Download A FORMULA SEPARATING TWO THEORIES

数理解析研究所講究録
912 巻 1995 年 82-84
82
A FORMULA SEPARATING TWO THEORIES
Takeshi Yamaguchi
(e-mail: y-takesi@is titech .
(山口
$\mathrm{a}\mathrm{c}.\mathrm{j}_{\mathrm{P})}$
武志)
(東工大・情報理工)
Abstract
is incomplete, then there is a formula such that
If theory
, Rosser sentence of is a separating formula.
. In special case
$\mathrm{T}+\neg\psi$
$\varphi$
$\psi=\mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T})$
$\varphi\subset\wedge \mathrm{T}+\psi$
$0$
.
$\mathrm{T}\subseteq \mathrm{T}+$
$\mathrm{T}$
Preliminaries
We consider only forlnulas in Arithlnetic. A theory is a set of sentences that contains
is
denotes the least theory which contains
all of its logical consequences.
sentence).
is
a
theory,
a
means “ is provable in ”
$\mathrm{T}\mathrm{U}\{\varphi\}(\mathrm{T}$
$\mathrm{T}+\varphi$
$\varphi$
$\mathrm{T}$
$\mathrm{T}\vdash\varphi$
$\varphi$
1. A formula separating two theories
be theories.
Definition Let
A sentence separates from
$\mathrm{T},$
$\mathrm{S}$
$\mathrm{T}$
$\mathrm{s}\mathrm{g}_{\mathrm{T}}\mathrm{f}\subseteq \mathrm{T}+\varphi\subset\wedge \mathrm{S}$
$\varphi$
Lemma 1 Let be a theory,
on are equivalent.
$\mathrm{T}$
$\psi$
be a sentence, and
$\mathrm{T}\subset\wedge \mathrm{T}+\psi$
. The following conditions
$\varphi$
1.
$\varphi$
2.
separates
(i)
(ii)
Proof)
$\mathrm{T}\mu_{\varphi}$
from
$\mathrm{T}+\psi$
.
, and
$\mathrm{T}\vdash\psiarrow\varphi,$
$\mathrm{T}\subseteq \mathrm{T}+\varphi$
Remark
theory.)
$\mathrm{T}$
$\mathrm{T}\mu_{\varphi}arrow\psi$
.
is equivalent to (i).
$\mathrm{T}+\varphi\subset\wedge \mathrm{T}+\psi$
is equivalent to (ii).
$\mathrm{T}\subseteq \mathrm{T}+\varphi\subseteq \mathrm{T}+\psi\Leftrightarrow \mathrm{T}+\neg\psi\subseteq \mathrm{T}+\neg\varphi\subset\wedge$
$\square$
Incon. (Incon is the inconsistent
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is incomplete, then there is a formula separating froln
Theorem If
because
and
such that
Proof) There is a sentence
. It is easily verified that satisfy the conditions of the
incolnplete. Let
.
remark. So this is a separating sentence from
$\mathrm{T}+\psi$
$\mathrm{T}$
$\mathrm{T}+\urcorner\psi$
$\varphi$
$\varphi\underline{=}\varphi’\vee\psi$
$\mathrm{T}+\neg\psi$
$\mathrm{T}+\neg\psi\mu\neg\varphi’$
$\mathrm{T}+\urcorner\psi^{\mu}\varphi’$
$\varphi’$
.
is
$\varphi$
$\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{v}_{\square }\mathrm{e}$
$\mathrm{T}+\psi$
$\mathrm{T}$
$\varphi$
Con(T) by second
is a consistent primitive rec.ursive theory, then
is a consistent primitive recursive theory,
incompleteness theorem [2]. So
is incomplete by Rosser’s theoreln [2]. So there is a
too. We have that
theory between and T+Con(T) by above theoreln.
Remark If
$\mathrm{T}\mu$
$\mathrm{T}$
$\mathrm{T}+\neg \mathrm{C}_{0}^{l}\mathrm{n}(\mathrm{T})$
$\mathrm{T}+\neg \mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T})$
$\mathrm{T}$
2. On a formula separating
$\mathrm{T}$
from
$\mathrm{T}+\mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T})$
In the previous section, we showed that there is a theory between and T+Con(T)
,
is consistent and primitive recursive). Actually, let be a Rosser sentence of
are separating sentences. In this section, we show that
and
then
Rosser sentence of itself is a sentence separating from T+Con(T). We will use certain
results on the provability predicate. See [1], [2].
$\mathrm{T}$
$(\mathrm{T}$
$\mathrm{T}+\neg \mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T})$
$\varphi$
$\varphi\vee \mathrm{C}_{\mathrm{o}\mathrm{n}}(\mathrm{T})$
$\urcorner\varphi\vee \mathrm{C}^{\mathrm{t}}\mathrm{o}\mathrm{n}(\mathrm{T})$
$\mathrm{T}$
$\mathrm{T}$
Definition
Rosser sentence
$\varphi$
of
$\mathrm{T}$
is a sentence such that
(1)
$\mathrm{T}\vdash\varphirightarrow\urcorner \mathrm{P}\mathrm{r}^{*}(\ulcorner\urcorner\varphi)$
where
$\mathrm{P}\mathrm{r}^{*}(x)\underline{=}\exists y(\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{v}\mathrm{T}(X, y)$
A
.
$\forall z\leq y(\urcorner \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{v}_{\mathrm{T}}(\mathrm{n}\mathrm{o}\mathrm{t}(X), z)))$
Theorem Rosser sentence is a sentence separating
Proof) We show (i)
implies that is the separating sentence.
(i)
is well-known as Rosser’s theorem.
(ii) By (1), we have
$\mathrm{T}$
$\varphi$
$\mathrm{T}\mu_{\varphi},$
$(\mathrm{i}\mathrm{i})\mathrm{T}\vdash \mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T})arrow\varphi,$
from T+Con(T).
Con(T), and then lemma 1
$(\mathrm{i}\mathrm{i}\mathrm{i})\mathrm{T}\mu_{\varphi}arrow$
$\varphi$
$\mathrm{T}\mu_{\varphi}$
$\mathrm{T}+\neg\varphi\vdash\exists y(\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{v}_{\mathrm{T}}(\ulcorner\urcorner\varphi, y))$
.
That is
$\mathrm{T}+\urcorner\varphi\vdash \mathrm{P}\mathrm{r}(\ulcorner\urcorner\varphi)$
(2)
.
On the other hand, we have
$\mathrm{T}\vdash \mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T})arrow\neg \mathrm{P}\mathrm{r}(\ulcorner\urcorner\varphi)$
$\mathrm{T}\vdash \mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T})arrow\urcorner \mathrm{P}\mathrm{r}(\ulcorner_{\neg\varphi^{\urcorner}})$
(3)
.
.
by formalized Rosser’s theorem. Frolll (2) and (3), we have
$\mathrm{T}+\urcorner\varphi\vdash_{\neg}\mathrm{c}^{1}\mathrm{o}\mathrm{n}(\mathrm{T})$
$\mathrm{T}\vdash\neg\varphiarrow\neg \mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T})$
,
,
$\mathrm{T}\vdash \mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T})arrow\varphi$
.
(4)
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(iii) Assume that
, we have
predicate
$\mathrm{T}\vdash\varphiarrow \mathrm{C}_{\mathrm{o}\mathrm{n}}(\mathrm{T})$
. By the derivability conditions of the provabiity
$\mathrm{P}\mathrm{r}()$
$\mathrm{T}\vdash \mathrm{P}\mathrm{r}(\ulcorner_{\neg \mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T}})arrow\neg\varphi^{\urcorner})$
,
$\mathrm{T}\vdash \mathrm{P}\mathrm{r}(\ulcorner_{\neg \mathrm{C}\mathrm{o}\mathrm{n}()^{\urcorner}}\mathrm{T})arrow \mathrm{P}\mathrm{r}(\ulcorner_{\urcorner}\varphi^{\urcorner})$
(5)
.
By (4) and (5),
$\mathrm{T}\vdash \mathrm{P}\mathrm{r}(\ulcorner_{\neg \mathrm{C}\mathrm{o}\mathrm{n}()^{\urcorner}}\mathrm{T})arrow\urcorner \mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T})$
.
Then we have
$\mathrm{T}\vdash\urcorner \mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{T})$
by L\"ob’s theorem, and by assumption
$\mathrm{T}\vdash\urcorner\varphi$
This contradicts that
$\varphi$
.
is Rosser sentence.
$\square$
References
[1] G. E. Boolos and R. C. Jeffery. Computability and Logic 3rd ed.
VERSITY PRESS 1989
.
CAMBRIDGE
UNI-
[2] C. Smorytski. The Incompleteness Theorems, Handbook of Math. Logic (J. Barwise
ed.). North-Holland 1977. pp.821-865