Aren’t you just begging the question by assuming ZFC is consistent to demonstrate that arithmetic is consistent?
Suppose for a moment that it makes sense to say arithmetic is part of the fabric of the universe (whatever that is supposed to mean). How would one know if arithmetic is consistent or complete within the context of being part of the universe? Suppose ZFC is part of the universe. Then arithmetic (the model of it as being part of the universe) is complete and consistent. Now what? I claim nothing of consequence follows from this in relation to whether or not space is continuous.
The same way I know that it’s incomplete when formulated in ZFC (assuming ZFC is consistent), from the first incompleteness theorem.
> Suppose ZFC is part of the universe. Then arithmetic (the model of it as being part of the universe) is complete and consistent.
Why does this conclusion follow?
Are you raising the question of whether mathematical propositions can be justified a posteriori?
Or are you arguing that the incompleteness theorems don’t necessarily apply to theories formulated ‘as part of the universe’?
If ZFC (or some other theory) implements arithmetic, then the first incompleteness theorem says that if ZFC is consistent then there must be true sentences in ZFC (not necessarily sentences of arithmetic) that can’t be proved in ZFC. Correct?
I had two points in these posts. One is that none of this pertains to whether or not space is continuous. The other is that the statement “arithmetic is consistent” is provable in some contexts. It depends on what the actual theory one is dealing with. In PA it’s not provable but in ZFC it is.
If the universe “contains” a model of PA then is that model consistent or not? How does one know? (I doubt it’s meaningful to say that the universe contains PA though.)
To me, “space is continuous” seems like a proposition that must be demonstrated a posteriori and I wouldn’t expect properties of formal systems to serve as evidence for the claim. So I agree.
> The other is that the statement “arithmetic is consistent” is provable in some contexts.
Agreed.
> If the universe “contains” a model of PA then is that model consistent or not? How does one know? (I doubt it’s meaningful to say that the universe contains PA though.)
Okay, yes, how can we interact with or measure PA as implemented by the universe? How do we (or can we) meaningfully talk about the universe implementing PA?
Yes those questions seem interesting to me, but I don’t have anything intelligible to say about them.
I don’t either!