The Paradox of Gödel’s Numbering and the Philosophy of Modern Metamathematics


  • Robert DJIDJIAN Department of Philosophy and Logic Named after Academician Georg Brutian at Khachatur Abovian ASPU



formalized theories, metatheory, Gödel’s theorem, Gödel’s numbering, formalized metamathematics, inconsistency, paradoxes


The author of this article critically analyses the proof of Gödel’s famous theorem on the incompleteness of formalized arithmetic. It is shown that Gödel’s formalization of meta-mathematics provides a proof of the incompleteness not of mathematical science but of the system of formalized meta-mathematics developed by Gödel himself. The arguments against the idea of the formalization of meta-mathematics are presented. The article suggests also an interpretation of the essence of mathematical truth. It is noted that the refutation of Gödel’s proof does not suggest returning to Hilbert’s program of formalism since the formalization of an axiomatic theory can’t exclude the appearance of paradoxes within its framework. It is shown that the use of self-referential Gödel’s numbering in a formalized system leads to the emergence of a Liar type paradox – a self-contradic­tory formula that demonstrates the inconsistency of that same system.


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Author Biography

Robert DJIDJIAN, Department of Philosophy and Logic Named after Academician Georg Brutian at Khachatur Abovian ASPU

Robert DJIDJIAN (Dr.) is Professor of the Department of Philosophy and Logic Named after Academician Georg Brutian at Khachatur Abovian ASPU, Yerevan, Armenia, member of the editorial board of the journal WISDOM. His areas of interest include logic of scientific research, axiomatic philosophy, artificial intellect. R. Djidjian is the author of 8 monographs and 66 scientific articles. Recent publications: “Getting Ready for Great Discoveries”, “Optimal sociology”, “Building the Logic of Scientific Discoveries”, “Building the General Theory of Meta-argumentation”.


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How to Cite

DJIDJIAN, R. (2017). The Paradox of Gödel’s Numbering and the Philosophy of Modern Metamathematics. WISDOM, 9(2), 18–28.