Say that a logic $\mathcal{L}$ is **directed** iff whenever $\mathfrak{A}\equiv\_\mathcal{L}\mathfrak{B}$ there is some $\mathfrak{C}$ with $\mathcal{L}$-elementary substructures $\mathfrak{A}'\preccurlyeq\_\mathcal{L}\mathfrak{C}$, $\mathfrak{B}'\preccurlyeq\_\mathcal{L}\mathfrak{C}$ with $\mathfrak{A}\cong\mathfrak{A}',\mathfrak{B}\cong\mathfrak{B}'$. It's a standard exercise to show that $\mathsf{FOL}$ is directed - or more generally, that every *compact* logic is directed. On the other hand, it's easy to whip up artificial logics demonstrating that this joint embeddability isn't *equivalent* to compactness.


I'm curious about the situation with second-order logic $\mathsf{SOL}$. It's consistent with $\mathsf{ZF}$ that there are $\mathsf{SOL}$-equivalent structures which do not $\mathsf{SOL}$-elementarily embed into the same structure (see below), but I don't see how to get this result outright in $\mathsf{ZFC}$ (much less $\mathsf{ZF}$). However, I recall seeing an easy argument (due to Mostowski?) that in fact this is a $\mathsf{ZF}$-theorem.



> 
> **Question**: Does $\mathsf{ZF}$ prove that $\mathsf{SOL}$ is not directed?
> 
> 
> 


Here's a proof that the non-directedness of $\mathsf{SOL}$ is *consistent* with $\mathsf{ZF}$. Suppose there is a family $\mathbb{A}$ of amorphous sets of pairwise incomparable cardinality such that there is no injection from $\mathbb{A}$ into $2^{\aleph\_0}$. Thinking of each element of $\mathbb{A}$ as a structure in the empty language, we must have $X,Y\in\mathbb{A}$ with $X\equiv\_\mathsf{SOL}Y$ but $X\not\cong Y$. But any set into which both $X$ and $Y$ inject must be non-amorphous, hence cannot satisfy $Th\_\mathsf{SOL}(X)=Th\_\mathsf{SOL}(Y)$ since amorphousness is second-order-expressible.


Of course this doesn't help at all without a background assumption of lots of amorphous sets, so it's not really relevant to the question I'm asking here, but it's still neat. Note that a positive answer will have to crucially involve uncountable structures, since [it's consistent with $\mathsf{ZFC}$ that $\equiv\_{\mathsf{SOL}}$ implies $\cong$ for countable structures](https://mathoverflow.net/a/161694/8133).



Failure of "directedness" for second-order logic?
The answer to the question is yes.


Let $\alpha\_0<\alpha\_1$ be the least ordinals (in the reverse lex order, say) such that $V\_{\alpha\_0}$ and $V\_{\alpha\_1}$ have the same second order theory $T$. Assume towards a contradiction that $V\_{\alpha\_0}$ and $V\_{\alpha\_1}$ are elementarily embeddable into a common structure $M$, which we may assume is transitive. Note that the embeddings fix $T$. Since $V\_{\alpha\_1}$ satisfies that there is an ordinal $\beta$ such that $V\_\beta$ satisfies $T$, so does $M$, and hence so does $V\_{\alpha\_0}$. But this means that some $\beta < \alpha\_0$ has the same theory as $V\_{\alpha\_0}$, contrary to the minimality of the pair $\alpha\_0 < \alpha\_1$.
(Edit: I now see this is very close to what Trevor was doing.)


