Failures of the Silver Dichotomy in the Generalised Baire Space

TL;DR

This paper demonstrates that Silver's dichotomy does not hold for Borel equivalence relations on the generalized Baire space when assuming the axiom V=L, challenging previous theoretical expectations.

Contribution

It provides the first known counterexamples to Silver's dichotomy in the context of the generalized Baire space under V=L.

Findings

Silver's dichotomy fails in the generalized Baire space under V=L

Constructs specific Borel equivalence relations that violate the dichotomy

Shows limitations of classical descriptive set theory results in larger spaces

Abstract

We prove results that falsify Silver's dichotomy for Borel equivalence relations on the generalised Baire space under the assumption V=L.