Grilliot's trick in Nonstandard Analysis

Recently, Dag Normann and the author have established a connection between higher-order computability theory and Nonstandard Analysis; new results in both fields are obtained by exploiting this connection ([28]). Now, on the side of computability theory, the technique known as Grilliot’s trick constitutes a template for explicitly defining the Turing jump functional (∃) in terms of a given effectively discontinuous type two functional ([14]). In light of the aforementioned connection, it is a natural question what corresponds to Grilliot’s trick in Nonstandard Analysis? In this paper, we discuss the nonstandard extensionality trick : a technique similar to Grilliot’s trick in Nonstandard Analysis. This nonstandard trick proceeds by deriving from the existence of certain nonstandard discontinuous functionals, the Transfer principle from Nonstandard analysis limited to Π01-formulas; from this (generally ineffective) implication, we obtain an effective implication expressing the Turing jump functional in terms of a discontinuous functional (and no longer involving Nonstandard Analysis). The advantage of our nonstandard approach is that one obtains effective content without paying attention to effective content. We also discuss a new class of functionals which fall outside the established categories. These functionals derive from the Standard Part axiom of Nonstandard Analysis.