Church-Turing thesis is a dualism Church-Turing thesis : Every effectively calculable function is a TM-computable function.

But, hypercomputation models are strictly more powerful than TM and can solve TM-uncomputable problems on the paper.

Does this imply that, for one who believes in the Church-Turing thesis, there is only two options :


There is two distincts worlds : The effective one (where Church-Turing thesis applies) and the paper
one (where Church-Turing thesis doesn't apply necessarily). That is the physical world and the ideas world of the classical
dualism theory.
All the hypercomputation models are inconsistent. Using an hyper-TM in a reasoning is the same than, for instance, using an integer both even and odd in a reasoning, this can lead to false conclusions.


?
There's another possibility: hypercomputation is not implementable in the real world, and is only an imaginary/theoretical concept.  If this is the case, then there is no contradiction with the Church-Turing thesis.

Of course, anyone can invent imaginary worlds where strange things are true.  For instance, there's the computer scientist's Superman: not only can he leap tall buildings in a single bound, he can also solve undecidable problems in his head!  I suppose one could imagine such a thing, but that doesn't mean it has any correspondence to the real world, and the possibility of imagining such a Superman doesn't contradict the Church-Turing thesis.  The study of hypercomputation can be thought of as asking, well, ok, so Superman might not be real, but what if he were?

The Church-Turing thesis is a hypothesis about the real world, so imaginary schemes that can't be implemented in the real world don't contradict it.