In
mathematics there are two non-equivalent definitions of when something is infinite, though the inventor of the second definition probably thought they were equivalent. The first definition is that a
set is infinite if it is not
finite, and a
finite set is one that is the same size as one of the
natural numbers 0, 1, 2, ... In
set theory the natural numbers are formalized as finite
ordinals, that is a kind of ordered set, where any
n = {0, 1, ...,
n−1}. If there is a
bijection between this and a set
S, then
S is finite (and of size
n), else it is called infinite.
In these terms, infiniteness is a purely negative property: there exists no mapping of such-and-such a nature. But infinite sets can be investigated, and hierarchies of ordinals and cardinals of various infinite sizes can be found. The term 'transfinite' means essentially the same as 'infinite', but with a positive sense that bijections can be found between the infinite set and some well-defined transfinite size.
The existence of infinite sets does not follow from the axioms of set theory used to construct finite sets: it requires a special Axiom of Infinity to assert explicitly that a certain infinite set (the set of the natural numbers) exists.
The nineteenth-century mathematician Richard Dedekind formulated a different definition of the infinite: a set is said to be infinite if there is a bijection from itself to a proper subset of itself. An infinite set is the same size as something smaller than itself, just as there are the same number of even numbers as natural numbers, or numbers in {2, 3, 4, ...}. This is the basis for the Hilbert Hotel metaphor.
These two definitions are equivalent for a set S if the set can be well-ordered, that is (roughly) arranged so that it begins at one end and goes onwards, the way {0, 1, 2, ...} does. If S is infinite, map each finite n-th element to the (n+1)-th, and thus construct a bijection from S to itself without its 0-th element.
So if all sets are well-orderable, 'infinite' and what is now called 'Dedekind infinite' are equivalent. However, this condition is the Well Ordering Theorem, and is equivalent to the Axiom of Choice, so doesn't follow from plain ZF axioms.
Non-mathematicians get all het up about 'infinity', as if it should be a strange kind of thing you could see or a place you could get to. (And all the writeups are over there.) But mathematicians don't use 'infinity' that way. There are sets that are infinite: it's a property of the set. Some of these sets can be used as sizes by which to measure other sets, so are in effect something like numbers. But the infiniteness is still a property. There's no one final or unique infinity that can be defined as any kind of mathematical thing.
Talk of infinite things is common; technical talk of 'infinity' is not, except in one usage: the limit as a variable goes to infinity. This means as it goes on forever, as it keeps going without stopping. It does not mean that it goes until it gets to a kind of place called infinity, then stops there. The whole point of the infinite is that you don't stop. You don't ever get to anything that can be labelled with the noun 'infinity'.
About the one exception to this is in the construction of a Riemann sphere, where a point is added to the complex plane, which is then wrapped round and joined to it. This point is labelled infinity, and does roughly correspond to the intuitive notion of being a thing at an infinite distance—except that, unintuitively, it's at that distance in whichever direction you go.