Restate the proposition

Suppose $\mathcal{M}_0$, $\mathcal{M}_1$, and $\mathcal{M}_2$ are $\mathcal{L}$-structures and $j_i ~:~ \mathcal{M}_0 \rightarrow \mathcal{M}_i, ~(i = 1,2)$ is an elementary embedding. Show that there is an $\mathcal{L}$-structure $\mathcal{N}$ and elementary embeddings $f_i ~:~ \mathcal{M}_i \rightarrow \mathcal{N}$ so that $f_1 \circ j_1 = f_2 \circ j_2$.

Problem 1

It was suggested that I should augment the original language by:

$$\mathcal{L}^+ = \mathcal{L} \cup \{ c_m ~:~ m \in \mathbb{M}_1 \sqcup \mathbb{M}_2, ~~ j_o(c_m) \wedge j_1(c_m) ~ defined \}.$$

And then it's sufficient to prove that the $\mathcal{L}^+$ theory:

$$T = Th(\mathcal{M}_1) \cup Th(\mathcal{M}_2)$$

is consistent, that is there is some model satisfying $T$. Could someone explain the rational of this thought?

Problem 2

Suppose I want to prove $T$ is consistent, then by the compactness theorem, it suffices to show all finite subsets of $T$ is consistent. However, I am not sure how to set up the argument here either.

  • $\begingroup$ This is not the language or the theory I gave you in the other thread. As per se, your post lacks meaning : $c_m$ is supposed to be a constant symbol, you cannot take its image by $j_0$ or $j_1$. $\endgroup$
    – Pece
    Feb 6 '15 at 6:57

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