Third line of the article

> No undecidable problem can ever be solved by a computer or computer program of any kind. In particular, there is no Turing machine to solve an undecidable problem.

This clearly not true. The halting problem can be solved for many problems from trivial to complicated.

For example, the busy beaver competitions show that the halting problem is routinely solved for thousands of Turing machines, some with extremely long and complicated running time.

Also all kinds of practical problems are routinely analyzed by static program analyzer. Many will point out halting problems to you.

What is probably meant, is that there is no Turing program that will work on any input.