[vacuumcl]

Half your examples are wrong, but maybe it is your point. Although it wasn't clear to me the first time I read your comment.

- The cardinality of the odd and even integers is the same.

- It is true there are more points on a plane then on a line (Cantor's theorem.)

- The circle is the compactified real line, i.e. it can be represented as the real numbers with one additional point (the point at infinity). In terms of cardinality they are the same since they just differ by one point which does not change the cardinality.

- There are not more points on a plane than a half-plane, you can find a bijective mapping between them easily.

- There are more rationals than integers: not true, they are both countable sets of the same cardinality.

- There are more reals than rationals, this is true (again Cantor.)

[function_seven]

Parent is making the same point you are. It’s a surprise to the intermediate mathematician when the last bullet point turns out to be true.

[jdkee]

"- It is true there are more points on a plane then on a line (Cantor's theorem.)"

There is a bijection between the points on a line and the points on a plane or in any n-dimensional space.

[vacuumcl]

Yes, that was a bad mistake on my part. Thanks for pointing it out!..