This question of mine arises from Kanamori's The Higher Infinite, where he introduces Vopenka's principle.
Let me give some definitions first(assume $\kappa$ is inaccessible throughout this post):
A sequence of structures $\langle \mathcal{M}_\alpha: \alpha \lt \kappa \rangle$ is natural iff each $ \mathcal{M}_\alpha$ has form $\langle V_{f(\alpha)}, \in, \{\alpha\}, R_\alpha \rangle$ where $R_\alpha \subseteq V_{f(\alpha)}$ and $\alpha \lt \beta \lt \kappa$ implies that $\alpha \lt f(\alpha) \le f(\beta) \lt \kappa$.
$X$ is Vopenka in $\kappa$ iff for any natural sequence $\langle \mathcal{M}_\alpha: \alpha \lt \kappa \rangle$ there is a $j: \mathcal{M}_\alpha \prec \mathcal{M}_\beta$ for some $\alpha \lt \beta \lt \kappa$ with critical point in $X$.
$\kappa$ is Vopenka iff $\kappa$ is Vopenka in $\kappa$.
$$F = \{ X \subseteq \kappa: \kappa - X \text{ is not Vopenka in } \kappa \}$$
Now in a short paragraph Kanamori gives some remarks which he says proves that the following are equivalent:
$(1)$ $F$ is a (proper) filter.
$(2)$ $V_\kappa \models \text{ Vopenka's principle}$.
$(3)$ $\kappa$ is Vopenka.
I will divide the paragraph into two parts to make everything more clear. This is the first one:
Note that $X \in F$ iff there is a natural sequence $\langle \mathcal{M}_\alpha: \alpha \lt \kappa \rangle$ such that for any $\alpha \lt \beta \lt \kappa$ and $j:\mathcal{M}_\alpha \prec \mathcal{M}_\beta$, it's critical point is in $X$. Also, if $X, Y \in F$, then $X \cap Y \in F$ by pointwise amalgamating the two corresponding natural sequences into one.
Now if we accept the above quote from the book then we can readily see $(1) \leftrightarrow (3)$.
$(*)$ So my first question is about the amalgamation he is talking about. I don't see how I can code two different $V_\alpha$'s into one, such that I can reduce the elementary embeddings between the amalgamations to the original structures. Is there some standard way to do this?
This is the second part of the paragraph which completes the circle:
Finally, if $\langle \mathcal{A}_\alpha: \alpha \lt \kappa \rangle \in {^\kappa V_\kappa}$ is any sequence of structures for the same language, then each $\mathcal{A}_\alpha$ can be encoded into an $\mathcal{M}_\alpha$ so that $\langle \mathcal{M}_\alpha: \alpha \lt \kappa \rangle$ a natural sequence, and so if $X$ is Vopenka in $\kappa$, there are $\alpha \lt \beta \lt \kappa$ and $j:\mathcal{A}_\alpha \prec \mathcal{A}_\beta$ such that if it has a critical point, it belongs to $X$.
$(**)$ This is the part which is more confusing for me. The main issue is size. Because each $\mathcal{A}_\alpha$ has size $\kappa$ and each $\mathcal{M}_\alpha$ must have size strictly less than $\kappa$ because of the definition of naturality. So two basic ideas came to my mind which I can't continue to complete:
$[1]$ Since our structures are sets, we use the Lowenheim-Skolem theorem to shrink each $\mathcal{A}_\alpha$ to some appropriate $V_{f(\alpha)}$. But this way I don't see how I can lift an elementary embedding $j: \mathcal{M}_\alpha \prec \mathcal{M}_\beta$ to some $j': \mathcal{A}_\alpha \prec \mathcal{A}_\beta$ and vice versa, to prove $(2) \leftrightarrow (3)$.
$[2]$ (I'm not really sure about this one) Since our goal is to prove $(2) \leftrightarrow (3)$ we may assume that $\langle \mathcal{A}_\alpha: \alpha \lt \kappa \rangle$ is a definable sequence of $V_\kappa$. So then it is definable by a single formula $\varphi$ and using reflection for each $\alpha$ (in $V_\kappa$) we can find an appropriate $V_{f(\alpha)}$ and appropriate relations and functions on $V_{f(\alpha)}$ such that it is elementarily equivalent to $\mathcal{A}_\alpha$. But then we have the same issue of $[1]$, I don't see how we can lift the elementary embeddings.
To sum it up, $(*)$ and $(**)$ are my questions. Also the material here can be found on page $336$ of Kanamori's The Higher Infinite, $2$nd edition.
Re: size issues, note that each $\mathcal{M}_\alpha$ and each $\mathcal{A}_\alpha$ is an element of (not just a subset of) $V_\kappa$; since $\kappa$ is inaccessible, this means that they're all of cardinality $<\kappa$.
Re: amalgamation and coding, I think what's going on is that we're coding into the "side relation" $R$. Let's ignore the "index parameter" for the moment and just suppose I have two structures $\mathcal{A}=(V_\alpha, R)$ and $\mathcal{B}=(V_\beta, S)$ with $R\subseteq V_\alpha$ and $S\subseteq V_\beta$. I can then consider the structure
$$\mathcal{C}_{\mathcal{A},\mathcal{B}}=(V_{\max\{\alpha,\beta\}}, (R\times1)\cup\{S\times 2\}\cup\{V_\alpha\times 3\}\cup\{V_\beta\times 4\}).$$ From this data I can definably decode both $(V_\alpha, R)$ and $(V_\beta, S)$, and so given $j:\mathcal{C}_{\mathcal{A},\mathcal{B}}\preccurlyeq\mathcal{C}_{\mathcal{A}',\mathcal{B}'}$ we get induced $j_0:\mathcal{A}\preccurlyeq\mathcal{A}'$ and $j_1:\mathcal{B}\preccurlyeq\mathcal{B}'$.
Now since this is "monotonic," when we throw in the "index parameter" as well things behave nicely (and we get naturality: the point being that we go up from $V_{f(\alpha)}$ and $V_{f(\beta)}$ to $V_{\max\{f(\alpha),f(\beta)\}}=V_{f(\max\{\alpha,\beta\})}$ and $\max\{\alpha,\beta\}\ge\alpha,\beta$).
Similarly, the $\mathcal{A}_\alpha$s we care about can be individually coded as $\mathcal{M}_\alpha$s: we first observe that a given structure $\mathcal{A}$ with domain $\subseteq V_{g(\mathcal{A})}$ can be coded by $(V_{g(\mathcal{A})}, U_\mathcal{A})$ for some $U\subseteq V_{g(\mathcal{A})}$, and then let $\mathcal{M}_\alpha=(V_{f(\alpha)}, \{\alpha\}, U_{\mathcal{A}_\alpha})$ where $f(\alpha)$ is defined recursively to be increasing and dominate $\alpha\mapsto g(\mathcal{A}_\alpha)$.