Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Observations concerning G\"odel's 1931   This article demonstrates the invalidity of Theorem VI of Godel's monograph
of 1931, showing that propositions (15) and (16), derived from definition
(8.1), in its proof, are false in PA. This is achieved in two steps.
  First, the predicate complementary to the well-known Godel's predicate Bew(x)
is ndefined by adding a new relation Wid(x), and new logical connections are
accordingly established, Lemma (6).
  Second, the negations of (15) and (16) are derived by definition (8.1) and
Lemma (6). It amounts to saying that (15) and (16) are false and unacceptable
for the system. On the account of that, the two well-known cases 1. 17 Gen r is
not k-PROVABLE, 2. Neg(17 Gen r) is not k-PROVABLE, can not be drawn, and
Theorem VI is therefore invalid.

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The categories of this paper are: adap-org
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The categories of this paper are: hep-ph
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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: ao-sci
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: stat