Logic and Logical Philosophy 15 (1):39-66 (2006)

Zbigniew Tworak
Adam Mickiewicz University
In this paper, I try to accomplish two goals. The first is to provide a general characterization of a method of proofs called — in mathematics — the diagonal argument. The second is to establish that analogical thinking plays an important role also in mathematical creativity. Namely, mathematical research make use of analogies regarding general strategies of proof. Some of mathematicians, for example George Polya, argued that deductions is impotent without analogy. What I want to show is that there exists a direct line leading from Cantor’s diagonal argument to constructions that underlies of the proofs of several important theorems of the mathematical logic (in particular, Church’s theorem concerning the undecidability of formal arithmetic, Gödel’s theorem concerning the incopleteness of formal arithmetic, Tarski’s theorem concerning truth, and Turing’s theorem concerning the Halting Problem), and that the line could be described as an analogical mapping. In other words, Cantor’s diagonal argument and the proofs of the limitative theorems are structurally the same. Hence they can be represented as instances (or special cases) of the same general scheme
Keywords undecidability  analogy  diagonal argument  limitative theorems  antinomy  provability  refutability  truth
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DOI 10.12775/LLP.2006.003
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References found in this work BETA

On Computable Numbers, with an Application to the N Tscheidungsproblem.Alan Turing - 1936 - Proceedings of the London Mathematical Society 42 (1):230-265.
An Unsolvable Problem of Elementary Number Theory.Alonzo Church - 1936 - Journal of Symbolic Logic 1 (2):73-74.
On a Puzzling Classical Validity.Robert L. Martin - 1977 - Philosophical Review 86 (4):454-473.
The Diagonal Argument and the Liar.Keith Simmons - 1990 - Journal of Philosophical Logic 19 (3):277 - 303.

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