Well-orderings and the perfect set property

set-theory descriptive-set-theory well-orders

Since you said, in a comment, "you can assume choice if necessary", I'll assume choice and fix a well-ordering $W$ of all the reals such that all the negative reals precede all the positive reals. Then $W$ includes the perfect set $\{(x,y):x<0<y\}$.

On the other hand, some such use of the axiom of choice is needed. It's consistent with ZF + DC that every well-orderable set of reals is countable.

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