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(Working in ZFC.) No (re $\mathcal{L}_{\omega_1,\omega}$). Suppose it is. Consider the signature $\Sigma$ with just one binary relation symbol $<$. Let $\Sigma',\eta$ witness SED-ness for $\Sigma$. Let $\pi:M\to V_\eta$ be elementary,with $\eta$ some sufficiently large limit ordinal, $M$ transitive, $M$ of cardinality $\kappa=2^{\aleph_0}$, with $\mathbb{... 8 (Working in ZFC.)$\omega_2$is not Fraissean. In fact, it is not Fraissean with respect to$\Sigma$, where$\Sigma$is the signature with a single binary relation$<$. To see this we use a variant of the argument you linked in the question. Suppose otherwise, and let$\Sigma'$and$\eta$witness this. Let$\gamma$be a large enough ordinal and let$\pi:M\...
It depends on what diamond sequence you have. $V=L$ proves there is a $\diamondsuit$-sequence of $L$-rank $\omega_1+1$: recall the famous way of proving the existence of $\diamondsuit$-sequence. The definition goes as follows (for example, Theorem 13.21 of Jech 3rd): $(S_\alpha,C_\alpha)$ is the $<_L$-least pair $S_\alpha\subseteq\alpha$, $C_\alpha\... 6 Um, for each variable$z_k = x_k + iy_k$we throw in the equation$x_k^2 + y_k^2 = 1$and rewrite everything in terms of$x_k$and$y_k$. I am missing something, I think. 5 If your main complexity objective is to keep the number of variables down, you can use a rational parametrization of the circle: for each variable$z_k\in S^1$, introduce a real variable$t_k$and rewrite everything using the substitution $$z_k=\frac{i-t_k}{i+t_k}=\frac{1-t_k^2}{1+t_k^2}+i\frac{2t_k}{1+t_k^2}.$$ (You will have to deal with$z_k=-1\$ somehow.)