Types of infinity
Infinity appears in several different senses: as the size of a set, as unboundedness, and as the limit of a divergent sequence. An infinite set has infinitely many elements, but it can be bounded. Infinite sets come in different sizes: countable and uncountable. A divergent sequence does not converge, but it doesn’t necessarily go to infinity.
Definitions
Missing definition: infinity
Missing definition: infinite_set
A set of numbers is bounded if it does not go on forever in either direction: there is some number that all of them are smaller than in size.
A set $S$ of real numbers is bounded if there is a number $M$ with $\lvert x \rvert \leq M$ for every $x \in S$.
A subset of a metric space is bounded if it is contained in some ball. A sequence or function is bounded if its set of values is bounded.
Missing definition: divergent
Missing definition: countably_infinite
Missing definition: uncountable
Examples
| Example | infinity | infinite_set | bounded | divergent | countably_infinite | uncountable | Why |
|---|---|---|---|---|---|---|---|
Missing example: naturals | |||||||
Missing example: rationals_intersect_0_1 | |||||||
Missing example: closed_unit_interval | |||||||
Missing example: sequence_alternating | |||||||
Missing example: the_sequence_1_2_3_etc | |||||||
Missing example: infinity | |||||||
Notes
- A set is unbounded if it does not fit in any ball: for every $M > 0$, it has two points farther than $M$ apart. This requires a notion of distance, whereas being infinite does not.
- “Infinitely many” and “arbitrarily large” are different. The set $\setdef{1/n \suchthat n \in \mathbb{N}}$ has infinitely many elements, none larger than $1$.