Wouldn’t it be: large ordinal → ZFC consistent → Cox’s theorem?
Maybe you then doubt that consequences follow from valid arguments (like Carroll’s Tortoise in his dialogue with Achilles). We could add a third premise that logic works, but I’m not sure it would help.
Believing that a mathematical system has a model usually corresponds to believing that a certain computable ordinal is well-ordered (the proof-theoretic ordinal of that system), and large ordinals imply the well-orderedness of all smaller ordinals.
I’m no expert in this—my comment is based just on reading the post, but I take the above to mean that there’s some large ordinal for ZFC whose existence implies that ZFC has a model. And if ZFC has a model, it’s consistent.
Wouldn’t it be: large ordinal → ZFC consistent → Cox’s theorem?
Maybe you then doubt that consequences follow from valid arguments (like Carroll’s Tortoise in his dialogue with Achilles). We could add a third premise that logic works, but I’m not sure it would help.
Can you elaborate on the first step?
I’m no expert in this—my comment is based just on reading the post, but I take the above to mean that there’s some large ordinal for ZFC whose existence implies that ZFC has a model. And if ZFC has a model, it’s consistent.