Is there a number before infinity?
In: Mathematics
“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.
Latest Answers