Positional strategies in long Ehrenfeucht-Fraiss\'e games

TL;DR

This paper explores the conditions under which the second player has a winning strategy in long Ehrenfeucht-Fraïssé games between models of size leph_2, revealing consistency results related to the Continuum Hypothesis.

Contribution

It demonstrates the relative consistency of the existence of models with specific game-theoretic properties under ZF + CH, and shows such models do not exist if CH fails.

Findings

Existence of models where the second player wins the game but no -closed back-and-forth set exists under ZF + CH.

Such models do not exist if the Continuum Hypothesis fails.

The results depend on set-theoretic assumptions related to CH and ZF.

Abstract

We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality \aleph_2 such that the second player has a winning strategy in the Ehrenfeucht-Fra\"iss\'e-game of length \omega_1 but there is no \sigma-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.