The Church-Turing thesis [1] states:
Every effectively computable function is Turing computable
That is, for every function which can be computed by physical means [2] there exists a Turing machine which computes exactly that.
The Church-Turing thesis is not a definite mathematical statement, but an inductive statement which affirms that every sensible model of computation we can come up with is equivalent or at least reducible to the model proposed by Turing.
Thus we cannot prove it in a mathematical sense, but we can gather evidence for it.
The consequences of the CT thesis are plenty and very significant for computer science in general, artificial intelligence, epistemology and other fields of knowledge.
often abbreviated as CT thesis
So no hypercomputers