It boggles my mind that a number (an uncomputable number, granted) like BB(748) can be "independent of ZFC". It feels like a category error or something.
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If you are refering to the idea that nothing that can't exist in the real universe "really exists", then the "Busy Beaver" portion of that idea is extraneous, as 100% of integers can't exist in the real universe, and therefore, 100% of integers are equally just "mathematical concepts". That one of them is identified by BB(748) isn't a particularly important aspect. But certainly, a very specific number is identified by that designation, though nothing in this universe is going to know what it is in any meaningful sense.
Many other numbers and functions are computable, including e, pi, 10^100, etc- these are fundamentally different than BB.
So in what sense is it actually a number? There is no algorithm which can resolve questions such as BB(748) < x given x. That doesn't seem like a number to me!
In fact, for some x, such questions will depend on the consistency of ZFC. All normal math we do is expressible in ZFC, but by incompleteness, ZFC cannot prove it's own consistency or is inconsistent. So, we cannot really ever know the value, we can only ever find lower bounds. Does this seem like a number to you? It's not in the English sense and neither is it in what I would consider a reasonable definition of numbers you actually encounter, the computable numbers. Real numbers are in fact, not very real at all.