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Author Raatikainen, Panu
Source CiteSeerX
Content type Text
File Format PDF
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Delian Argument ♦ Turing Machine ♦ Sentence Gt ♦ Human Being ♦ Arbitrary Theory ♦ Delian Anti-mechanist Argument ♦ Logical Fact ♦ Diagonalization Lemma ♦ Mccall Argumentation ♦ Provability Part Company ♦ Storrs Mccall ♦ Mccall Note ♦ Turing Machine Cannot ♦ Uman Being ♦ Provability Predicate ♦ Self-referential Trick ♦ Machine Thinking ♦ Peano Arithmetic Pa ♦ Weakly Represent Provability ♦ Unproved Assumption ♦ Chosen Theory ♦ Incompleteness Theorem ♦ Sharp Dividing Line ♦ Supercompact Cardinal
Abstract Storrs McCall continues the tradition of Lucas and Penrose in an attempt to refute mechanism by appealing to Gödel’s incompleteness theorem (McCall 2001). That is, McCall argues that Gödel’s theorem “reveals a sharp dividing line between human and machine thinking”. According to McCall, “[h]uman beings are familiar with the distinction between truth and theoremhood, but Turing machines cannot look beyond their own output”. However, although McCall’s argumentation is slightly more sophisticated than the earlier Gödelian anti-mechanist arguments, in the end it fails badly, as it is at odds with the logical facts. McCall’s reasoning differs from the earlier Gödelian arguments in his admission that the recognition of truth of Gödel sentence GT for a theory T depends essentially on the unproved assumption that the theory T under consideration is consistent. But, so the argument continues, still human beings, but not Turing machines, can see that truth and provability part company. For, McCall notes, we can argue by cases: Case 1. T is consistent. GT is unprovable, but true. Case 2. T is inconsistent. GT is provable, but false. Whichever alternative holds, McCall concludes, truth and provability fail to coincide. According to McCall, human beings can see this, but a Turing machine cannot. The conclusion, however, is simply false. McCall does not seem to realize that e.g. in Peano Arithmetic PA one can formalize and prove Gödel’s theorem for an arbitrary theory, however strong (McCall considers Zermelo-Fraenkel set theory ZFC, but one can take just any theory, e.g. ZFC + there exist supercompact cardinals, i.e. the sky is the limit, as long as the theory is effectively axiomatizable). More exactly, one can, in PA, formalize and weakly represent provability in any chosen theory T; let us denote such a provability predicate by ProvT(x). Gödel’s self-referential trick (the diagonalization lemma) then provides a sentence GT such that
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