# Grilliot's trick in Nonstandard Analysis

## TL;DR

This paper explores a nonstandard analysis technique similar to Grilliot's trick, deriving the Turing jump functional from nonstandard discontinuous functionals via the Transfer principle, providing effective content without explicit focus on effectiveness.

## Contribution

It introduces a nonstandard analysis method to extract effective content from nonstandard discontinuous functionals, expanding the understanding of the Standard Part axiom's computational implications.

## Key findings

- Derived the Turing jump functional from nonstandard discontinuous functionals.
- Showed the Transfer principle can yield effective implications in nonstandard analysis.
- Identified new classes of functionals related to the Standard Part axiom.

## Abstract

The technique known as Grilliot's trick constitutes a template for explicitly defining the Turing jump functional $(\exists^2)$ in terms of a given effectively discontinuous type two functional. In this paper, we discuss the standard 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 $\Pi_1^0$-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 all seem to fall outside the established categories. These functionals directly derive from the Standard Part axiom of Nonstandard Analysis.