Aleph, AlephZero, AlephOne

I was wondering what I am learning these days !! As I wrote earlier discussion on Infinity was wonderful. Infact I will have to edit what I said earlier about cardinal numbers. The cardinality of natural numbers is alephZero and it is same as the cardinality of N*N which is to be noted. So I repeat again. The cardinality of natural numbers is alephZero and cardinality of N*N is also alephZero. Now the next question one may ask what is the cardinality of N*N*N*...N. Well its also alephZero. As long as the number of Ns are finite its alephZero. Then we have a transition we have alephOne. Which is equal to the all possible subsets of Natural number and its same as cardinality of real numbers.
So
Cardinality of Natural numbers = alephZero
Cardinality of N*N = alephZero
Cardinality of N*N*... N = alephZero if Ns are finite

Cardinality of powerset of Natural number = alephOne
Cardinality of Real numbers = alephOne

I haven't swam for a while and tommorow I will definitely swim. Yesterday I was glad that I could solve the general fibonacci series solution. That the ratio of consecutive terms approach Golden Ratio. That thing is neat. Start with any seed, infact the first two terms can be different instead of assuming the seed to be the single number. The proof can be had by induction method. Assuming that lim n-> infinity Fn+1/Fn approaches golden ratio. Dr Jerzy Kocik also discussed some more properties of golden ratio. Now I should spend some time learning complex analysis stuff.