Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Godel-Rosser's Incompleteness Theorems for Non-Recursively Enumerable
  Theories   Godel's First Incompleteness Theorem is generalized to definable theories,
which are not necessarily recursively enumerable, by using a couple of
syntactic-semantic notions, one is the consistency of a theory with the set of
all true $\Pi_n$-sentences or equivalently the $\Sigma_n$-soundness of the
theory, and the other is $n$-consistency the restriction of
$\omega$-consistency to the $\Sigma_n$-formulas. It is also shown that Rosser's
Incompleteness Theorem does not generally hold for definable non-recursively
enumerable theories, whence Godel-Rosser's Incompleteness Theorem is optimal in
a sense. Though the proof of the incompleteness theorem using the
$\Sigma_n$-soundness assumption is constructive, it is shown that there is no
constructive proof for the incompleteness theorem using the $n$-consistency
assumption, for $n\!>\!2$.

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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con