# arbitrarily large vs infinite large

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?

• What do you mean by 'an arbitrarily large positive integer'? – paw88789 Sep 25 '14 at 3:01
• I mean it is arbitrarily large. Sometimes we see some statement like, arbitrarily large, arbitrarily long, etc.. – Ocean Yu Sep 25 '14 at 3:11
• I would say one integer can't be arbitrarily large. You could talk about a set of integers having arbitrarily large elements, meaning that the set is unbounded. – paw88789 Sep 25 '14 at 3:20
• It is used in some statements, like as you say, "arbitrary long arithmetic progressions". But this language ALWAYS is in the context of a SET of numbers, e.g., the set of all lengths of arithmetic progressions of primes. The language that the numbers in the set get arbitrary large then means that the set is not bounded above. Your statement "if N is arbitrarily large" does not seemingly refer to a representative of a well-defined set of numbers, so is misuse of the language. – Barry Smith Sep 25 '14 at 3:20
• Awww, paw beat me by 3 seconds – Barry Smith Sep 25 '14 at 3:20

All the elements of $L$ are naturals, so none of them are infinite. If $L$ contains an infinite number of elements (which is required when you say its elements can be arbitrarily large) it is incorrect to say the elements can range from $2$ to $n$. You have not defined $n$ in any proper way. If you think of $L$ as a sequence, so $a_i$ is the $i^{\text{th}}$ smallest element of the set, you can say $\lim a_i = \infty$ (assuming you allow infinite limits). It shows that $|L|=\aleph_0$, not that $L$ is $\aleph_0$ as there may well be some elements missing (like $1$).
When we say $L$ contains arbitrarily large numbers, it means that whatever number you name, the set contains a larger one. There is no single "arbitrarily large number", "arbitrarily large constant", or $\aleph_x$