Why does transfinite cardinal arithmetic differ from ordinary arithmetic?

Transfinite cardinal arithmetic differs from ordinary arithmetic because it is designed to measure the size of infinite sets rather than finite collections of objects. While ordinary arithmetic follows rules based on natural numbers, transfinite arithmetic—introduced by Georg Cantor—operates under different principles where the addition or multiplication of infinite quantities does not always result in a larger value.12

Key differences include:

• Absorption of Values: In transfinite arithmetic, adding or multiplying by a smaller infinite cardinal often leaves the original value unchanged. For example, ${\aleph }_0+{\aleph }_0={\aleph }_0$ and ${\aleph }_0·{\aleph }_0={\aleph }_0$, because the resulting sets can still be put into a one-to-one correspondence with the original set of natural numbers.12
• Subset Equivalence: Unlike finite sets, an infinite set can have the same cardinality as one of its proper subsets. For any transfinite cardinal $c$, the equation $c+{\aleph }_0=c$ holds true because every infinite set contains a subset equivalent to the set of natural numbers.312
• Hierarchy of Sizes: Cantor proved that there is not just one "infinity," but an unending hierarchy of transfinite numbers, such as ${\aleph }_0,{\aleph }_1,{\aleph }_2\dots$, where each subsequent power set is strictly larger than the one before it.32