1) Starting with arithmetics, whole mathematics must be founded in the continuum of geometry. Even Frege admitted that he had spent his whole life on the wild goose
chase, trying to found it in the discreteness of the Set Theory.
2) let's consider the famous Goedel sentence G: "This sentence is not provable" and the theorem: "G is true but not provable in the theory".
G is neither false,nor true for the simple reason that it is NO statement at all. By the standards of Goedell's own Predicate Logic a statement is a predication,
an assignment of a property to a subject. "Truth/Falsity" qualifies the predication itself and not the subject of predication. A statement "all cars are red" is a valid
predication or statement which may be true or false and by virtue of observations turns out to be false.
Now, G does not assign any property to any subject, thus is not a predication, not a statement at all, a meaningless chain of characters that may not be true or false.
Details in my site http://findgeorges.com/ chapter C4_LIAR,_RUSSELL_AND_GOEDEL.
Georges Metanomski