High praise, thank you! Also, I realized there may indeed be a class of reals that are a subset of the reals, which might not actually exist. Within the reals,

High praise, thank you! Also, I realized there may indeed be a class of reals that are a subset of the reals, which might not actually exist. Within the reals, you have: - Reals equal to integers and rationals: Exist insofar as integer/rationals exist - Algebraic vs Transcendental: Algebraic reals exist insofar as finite algebraic expressions exist. I believe √2 is approximately as real as 2. - Special-case transcendentals like pi, e: Same "realness cardinality" as √2. - closed-form transcedentals like 2^√2 (Gelfond–Schneider constant): Also same "realness cardinality" as √2. From there, I think there is a "realness cardinality" transition, much like the countable to uncountable transition. These "boring transcendentals" have absolutely nothing notable about them, no way to label them using finite information, they are only expressible by the full, infinite description of themselves. these bad boys are ones that might not exist a priori, and the real heart of the question in TFA. I still contend the "boring transcendentals" do exist, because they allow dense cover of the [0,1] interval. In the same way that you can have a "lazy list" of the primes and can do a containment check of said list without having to generate all the primes in between, you could do a containment check of any piece of information on such a lazy set of transcendentals, including an infinite string of decimals, and it would always return True. So in a sense, every real must exist in that container (that's kind of how we arrived at the set of reals - we can prove their absence would be a contradiction in the rules of algebra).