Universal Graphs at Aleph_{omega_1+1} and Set-Theoretic Geology

摘要: This thesis consists of two parts: the construction a jointly universal family graphs, and then an exploration set-theoretic geology. Firstly we shall construct model in which $2^{\aleph_{\omega_1}}=2^{\aleph_{\omega_1+1}}=\aleph_{\omega_1+3}$ but there is size $\aleph_{\omega_1+2}$ graphs on $\aleph_{\omega_1+1}$. We take supercompact cardinal $\kappa$ will use Radin forcing with interleaved collapses to change into $\aleph_{\omega_1}$. Prior perform preparatory iteration add functions from $\kappa^+$ names for what become members $\kappa^+$. The same technique can be used any uncountable place $\omega_1$. Secondly explore various topics begin by showing that class Easton support $\mathrm{Add}(\kappa,1)$ at regular results universe its own generic mantle. consider set forcings $\mathbb{P}$, $\mathbb{Q}$, $\mathbb{R}$ $\mathbb{S}$ respective generics $G$, $H$, $I$ $J$ such $V[G][I]=V[H][J]$ show $V[G]$ $V[H]$ must have shared ground via $(|\mathbb{R}|+|\mathbb{S}|)^+$-cc forcing. allows similar analysis related situation when $\mathbb{P}$ replaced $V[G]$. conclude simple characterisation mantle extension, investigation possibilities version intermediate theorem applies