  In 1960s, Dana Scott gave a recursion theoretic characterization of standard
systems of countable non-standard models of arithmetic, i.e., collections of
sets of standard natural numbers coded in non-standard models. Later, Knight
and Nadel proved that Scott's characterization also applies to non-standard
models of arithmetic with cardinality $\aleph_1$. But the question, whether the
limit on cardinality can be removed from the above characterization, remains a
long standing question, known as the Scott Set Problem. This article presents
two constructions of non-standard models of arithmetic with non-trivial
uncountable standard systems. The first one leads to a new proof of the above
theorem of Knight and Nadel, and the second proves the existence of models with
non-trivial standard systems of cardinality the continuum. A partial answer to
the Scott Set Problem under certain set theoretic hypothesis also follows from
the second construction.
