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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

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?

syki 5y ago
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.)
- burrows 5y ago
  > 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.
  - syki 5y ago
    …I don’t have anything intelligible to say about them. I don’t either!

syki 5y ago

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.)

burrows 5y ago
> 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.
- syki 5y ago
  …I don’t have anything intelligible to say about them. I don’t either!