A turing machine is a mathematical abstraction that is the simplest machine that can compute things. Due to it&#x27;s nature of &quot;mathematical abstraction&quot;, it&#x27;s actually impossible to build one, though you can restrict it and build one of course. In particular, we have real computers (more complex than Turing machines, and with other restrictions).<p>These particular Turing machines that were built, have special properties that allow them to test mathematical axioms&#x2F;theorems, so for example, if the Z machine halts, then it means that the ZFC axioms are wrong. In the article, this is stated in the negative form (&quot;Z will run forever if ZFC is right&quot;).<p>Disclaimer: it&#x27;s been a long time since I last read the formal definitions of turing machines, and my explanation may be sloppy -- but I aimed for a more general thing to give you an idea.