There are many different things called infinite in mathematics.
There are the ordinal numbers from ω onwards.
There are the infinite cardinals.
There are infinitely many rational numbers, and infinitely many reals, but more reals than rationals.
Sometimes it is convenient to add a “point at infinity” at each end of the real line.
Sometimes it is convenient to bend it into a circle and add a single “point at infinity” that joins the ends — which actually makes it a compact topological space, which seems like a smaller sort of thing than the real line.
In the plane one may add a single point at infinity to make the whole thing into something homeomorphic to a sphere. This is useful in projective geometry and complex analysis.
The delta function is notionally “infinite” at zero and zero everywhere else, which makes little sense literally as stated, but can be understood as an informal way of talking about a certain distribution.
Surreals introduce in some sense as many “infinities” as possible.
Non-standard analysis introduces infinite numbers in yet another way; but arguments in that system can be translated back into traditional epsilon-delta forms.
When extending a space with objects whose existence would be convenient, the question to ask is not, “Does such an object really exist?” but “What properties does this thing need to have, to do what I want it to do? Can it be consistently axiomatised?” For it is said that in mathematics, existence is freedom from contradiction.
There are many different things called infinite in mathematics.
There are the ordinal numbers from ω onwards.
There are the infinite cardinals.
There are infinitely many rational numbers, and infinitely many reals, but more reals than rationals.
Sometimes it is convenient to add a “point at infinity” at each end of the real line.
Sometimes it is convenient to bend it into a circle and add a single “point at infinity” that joins the ends — which actually makes it a compact topological space, which seems like a smaller sort of thing than the real line.
In the plane one may add a single point at infinity to make the whole thing into something homeomorphic to a sphere. This is useful in projective geometry and complex analysis.
The delta function is notionally “infinite” at zero and zero everywhere else, which makes little sense literally as stated, but can be understood as an informal way of talking about a certain distribution.
Surreals introduce in some sense as many “infinities” as possible.
Non-standard analysis introduces infinite numbers in yet another way; but arguments in that system can be translated back into traditional epsilon-delta forms.
When extending a space with objects whose existence would be convenient, the question to ask is not, “Does such an object really exist?” but “What properties does this thing need to have, to do what I want it to do? Can it be consistently axiomatised?” For it is said that in mathematics, existence is freedom from contradiction.