Perhaps surprisingly, we work in the language of second-order arithmetic. I was wondering if the strength of the following statement LP was known:


*An uncountable closed set in $\mathbb{R}$ has a limit point*.


For reference, the perfect set theorem (equivalent to ATR$\_0$) implies LP, while LP implies ACA$\_0$ (using Specker sequences). Hence we have


ATR$\_0 \rightarrow$ LP $\rightarrow$ ACA$\_0$.


Do any of these reverse?



What is the strength of the second-order statement 'an uncountable closed set in $\mathbb{R}$ has a limit point'?
I'm pretty sure it follows from ACA$\_0$. The point is that since you're not trying to prove the full perfect set property, you only need one iteration of the Cantor-Bendixson derivative.


Given a closed set $C$, two or three jumps can obtain the derivative $C'$. If this is empty, then you can index the elements of C by basic open neighborhoods, and some finite number of jumps can turn this into whatever sort of countable listing you desire. Otherwise, one more jump can find an element of $C'$, and then a sequence in $C$ converging to it.


