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sav08:isomorphism_of_interpretations [2008/03/19 17:34] vkuncak |
sav08:isomorphism_of_interpretations [2008/03/20 13:37] vkuncak |
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| and for every first-order logic formula $F$ we have $e_F(F)(I_1)=e_F(F)(I_2)$. | and for every first-order logic formula $F$ we have $e_F(F)(I_1)=e_F(F)(I_2)$. | ||
| - | **Proof:** ++|Induction on the structure of terms and formulas. | + | **Proof:** ++++|Induction on the structure of terms and formulas. |
| Case for $F_1 \land F_2$. | Case for $F_1 \land F_2$. | ||
| - | Case for $\exists x.F$. Induction issues. | + | Case for $\exists x.F$. Induction issues, function update on isomorphic interpretations. |
| - | ++ | + | ++++ |
| **Lemma:** If $(D_1,\alpha_1)$ is an interpretation for language ${\cal L}$, if $D_2$ is a set and $s : D_1 \to D_2$ a bijective function, then there exists a mapping $\alpha_2$ of symbols in ${\cal L}$ such that $(D_2,\alpha_2)$ is an interpretation for ${\cal L}$ and $(D_2,\alpha_2)$ is isomorphic to $(D_1,I_1)$. | **Lemma:** If $(D_1,\alpha_1)$ is an interpretation for language ${\cal L}$, if $D_2$ is a set and $s : D_1 \to D_2$ a bijective function, then there exists a mapping $\alpha_2$ of symbols in ${\cal L}$ such that $(D_2,\alpha_2)$ is an interpretation for ${\cal L}$ and $(D_2,\alpha_2)$ is isomorphic to $(D_1,I_1)$. | ||