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Comment by burrows | Hacker News Reader

> How would one know if arithmetic is consistent or complete within the context of being part of the universe?

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’?

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Arithmetic is complete in ZFC. Well, in each model of ZFC sits a model of arithmetic (first order Peano axioms) and that model is complete. The incompleteness theorem doesn’t apply in this case because the incompleteness theorem is a statement about the first order Peano axioms and not about the situation in which they are residing in a larger theory (which in our case is ZFC). The Peano axioms are not able to prove their own completeness but if they reside in a larger theory then that larger theory may be able to prove their completeness.

Okay, yeah I was confused.

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?

Yes! ZFC can’t prove it’s own consistency or completeness but a larger theory can do this. I think ZFC + Inaccessible Cardinal can prove ZFC is consistent.

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.)

> I had two points in these posts. One is that none of this pertains to whether or not space is continuous.

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 have anything intelligible to say about them.

I don’t either!

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