This question is an extension question of previous one.
link: Löb's theorem and provability
Now, there is three sentences, P, Q and R.
sort-of-says, they are like following.
P: P, Q and R is provable all.
Q: P, Q and R is provable all.
R: P, Q and R is provable all.
By previous approach, by Löb's theorem, P,Q,R is provable all.
Now expand this story to many sentences.
P(1) : P(1), P(2).... P(n) is provable all
p(2) : P(1), P(2).... P(n) is provable all
....
p(n) : P(1), P(2).... P(n) is provable all
By previous approach, by Löb's theorem, P(i) is provable from i=1 to i=n.
My question: Is this story valid when n goes infinite?
It means,
P(1) : P(1), P(2)......P(i)...... is provable for all i
p(2) : P(1), P(2)......P(i)...... is provable for all i
....
p(i) : P(1), P(2)......P(i)...... is provable for all i
.....
By previous approach, by Löb's theorem, is P(i) provable for all i ?
