Show that{∀x(α∧β)} `(∀xα)∧(∀xβ)whereα,βare anyL-formulas.
Question Description
Problem
1 (5 pts)
.
Show that
{∀
x
(
α
∧
β
)
} `
(
∀
xα
)
∧
(
∀
xβ
)
where
α,β
are any
L
-formulas.
Problem
2 (5pts)
.
Fix a first order language
L
and let
α
be an
L
-sentence. Suppose that for every
N
∈
N
there is an
L
-structure with at least
N
elements that satisfies
α
. Show that there is an infinite
L
-structure
that satisfies
α
. (You will need to use the completeness theorem.)
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