User blog comment:Simplicityaboveall/The Construction of Extremely Large Numbers/@comment-5529393-20160724103034/@comment-5529393-20160725125123

@Simplicityaboveall: You are very welcome! Concerning SGH, yes it is true that below epsilon_0  there is a nice "replace w with n" correspondence, however if you wish to do the same beyond epsilon_0, then precise definitions must be made. First of all, to define SGH up to an ordinal alpha, we need to define fundamental sequences for all limit ordinals below alpha; otherwise, the SGH is not well-defined. Secondly, even if we have defined the SGH, that doesn't mean that we have a unique "decomposition" of ordinals into w; again, we must define how we are representing a particular SGH function in terms of n, so that we can replace n with w. TLDR: we can't magically proceed through all the countable ordinals, we must define everything as we go.

I thought we were in agreement that R needed a proper definition, but then in your reply to Chronolegends you say that "everything is well defined in my work." Do you agree that R still needs to be properly defined, or not?

@PsiCubed: For some reason, we can only reply to top-level comments. So yeah, we just keep responding to the parent post.