Is there a number before infinity?
In: Mathematics
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.
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