The Scott set problem.

A Scott set is a set of sets of natural numbers $\mathfrak X$ such that $(\omega, {\mathfrak X})\models {\sf WKL}_0$, Assume $\lnot{\sf CH}$. Is every Scott set the standard system of a nonstandard model of PA?

The origins of the problem goes back the paper of Scott, where it is shown that the answer is positive for countable sets $\mathfrak X$.

Knight and Nadel, and independently Ehrenfeucht (unpublished), gave positive answer for $\mathfrak X$ of cardinality $\aleph_1$.

Jim Schmerl gave positive answer for arithmetic closures of sets of (arithmetic) Cohen generics of any cardinality.

References:

D. Scott, Algebras of sets binumerable in complete extensions of arithmetic, Proceedings of the symposium on pure mathematics vol. V, American Mathematical Society, Providence, R.I., 1962, pp. 117--121.

Knight J. and M. Nadel, Models of Peano arithmetic and closed ideals, Journal of Symbolic Logic, vol. 47 (1982), no. 4, pp. 833--840.

Schmerl, James H. Peano models with many generic classes. Pacific J. Math. 46 (1973), 523–536.

For more recent attempts employing forcing axioms see: Gitman V., Scott’s problem for Proper Scott sets, J. Symbolic Logic Volume 73, Issue 3 (2008), 845-860.