“Infinite” does not mean that every infinite set has the same size. In set theory, two sets have the same cardinality when their elements can be paired one-to-one with none left over—even when both sets are infinite.
A countable example: natural numbers and even numbers
The natural numbers 1, 2, 3, … can be paired with the positive even numbers 2, 4, 6, … using n ↔ 2n. Each natural number gets exactly one even number, and every positive even number is matched. This is a countably infinite set. The rational numbers are also countable, although their pairing requires a more elaborate enumeration.
Why the real numbers are uncountable
Cantor’s diagonal argument shows that no proposed list can contain every real number between 0 and 1. Imagine writing each number as a decimal in a row. Construct a new decimal by changing the first digit of the first row, the second digit of the second row, and so on. The new number differs from every row in at least one digit, so it was not on the list. Therefore the real numbers in that interval cannot be listed completely.
The referenced video introduces the idea through a number-matching game. Pairing is an intuition for the formal definition; the diagonal construction is the key argument for uncountability. This is a simplified explanation; consult a standard set-theory text for a formal treatment.
Video and references
- Watch the original video on YouTube: อินฟินิตี้กว่า? | Uncountable Infinity
- Browse the source channel: แมทเล่าให้ฟัง | MLHF
- Browse the source playlist: InFUNity
- Browse the source playlist: เล่าตัวเลข
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