We prove independence results concerning the number of nonisomorphic models (using the $\mathbf{S}$-chain condition and $\mathbf{S}$-properness) and the consistency of "$ZCF + 2^{\aleph_0} = \aleph_2 + \text{there is a universal linear order of power} \aleph_1$". Most of these results were announced in [Sh 4], [Sh 5]. In subsequent papers we shall prove an analog f MA for forcing which does not destroy stationary subsets of $\omega_1$, investigate $\mathscr{D}$-properness for various filters and prove the consistency with G.C.H. of an axiom implying SH (for $\aleph_1$), and connected results.