let T1 and T2 be two theories in language L and none of the symbols of L occur in both T1 and T2, also both T1 and T2 has infinite model, how can i show the union of T1 and T2 has a model? i first tried to take m1 as a model of T1 and m2 as a model of T2, i want to show that the union of m1 and th(m1) and m2 and th(m2) has a model, if not then there exists p in the union of m1 and th(m1) and a -p in the union of m2 and th(m2) since T1 and T2 doesn't contain same symbols so this can not happen and the union of T1 and T2 must have a model. or if i use the LST theorem then I have both T1 and T2 has model of cardinality $X_0$ but I don't know how to use this...
2026-04-12 03:13:47.1775963627
union of two theories which doesn't contain same symbols of language has a model
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By the Löwenheim-Skolem theorem, both $T_1$ and $T_2$ have a model of size $\kappa$ for some infinite $\kappa$. By transferring the structure, one may as well assume that the model has underlying set $\kappa$ for both.
Your new structure should be $\kappa$ with the symbols of $L$ appearing in $T_1$ interpreted as in the model of $T_1$, and the symbols of $L$ appearing in $T_2$ interpreted as in the model of $T_2$. Given the hypothesis, this is consistant. This new model should work