Stateful Realizers for Nonstandard Analysis

In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with analysis in the context of (semi)intuitionistic realizability, focusing on Lightstone-Robinson construction model through an ultrapower. particular, consider extension $\lambda$-calculus memory cell, that contains integer (the state), order indicate which slice ultrapower $\cal{M}^{\mathbb{N}}$ computation is being done. pay attention principles (and their computational content) obtainable setting. give non-trivial realizers Idealization and non-standard version LLPO principle. then discuss how quotient product mimic construction.