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On Non-standard Models of Arithmetic with Uncountable Standard Systems

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.