People complain about infinite information, but the theory of real fields is complete and consistent, while arithmetic is not:
https://en.wikipedia.org/wiki/Decidability_of_first-order_th...
https://en.wikipedia.org/wiki/Decidability_of_first-order_th...
People tend to mentally construct the natural numbers from set theory, wholes from naturals, rationals from wholes, reals from rationals and so on.
But what if there is some universe (in the math sense), which is actually complete and decidable, it's just that the moment you take discrete subsets of it, you also remove the connections that make it consistent or complete.
The very act of formalising mathematical concepts into words and paper is a quantisation step after all, because both are symbols. Maybe there are proofs that can be inuitioned about (assuming brains are continuuous in some sense) but neither verbalised nor formalised.
Presumably this hypothetical universe implements arithmetic, so it’s not complete.
Incompleteness means that there are true statments for which there are no proofs.
But that doesn't preclude proofs that are beyond the proof system that you proofed incompleteness for.
A non discete/symbolic proof might exist after all.
It would therefore not be the existence of the natural number subset that causes undecidability, but the missing parts of the non-natural superset required to talk about the proofs that cause undecidability.
Then the system you are proposing, does it implement arithmetic or not?
But proof by contradiction itself is problematic, because it relies on the law of the excluded middle, which only holds in classical two valued logic.
If you construct math from the top down rather than from the bottom up, then it is natural that it also has a multi valued logic.
In fact, it also would have infinitely valued logic. Infinite sentences, infinite theorems.
Such math is non expressible for us, because we rely on discrete descriptions.
But that doesn't preclude its existence.
Why should it though? What evidence do we have, if we can't express it or fit it into our existing mathematical framework.
To illustrate, think of Goedelsz approach and turn it backwards for a second. Instead of taking predicate logic and assigning each sentence a natural number, imagine that predicate/classical logic is a different view on the natural numbers. Now that means that there might also be logical interpretation of the reals, the hyperreals and so on. (We could do the same with Alephs, in fact they might be the more fundamental objects.)
Perhaps that is exactly what mathematics is? and all it can be? Or perhaps there is a 'higher mathematics' that we cannot reach yet? (or ever?)
At any rate, what does any of this have to do with information capacity in the universe? Is the information capacity of the universe related to the consistency/completeness of arithmetic?
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!