How does cardinal arithmetic differ from ordinary arithmetic for infinite numbers?

Cardinal arithmetic differs significantly from the arithmetic of natural numbers because it involves transfinite numbers, which describe the sizes of infinite sets. While ordinary arithmetic follows standard rules for finite quantities, cardinal arithmetic behaves differently when infinite sets are involved.1

Key differences include:

• Absorption of Values: In cardinal arithmetic, adding or multiplying infinite cardinals often results in the same infinite value. For example, ${\aleph }_0+{\aleph }_0={\aleph }_0$ and ${\aleph }_0·{\aleph }_0={\aleph }_0$.1
• Dominance of Larger Infinities: When adding a finite or smaller infinite cardinal to a larger one, the larger cardinal "absorbs" the smaller one. Specifically, $c+{\aleph }_0=c$ for any transfinite cardinal $c$, because every infinite set contains a subset equivalent to the set of natural numbers.1
• Hierarchy of Sizes: Unlike finite arithmetic where adding or multiplying positive numbers always increases the total, cardinal arithmetic recognizes that some infinite sets are strictly larger than others. Georg Cantor proved that the power set of any set is always larger than the original, creating an infinite hierarchy of increasing transfinite numbers.23

These rules reflect the unique nature of infinity in set theory, where the size of a set can be equal to that of its proper subset.12