On the Σ ^0_1 -Fragments of the Kreisel-Putnam Axiom and Two Variants of the Linearity Axiom in Intuitionistic Arithmetic and Analysis

We study the Σ ^0_1 -fragment of the Kreisel-Putnam axiom and its Σ ^0_n -extensions in the context of intuitionistic arithmetic and analysis. Among other things, we show that the Σ ^0_1 -fragment of the Kreisel-Putnam axiom is exactly the principle that fills the gap between the lesser limited principle of omniscience and the disjunctive Markov principle in constructive reverse mathematics. In addition, we introduce two variants of the linearity axiom, which are related to the Kreisel-Putnam axiom, and show that the Σ ^0_1 -fragments of these three axioms do not lie in any previously known layers in the arithmetical hierarchy of logical axioms.