Comment by bo1024

Many replies don't seem to understand Godel and independence (and one that might is heavily downvoted). Cliff notes:

* ZFC is a set of axioms. A "model" is a structure that respects the axioms.

* By Godel, we know that ZFC proves a statement if and only if the statement is true in all models of ZFC.

* Therefore, the statement "BB(748) is independent of ZFC" is the same as the statement "There are two different models of ZFC where BB(748) are two different numbers.

* We can take one of these to be the "standard model"[1] that we all think of when we picture a Turing Machine. However, the other would be a strange "non-standard" model that includes finite "natural numbers" that are not in the set {0,1,2,3,...} and it includes Turing Machines that halt in "finite" time that we would not say halt at all in the standard model.

* So BB(748) is indeed a number as far as the standard model is concerned, the problem only comes from non-standard models.

TL;DR this is more about the fact that ZFC axioms allow weird models of Turing Machines that don't match how we think Turing Machines usually work.

[1] https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet...