Set Theory Theorem: Existence of Natural Number Sequences & Large Cardinals

[Dragonfall]

Is there a theorem which says that if certain natural number sequences exist, then some large cardinals exist. Can anyone tell me if it's true and what it says?

I vaguely remember my set theory professor mention this theorem years ago.

 

[aleph-aleph]

If you mean theorem that produce larger cardinality, it's the power set P(n)=2^n where n is cardinal number.
http://mathworld.wolfram.com/PowerSet.html