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Paper   IPM / M / 12784
School of Mathematics
  Title:   Godel's incompleteness phenomenon-computationally
  Author(s):  S. Salehi
  Status:   Published
  Journal: Philosophia Scientiae
  Vol.:  18
  Year:  2014
  Pages:   23-37
  Supported by:  IPM
We argue that Gödel's completeness theorem is equivalent to completability of consistent theories, and Gödel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enumerable theory. Though any consistent and decidable theory can be extended to a complete and consistent and decidable theory. Thus deduction and consistency are not decidable in logic, and an analogue of Rice's Theorem holds for recursively enumerable theories: all the non-trivial properties of them are undecidable.

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