Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
sav08:isomorphism_of_interpretations [2008/03/19 17:23] vkuncak |
sav08:isomorphism_of_interpretations [2008/03/19 17:29] vkuncak |
||
---|---|---|---|
Line 1: | Line 1: | ||
====== Isomorphism of First-Order Logic Interpretations ====== | ====== Isomorphism of First-Order Logic Interpretations ====== | ||
+ | |||
+ | (Building on [[First-Order Logic Semantics]].) | ||
**Example:** How many models does this formula have? | **Example:** How many models does this formula have? | ||
Line 51: | Line 53: | ||
**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 $\exists x.F$. | ||
++++ | ++++ |