Is the cardinality of the set of points on the circumference of a circle equal to the cardinality of the set of points on the interior of a circle?

Question

Is the cardinality of the set of points on the circumference of a circle equal to the cardinality of the set of points on the interior of a circle? They're both infinite, but are they the same aleph?

Answer 1

Look at it this way: you're comparing the cardinality of the points on a line, with those in an area of the plane.

Obviously both are finite (finite length and area I mean, not cardinality), but you know that the interval $(0,1)$ has the same cardinality as that of the entire number line $\mathbb{R}$ - both having cardinality of the continuum, $c$. It should then come to no surprise - someone could probably flesh out the details - that the cardinality of points enclosed by a circle in the plane with nonzero area would have the same cardinality as the plane itself.

In this sense, then, the circle's circumference is analogous to a finite interval of $\mathbb{R}$, and similarly the circle encloses a finite area of $\mathbb{R} \times \mathbb{R}$. The cardinalities we're concerned with then, in the end, are the cardinalities of $\mathbb{R}$ and $\mathbb{R} \times \mathbb{R}$, right?

For a Cartesian product, we note that $|A \times B | = |A| \cdot |B|$. Thus,

$$|\mathbb{R} \times \mathbb{R}| = |\mathbb{R}| \cdot |\mathbb{R}| = c \cdot c = c = |\mathbb{R}|$$

(Counterintuitive as $c \cdot c = c$ would be, it's true. In fact, in general, $| \mathbb{R}^n | = c$ for all $n \in \mathbb{N}$. That's another detail someone else more qualified might want to hash out if you're confused though.)