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

Thanks for the clarification, Joe, i post this new comment as a separate response to symbolize the withdrawal of my previous comment which wrongly suggested your set K caps out at ω^ω, i made that claim solely based on your (1..2...) description and did not pay attention to the R function.

Upon further analysis it seems that it caps out at ε_0,and not ω^ω as i originally claimed.

The key is to realize that the hereditary notation will always produce some number in the form of a power tower of finite height (otherwise your R wouldnt work), and as such it can always be represented as  10^^n with a finite n

I produce for you a new table:

nth member / ordinal / nth member of a set which has ε_0 as supremum and starts with ω

10 / ω / 1

10^10 / ω^ω / 2

10^^3 / ω^^3 / 3

...

10^^9 / ω^^9 / 9

By this point i'm sure you havent got a problem with the table, however, this following element might upset you

10^^10 / ω^^10 / 10

The reasoning, however is clear with the next element: 10^^11

10^^11 in hereditary base 10? , thats a tower of 10s, 11 tall. It isnt 10^^(10+1), as R has no way of accessing the tetration controller and thats your main problem. R will transform  10^^11  to ω^^11 and not ω^^(ω+1) which is by the way is also not canonically defined (though some people including me have proposed various ways of settling this Link )

_Even if_ you define hyperators on ordinals, your accessing method only reaches ^^.

I now conclude with the rest of the table

10^^250 / ω^^250 / 250

10^^10,000,000,000 / ω^^10,000,000,000 / 10,000,000,000

10^^10^^11/ ω^^(10^^11) /10^^11

10^^10^^10^^10/ ω^^(10^^10^^10) /10^^10^^10

10^^^10 / ω^^(10^^^9) / 10^^^9

As you can see, it doesn't matter that the set K is infinite, as my list can also contain an infinity of  numbers yet no matter how huge the finite index gets the ordinals in K will all be under ε_0.