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 $\mathfrak X$. 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Knight and Nadel, and independently Ehrenfeucht (unpublished), gave positive answer for $\mathfrak X$ of cardinality $\aleph_1$. Knight J. and M. Nadel, Models of Peano arithmetic and closed ideals, Journal of Symbolic Logic, vol. 47 (1982), no. 4, pp. 833--840.

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