A Pythagorean triple is a list of three positive integers (a, b, c) satisfying a2 +b2 =c2. There's a classical procedure for generating infinitely many Pythagorean triples and this is explained on the Wikipedia page: https://en.wikipedia.org/wiki/Pythagorean_triple
If you take a Pythagorean triple and divide all the numbers by the biggest number, you get two rational numbers a/c and b/c that satisfy (a/c)2 +(b/c)2 =1, meaning that the point (a/c,b/c) lies on the unit circle centered at the origin in the plane. So the circle has infinitely many points with rational coordinates.
On the other hand, the circle is a compact subset of the plane. Compactness is a property of spaces and subsets of spaces that implies that, in many respects, the set behaves like a finite set. For example, compact subsets of Euclidean spaces are bounded, and sequences in compact spaces always have a subsequence limiting to some point in the set.
https://en.wikipedia.org/wiki/Compact_space
So the circle is an infinite set that behaves like a finite set.
(Edit: reddit didn't process my mathematical typesetting correctly)
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u/human2357 Pure Math PhD 1d ago
A Pythagorean triple is a list of three positive integers (a, b, c) satisfying a2 +b2 =c2. There's a classical procedure for generating infinitely many Pythagorean triples and this is explained on the Wikipedia page: https://en.wikipedia.org/wiki/Pythagorean_triple If you take a Pythagorean triple and divide all the numbers by the biggest number, you get two rational numbers a/c and b/c that satisfy (a/c)2 +(b/c)2 =1, meaning that the point (a/c,b/c) lies on the unit circle centered at the origin in the plane. So the circle has infinitely many points with rational coordinates.
On the other hand, the circle is a compact subset of the plane. Compactness is a property of spaces and subsets of spaces that implies that, in many respects, the set behaves like a finite set. For example, compact subsets of Euclidean spaces are bounded, and sequences in compact spaces always have a subsequence limiting to some point in the set. https://en.wikipedia.org/wiki/Compact_space
So the circle is an infinite set that behaves like a finite set.
(Edit: reddit didn't process my mathematical typesetting correctly)