Infinity
Infinity 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, but the idea of something with no largest value. Symbolized by the lemniscate (a sideways $8$), infinity is not itself a real number; it describes an unbounded quantity, as in $\lim_{x \to \infty} f(x)$ meaning $f(x)$ grows without bound. Set theory distinguishes different sizes of infinity by cardinality: $|\mathbb{N}| = \aleph_0$ (countable) while $|\mathbb{R}| = c = 2^{\aleph_0}$ (uncountable), and the Continuum Hypothesis, that there is no cardinality strictly between $\aleph_0$ and $c$, was shown by Godel (1940) and Cohen (1963) to be independent of ZFC set theory.
Example
The natural numbers go on forever, $1, 2, 3, 4, \ldots$, with no "last" one, and the number line extends infinitely in both directions. $\lim_{x \to \infty} 1/x = 0$, and the series $1 + 1/2 + 1/4 + \ldots = 2$ converges to a finite sum despite having infinitely many terms, while $1+1+1+\ldots$ diverges to infinity. Hilbert's Hotel, with countably infinite rooms all full, can still accommodate a new guest by shifting every current guest from room $n$ to room $n+1$, illustrating the non-intuitive arithmetic of infinite sets.
Key Insight
No matter how large a number you name, you can always add $1$ to get a larger one, the core idea of infinity. Not all infinities are equal: Cantor proved the real numbers are "more infinite" than the natural numbers, since there is no one-to-one correspondence between them (Cantor's diagonal argument), a discovery that was controversial in his time but is now foundational. The existence of transcendental numbers, and hence most irrational numbers, follows immediately from this: there are only countably many algebraic numbers but uncountably many real numbers.