We investigate Maker-Breaker games on graphs of size $\aleph_1$ in which Maker's goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic framework. Relating to this, we prove that there is a winning strategy for Maker in the $K_{ω,ω_1}$-game under ZFC+MA+$\neg$CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{ω_1}$-game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.