Suppose there is a set L which contains all lengths of arithmetic progressions of primes, the elements inside of L can vary from 2 to n, where n can be arbitrarily large.

However, n can not be infinite large.

Is it true that lim(n)=∞ ?

On the other word, n can reach infinity large but can never be infinite, is this right?

And it shows that set L is ℵ0, if we assume 'arbitrarily large' as a constant, say ℵx, which one can not assign a certain value because it is always larger than it, will ℵx<ℵ0 hold?

Or we say there is one constant ℵx, it has,

(1) lim(ℵx)=∞
(2) ℵ0 is equivalent to ∞
(3) ℵx<ℵ0
My question is, is there any value about this? Is it true? Can someone explain more deeper meaning about arbitrarily large and infinite large?

存在任意长的等差素数列已经被证明，但显然不存在无穷长的等差素数列。

任何等差数列都可以写成x[n]=a+bn，那么x[a]=a+ab=a(b+1)必然不是素数。

好像没回答我的问题。。。