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slides
Finding the limit of incompleteness I
Authors
Yong Cheng
Publication date
5 April 2020
Publisher
'Cambridge University Press (CUP)'
Doi
Cite
View
on
arXiv
Abstract
In this paper, we examine the limit of applicability of G\"{o}del's first incompleteness theorem (
G
1
\sf G1
G1
for short). We first define the notion "
G
1
\sf G1
G1
holds for the theory
T
T
T
". This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which
G
1
\sf G1
G1
holds. To approach this question, we first examine the following question: is there a theory
T
T
T
such that Robinson's
R
\mathbf{R}
R
interprets
T
T
T
but
T
T
T
does not interpret
R
\mathbf{R}
R
(i.e.
T
T
T
is weaker than
R
\mathbf{R}
R
w.r.t. interpretation) and
G
1
\sf G1
G1
holds for
T
T
T
? In this paper, we show that there are many such theories based on Je\v{r}\'{a}bek's work using some model theory. We prove that for each recursively inseparable pair
β¨
A
,
B
β©
\langle A,B\rangle
β¨
A
,
B
β©
, we can construct a r.e. theory
U
β¨
A
,
B
β©
U_{\langle A,B\rangle}
U
β¨
A
,
B
β©
β
such that
U
β¨
A
,
B
β©
U_{\langle A,B\rangle}
U
β¨
A
,
B
β©
β
is weaker than
R
\mathbf{R}
R
w.r.t. interpretation and
G
1
\sf G1
G1
holds for
U
β¨
A
,
B
β©
U_{\langle A,B\rangle}
U
β¨
A
,
B
β©
β
. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree
0
<
d
<
0
β²
\mathbf{0}< \mathbf{d}<\mathbf{0}^{\prime}
0
<
d
<
0
β²
, there is a theory
T
T
T
with Turing degree
d
\mathbf{d}
d
such that
G
1
\sf G1
G1
holds for
T
T
T
and
T
T
T
is weaker than
R
\mathbf{R}
R
w.r.t. Turing reducibility. As a corollary, based on Shoenfield's work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which
G
1
\sf G1
G1
holds.Comment: 18 pages. Accepted and to appear in Bulletin of Symbolic Logi
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oai:arXiv.org:1902.06658
Last time updated on 02/06/2019