I think the current record is 1919 states, so if you run a certain Turing Machine for BB(1919) steps and it doesn't halt then you know that ZFC is consistent. Godelian considerations might make it reasonable to say that integers BB(1919) or larger don't exist.
I get skeptical of integers so large that you need weird Turing Machines to enumerate their digits whose halting proof is independent of set theory, but I don't yet disbelieve in their existence.
http://www.scottaaronson.com/blog/?p=2725
I think the current record is 1919 states, so if you run a certain Turing Machine for BB(1919) steps and it doesn't halt then you know that ZFC is consistent. Godelian considerations might make it reasonable to say that integers BB(1919) or larger don't exist.
I get skeptical of integers so large that you need weird Turing Machines to enumerate their digits whose halting proof is independent of set theory, but I don't yet disbelieve in their existence.