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In-Depth Information
a
+
b
i
p
i
(1
a−
1
p
)
a
+
b−i
−
≤
1
−
α
i
=0
and it finishes the proof.
Lemma 6.
1. The 4ft-quantifier
⇒
p,Base
of founded implication satisfies the condition
b from the lemma 1 for each
0
<p
≤
1
and Base >
0
.
!
p,α,Base
of lower critical implication satisfies the con-
dition b from the lemma 1 for each
0
<p<
1
,
0
<α<
0
.
5
and
Base >
0
.
2. The 4ft-quantifier
⇒
Proof.
1. We have to prove that for each a
≥
0
, b
≥
0
such that
⇒
p,Base
(
a,b
)=0
there is a
≥
⇒
p,Base
(
a
,b
)=1
. The proof is trivial, we
a for that it is
a
a
+
b
=1
.
2. We have to prove that for each a
use the fact that
lim
a→∞
≥
0
and b
≥
0
such that it is satisfied
!
p,α,Base
(
a,b
)=0
there is a
≥
!
p,α,Base
(
a
,b
)=1
.
⇒
a for that it is
⇒
!
p,α,Base
Let us suppose that
⇒
(
a,b
)=0
. It means that a < Base or
i
=
a
a
+
i
p
i
(1
a
+
b
p
)
a
+
b−i
>α.
According to the Lemma 4 there is natural n such that for each e, e>n
and k
=0
,...,b it is
e
+
b
e
+
k
−
p
e
+
k
(1
α
b
+1
·
p
)
b−k
<
−
Let us choose a
=max
. Then it is a
{
a,n,Base
}
≥
Base and also
a
+
b
i
p
i
(1
a
+
b
p
)
a
+
b−i
−
i
=
a
a
+
b
a
+
k
p
a
+
k
(1
b
p
)
b−k
<α.
=
−
k
=0
!
p,α,Base
(
a
,b
)=1
and it finishes the proof.
Thus it is
⇒
Theorem 6.
The 4ft-quantifier
⇒
p,Base
of founded implication is not classi-
cally definable for each
0
<p
≤
1
and Base >
0
.
!
p,α,Base
of lower critical implication is not classically
definable for each
0
<p<
1
,
0
<α<
0
.
5
and Base >
0
.
The 4ft-quantifier
⇒
