What are the implications of Godel's incompleteness theorems

those theorems are why string theory is true but unproven

none, practically speaking.

you can cook up some very goofy models of phenomena (that are nowhere near reasonable to assume) and there you'll find that you can pose questions about the systems (and by extension, about the described phenonomena) that can provable not be answered.
(Things like questions regarding the eigenvalues of very exotic hamiltonians)

For example it directly follows from the incompleteness theorem that in three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.

>>9070130
There's no general theory of PDE. That's it

>>9070155
No.

Given that any theory of physics will include some number theory (e.g. [math]\pi _1 [/math]), Gödel's first incompleteness theorem will also apply to them. However, this means only that there will be some arithmetical truths that the theory won't be able to prove. It could well be that it proves all relevant physical truths.

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26KB, 318x500px

>>9070268
>Given that any theory of physics will include some number theory
Not necessarily true.

>>9070508

[math]\pi_1(S^1) = {\bf Z} [/math]

[math]\pi_1(T^2) = {\bf Z}^2[/math]

> make circle / torus
> think about what happens when you do stuff with it
> prove stuff about integers

>>9070224
Yes.

>>9070224
>There's no general theory of PDE
What do you mean exactly?

>>9071307
Every single ODE can be solved exactly by utilizing some very simple algorithms. You can't solve a single PDE with any existing method and this is guaranteed by Gödels Incompleteness Theorem.

>>9071310
>Every single ODE can be solved exactly by utilizing some very simple algorithms
Aren't there a lot of ODEs that don't have an explicit solution in terms of transcendental functions? Or are you talking about something else?
>You can't solve a single PDE with any existing method and this is guaranteed by Gödels Incompleteness Theorem.
Where can I read more about this?

>>9070130
Your mum's existence

>>9071314
>Where can I read more about this?
Do you know any axiomatic set theory and/or "advanced" mathematical logic?

>>9070130
none