User blog comment:Simplicityaboveall/Insanely Fast-Growing Functions/@comment-28606698-20171025182658/@comment-28606698-20171029183145

Yes, Joe, even if you are an infinitist and don't believe in existance of infinite ordinals you can look at them as at some technical tools that allows to create googologically large numbers, as was said in previous comment. There are a lot of googological notations where authors introduce some separators between natural numbers and those separators not being ordinals work exactly as infinite ordinals. For example Saibian's Cascading-E notation simulates limit ordinals written in Cantor normal form (for this reason I have some mistrust when Saibian is comparing his notation with ordinals beyond $$epsilon_0$$) or for example I-notation simulates limit ordinals of Buchholz's notation. Using of ordinal notations, which came from professional math, very simplifyes working with numbers, especially beyond $$f_{\omega^\omega}(n)$$.