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sav08:isomorphism_of_interpretations [2008/03/19 17:23] vkuncak |
sav08:isomorphism_of_interpretations [2008/03/19 17:34] vkuncak |
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**Lemma:** If $s$ is isomorphism from $I_1$ to $I_2$, then for every first-order term $t$ we have | **Lemma:** If $s$ is isomorphism from $I_1$ to $I_2$, then for every first-order term $t$ we have | ||
\[ | \[ | ||
- | s(e_T(I_1)(t))=e_T(I_2)(t) | + | s(e_T(t)(I_1))=e_T(t)(I_2) |
\] | \] | ||
- | and for every first-order logic formula $F$ we have $e_F(I_1)(F)=e_F(I_2)(F)$. | + | and for every first-order logic formula $F$ we have $e_F(F)(I_1)=e_F(F)(I_2)$. |
- | **Proof:** | + | **Proof:** ++|Induction on the structure of terms and formulas. |
- | ++++|Induction on the structure of terms and formulas. | + | |
+ | Case for $F_1 \land F_2$. | ||
+ | Case for $\exists x.F$. Induction issues. | ||
- | ++++ | + | ++ |
**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 $(D_1,I_1)$. |