By moving to the Second Order, on the other hand, the True/False OR False/True oscillation is, sometimes, capable of resolution.

But that, very often, takes you outside the original axiomatic system.

She believed in nothing; only her skepticism kept her from being an atheist. -- Jean-Paul Sartre

That was the sailing-over-the-edge moment.  

Now it may be that arithmetic really is consistent, but that is another truth function that is "uncomputable."  

The Fates are kind.