinity” available. Here are two examples that should be relatively understandable even if you don’t have much background in theoretical math:
* Suppose we think of “infinity” as “the size of the set of natural numbers N = {1, 2, …}”. Then your question becomes “is there a set S so that (a) that set is smaller than N and (b) there is no set that is smaller than N but bigger than S”. The answer is no, there isn’t – or rather, the weakest axioms of mathematics don’t allow us to prove the existence or nonexistence of such a set. The stronger versions of the usual axioms allow us to prove that one does not exist.
* Suppose we think of “infinity” as “a single number attached to the real numbers, such that that number is bigger than all real numbers”. Then your question becomes “is there a number *x* such that (a) x < infinity and (b) there is no other number y such that x < y < infinity”. The answer is no. Since infinity is the *only* infinite element in this set, x < infinity implies that x is a finite number, and therefore there is a number (say, x+1) that is still smaller than infinity but also bigger than x.
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.]
“Infinity” is a name shared by lots of things that are technically different. My answer will apply to most of the ways mathematicians use the word “infinity,” but it might not apply to everything called infinity. (I’m not enough of an expert on infinity to say for sure.)
I’ll use the word “number” to refer to counting numbers (positive integers) only. So we’ll not worry about negative numbers, fractions, or irrational numbers.
You don’t precisely define what you mean by “before.” There are two concepts you might mean:
– (a) By “before infinity,” you mean (in standard mathematical terminology) “less than infinity.”
– (b) By “before infinity,” you mean (in standard mathematical terminology) a “predecessor of infinity.”
These are different questions with different answers.
Say instead of talking about infinity (which is tricky to think about), we instead talk about 1,000,000.
– (a) Is there a number less than 1,000,000? Yes. 1 is less than 1,000,000. Also, 2 is less than 1,000,000. Also, so are 3, and 4, and 5. Also, 137,649 is less than 1,000,000. So is 544,203. In fact there are 999,999 different examples of numbers less than 1,000,000.
– (b) Is there a predecessor of 1,000,000? Yes. 999,999 is a predecessor of 1,000,000. It’s the *largest* number that’s smaller than 1,000,000. There is only one predecessor of 1,000,000: it’s 999,999 [1] [2].
Now infinity.
– (a) Is there a number less than infinity? Yes. 1 is less than infinity. Also, 2 is less than infinity. Also, so are 3, and 4, and 5. Also, 137,649 is less than infinity. So is 544,203. In fact there are infinitely many examples of integers less than infinity, since every integer is less than infinity.
– (b) Is there a predecessor of infinity? No. Suppose you think you have some candidate that might be a predecessor of infinity, let’s call your candidate N. If you show me your candidate, I will point out that N+1 is also less than infinity. Therefore you must have been mistaken: N couldn’t possibly be the predecessor of infinity, as being the predecessor means you’re the *largest* number less than infinity, and I’ve just shown you that N isn’t the largest such number.
[1] Again, I’m talking about counting numbers only. If we allow decimals, there are lots of decimals / fractions before 1,000,000, such as 999999.5, 999999.9999, and so on, but we decided earlier that we’ll allow counting numbers only.
[2] Switching from talking about “a predecessor” to “the predecessor” brings up a technical issue: Uniqueness. It’s somewhat off-topic, somewhat technical, and also not very easy to ELI5, so I’ll give a short acknowledgement:
*When talking about integers*, it’s generally safe to switch from talking about “a largest integer with property P” to “the largest integer with property P”.
There are other kinds of mathematical objects where switching from talking about “*a* largest” to “*the* largest” is making an assumption which might be false.
No. It’s one of the more defining features of infinity.
But in many cases, there is a number after infinity, infinity + 1.
But yeah, if we talk of positive integers and ways to expand them to cover infinity, key facts to keep in mind are,
* there are infinitely many numbers
* no integer is infinite. Even though there are infinitely many integers bigger than any limit you set, none of them are infinite. This might seem obvious but in surprisingly many cases people end up messing up by assuming “infinitegers”
* the infinity that comes after all infinitely many integers is called ω
* If we expand integers to contain ω, and numbers like ω + ω and ω^ω + 5, we get so called *surreal numbers*, expansion of real numbers.
* with surreal numbers 1 + ω = ω, but ω + 1 > ω. Order matters, adding 1 after the infinity works different than adding it before infinity.
* with all that being said, you can keep counting from infinity with little trouble, but you cannot go backwards. You can’t take one-less-than-infinity.
All this being said, infinity gets used in various ways in maths, ω is from what I can tell closest to what you were asking about, out of things maths has to offer.
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