>>12468250>Because it's existence can be proven.Bold claim!
>You do not need an explicit construction to prove the existence of an object. I would actually believe. For example, I do believe that there is a smallest prime factor of the number 123413, even though I can't explicitly point it out right now.
>or show (in this case constructively) that there exist an infinite amount of unique natural numbersIf by an infinite you mean an ongoing amount of naturals, never completed, then yes I, and Wildberger, would agree.
After all, there is nothing more compelling like the argument that you can always add 1 to the natural number of your choice. This is called the POTENTIAL infinity, not COMPLETED infinity. It's ok to say there is an unending, potentially infinite amount of natural numbers. What is not OK is to claim you can somehow encompass all of it into some single infinite object. Showing that the sequence of natural numbers is unending DOES NOT show that you can put them all into a whole single object.
>an uncountable amount of unique real numbersYour "proof" of this "theorem" (which is of course completely meaningless because it talks about "real numbers") uses the assumption that
1. It makes sense to talk about completed sequences of real numbers (it's not viewed as an ongoing, ever evolving object but rather an already completed static set)
2. It asks you to do an infinite process to "generate" a new number by going across the diagonal. This also plainly indicates that the theorem is about some fairy tale fantasy, and not mathematics, since it's actually impossible to do such an infinite amount of work.
>, IS scientific empirical proof of their existence (a mathematical proof of existence is a scientific experiment that proves the existence of the structure).Where is the science? Did I miss it? So far all I've hard are some fairy tale descriptions of impossible things about impossible, meaningless objects.
