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 $\lim (n) =\infty$ ?
On the other word, $n$ can reach infinity large but can never be infinite, is this right?
And it shows that set $L$ is $\aleph_0$, if we assume 'arbitrarily large' as a constant, say $\aleph_x$, which one can not assign a certain value because it is always larger than it, will $\aleph_x < \aleph_0$ hold?
Or we say there is one constant $\aleph_x$, it has,
(1) $\lim (\aleph_x) =\infty$
(2) $\aleph_0$ is equivalent to $\infty$
(3) $\aleph_x < \aleph_0$
My question is, is there any value about this? Is it true? Can someone explain more deeper meaning about arbitrarily large and infinite large?