This article focuses on issues related to improving an argument about minds and machines given by Kurt Gödel in 1951, in a prominent lecture. Roughly, Gödel’s argument supported the conjecture that either the human mind is not algorithmic, or there is a particular arithmetical truth impossible for the human mind to master, or both. A well-known weakness in his argument is crucial reliance on the assumption that, if the deductive capability of the human mind is equivalent to that of a (...) formal system, then that system must be consistent. Such a consistency assumption is a strong infallibility assumption about human reasoning, since a formal system having even the slightest inconsistency allows deduction of all statements expressible within the formal system, including all falsehoods expressible within the system. We investigate how that weakness and some of the other problematic aspects of Gödel’s argument can be eliminated or reduced. (shrink)