$WKL\_0$ extends $RCA\_0$ with the statement that any infinite subset of the infinite binary tree has an infinite branch. Does $WKL\_0$ Prove that there are sets which are not proven to exist by the $\Delta\_0$-comprehension schema of $RCA\_0$? If so, to what level of comprehension do such sets belong?



Does $WKL_0$ provide more comprehension than $RCA_0$?
If I understand your question correctly, the answer is 
"no" in the following sense:


Fix a model $M=(\omega\_M, \mathbb{R}\_M)$ of $RCA\_0$, and let $\mathcal{C}\_M$ be the set of structures $N=(\omega\_N, \mathbb{R}\_N)$ such that


* $N\models WKL\_0$,
* $\omega\_N=\omega\_M$ (same first-order parts), and
* $\mathbb{R}\_N\supseteq\mathbb{R}\_M$ ($N$ is gotten by adding sets to $M$).


Then we have:


* $\mathcal{C}\_M$ is nonempty. (This is due to Harrington, and is how he shows that $WKL\_0$ is $\Pi^1\_1$-conservative over $RCA\_0$.)
* $\bigcap\_{N\in\mathcal{C}\_M} \mathbb{R}\_N=\mathbb{R}\_M$. That is, there is no *specific* set which must be added when we expand $M$ to a model of $WKL\_0$. (This is a direct consequence of the proof of Jockusch and Soare's Low Basis Theorem, but I don't know who first explicitly observed this.)


In particular, the intersection of all the Scott sets is precisely $REC$, the set of recursive sets.


Does this answer your question?


