Infinity
ArithmeticInfinity is the concept of a quantity without bound or end; it is not a real number but is used to describe values that grow without limit.
Definition
Infinity means "going on forever without end." It is not a regular number you can count to. It is the idea of something that has no largest value.
Example
The natural numbers go on forever: $1, 2, 3, 4, \ldots$ and there is no "last" one. The number line also extends infinitely in both directions.
Key Insight
No matter how large a number you name, you can always add $1$ to get a larger one. That is the idea of infinity: there is always a bigger number.
Definition
Infinity (symbolized by the lemniscate: a sideways $8$) is not a real number. It describes an unbounded quantity. In limits: $\lim_{x \to \infty} f(x)$ means $f(x)$ grows without bound. The set of natural numbers is infinite (countably infinite). The reals are a larger infinity (uncountably infinite).
Example
$\lim_{x \to \infty} 1/x = 0$. The series $1 + 1/2 + 1/4 + \ldots = 2$ (finite sum despite infinitely many terms). $1 + 1 + 1 + \ldots$ diverges to infinity (sum grows without bound).
Key Insight
Not all infinities are equal. Cantor proved the real numbers are "more infinite" than the natural numbers: there is no one-to-one correspondence between them (Cantor's diagonal argument).
Definition
Set theory distinguishes infinities by cardinality: $|\mathbb{N}| = \aleph_0$ (countable), $|\mathbb{R}| = c = 2^{\aleph_0}$ (uncountable). The Continuum Hypothesis (CH) states there is no cardinality strictly between $\aleph_0$ and $c$. Godel (1940) and Cohen (1963) showed CH is independent of ZFC set theory.
Example
Hilbert's Hotel: a hotel with countably infinite rooms, all full, can accommodate a new guest by shifting every current guest from room $n$ to room $n+1$. This illustrates the non-intuitive arithmetic of infinite sets.
Key Insight
Cantor's discovery of multiple sizes of infinity was controversial in his time but is now foundational. The existence of transcendental numbers (and hence most irrational numbers) follows immediately: there are only countably many algebraic numbers but uncountably many real numbers.