User blog comment:Simplicityaboveall/The Construction of Extremely Large Numbers/@comment-24920136-20160728195542/@comment-25912386-20160729083929

Hi Chronolegends ! Thanks for your kind comments man ! I see that Mathematics and the set thoery is a passion for you. Keep up !

I believe that all ordinals, no matter how large can be " decomposed " into w !

For example, if e(0) = w ^^ w, there must be a way of expressing e(1) or e(2) etc !

e(1) = e(0) ^^ w, so e(1) = ( w ^^ w ) ^^ w  which is approximately ( w ^^ w2 ). Otherwise, we use the formal definition :

which is equivalent to



e(1) = sup (  e(0) +1, e(0)w, e(0)^w, e(0)^e(0)^w, ... )