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過去の記録 ～04/24｜次回の予定｜今後の予定 04/25～
| 担当者 | 新井 敏康 |
|---|
過去の記録
2019年12月20日(金)
13:00-14:30 数理科学研究科棟(駒場) 156号室
池上 大祐 氏 (芝浦工業大学)
On supercompactness of $\omega_1$
池上 大祐 氏 (芝浦工業大学)
On supercompactness of $\omega_1$
[ 講演概要 ]
In ZFC, all the large cardinals are much bigger than $\omega_1$, the least uncountable cardinal,
while without assuming the Axiom of Choice, $\omega_1$ could have some large cardinal properties.
Jech and Takeuti independently proved that if the axiom system ZFC + There is a measurable cardinal is consistent,
then so is ZF + $\omega_1$ is a measurable cardinal.
Takeuti also proved that one can replace "measurable cardinal" above with "supercompact cardinal" as well as some other large cardinals.
Woodin proved that one can reduce the assumption, i.e., the consistency of ZFC + a supercompact cardinal,
to that of ZFC + There are proper class many Woodin cardinals which are limits of Woodin cardinals,
to obtain the consistency of ZF + $\omega_1$ is a supercompact cardinal.
Furthermore, the model he constructed also satisfies the Axiom of Determinacy (AD).
In this talk, after giving some background on the connections between large cardinals and determinacy, we discuss some consequences of the axiom system ZF + $\omega_1$ is a supercompact cardinal.
This is joint work with Nam Trang.
In ZFC, all the large cardinals are much bigger than $\omega_1$, the least uncountable cardinal,
while without assuming the Axiom of Choice, $\omega_1$ could have some large cardinal properties.
Jech and Takeuti independently proved that if the axiom system ZFC + There is a measurable cardinal is consistent,
then so is ZF + $\omega_1$ is a measurable cardinal.
Takeuti also proved that one can replace "measurable cardinal" above with "supercompact cardinal" as well as some other large cardinals.
Woodin proved that one can reduce the assumption, i.e., the consistency of ZFC + a supercompact cardinal,
to that of ZFC + There are proper class many Woodin cardinals which are limits of Woodin cardinals,
to obtain the consistency of ZF + $\omega_1$ is a supercompact cardinal.
Furthermore, the model he constructed also satisfies the Axiom of Determinacy (AD).
In this talk, after giving some background on the connections between large cardinals and determinacy, we discuss some consequences of the axiom system ZF + $\omega_1$ is a supercompact cardinal.
This is joint work with Nam Trang.
2019年11月21日(木)
13:30-15:00 数理科学研究科棟(駒場) 156号室
佐藤憲太郎 氏
Self-referential Theorems for Finitist Arithmetic
佐藤憲太郎 氏
Self-referential Theorems for Finitist Arithmetic
[ 講演概要 ]
The finitist logic excludes,on the syntax level, unbounded quantifiers
and accommodates only bounded quantifiers.
The following two self-referential theorems for arithmetic theories
over the finitist logic will be considered:
Tarski's impossibility of naive truth predicate and
Goedel's incompleteness theorem.
Particularly, it will be briefly explained that
(i) the naive truth theory over the finitist arithmetic with summation and multiplication
is consistent and proves its own consistency, and that
(ii) by the use of finitist arithmetic, the hierarchy of consistency strengths,
based on Goedel's second incompleteness theorem,
can be extended downward (to the area not reachable by first order predicate arithmetic).
This is a joint work with Jan Walker, and overlaps significantly with his doctoral dissertation.
The finitist logic excludes,on the syntax level, unbounded quantifiers
and accommodates only bounded quantifiers.
The following two self-referential theorems for arithmetic theories
over the finitist logic will be considered:
Tarski's impossibility of naive truth predicate and
Goedel's incompleteness theorem.
Particularly, it will be briefly explained that
(i) the naive truth theory over the finitist arithmetic with summation and multiplication
is consistent and proves its own consistency, and that
(ii) by the use of finitist arithmetic, the hierarchy of consistency strengths,
based on Goedel's second incompleteness theorem,
can be extended downward (to the area not reachable by first order predicate arithmetic).
This is a joint work with Jan Walker, and overlaps significantly with his doctoral dissertation.
2019年10月24日(木)
13:30-15:00 数理科学研究科棟(駒場) 156号室
大川 裕矢 氏 (千葉大学)
部分保存性に対する，Bennet の結果の一般化について (JAPANESE)
大川 裕矢 氏 (千葉大学)
部分保存性に対する，Bennet の結果の一般化について (JAPANESE)
[ 講演概要 ]
文 $\varphi$ が理論 $T$ 上 $\Gamma$-保存的であるとは,
任意の $\Gamma$ 文 $\psi$ について,
$T + \varphi \vdash \psi$ ならば $T \vdash \psi$ が成立することをいう.
1979 年 Guaspari は複数の理論に対して,
同時に $\Gamma$-保存的であり,
各理論では証明できない文の存在に関する部分的な議論を行ったが,
その一般的な状況を解明するという問いを残していた.
この問いに対し, 1986年 Bennet は特に2つの理論に対する分析を行い,
存在条件をある程度特徴付けることに成功した.
今回木更津工業高等専門学校の倉橋太志講師との共同研究により ,
この Bennet の結果は任意有限個の理論に拡張可能であることが判明した.
本講演ではその拡張した結果を紹介する.
文 $\varphi$ が理論 $T$ 上 $\Gamma$-保存的であるとは,
任意の $\Gamma$ 文 $\psi$ について,
$T + \varphi \vdash \psi$ ならば $T \vdash \psi$ が成立することをいう.
1979 年 Guaspari は複数の理論に対して,
同時に $\Gamma$-保存的であり,
各理論では証明できない文の存在に関する部分的な議論を行ったが,
その一般的な状況を解明するという問いを残していた.
この問いに対し, 1986年 Bennet は特に2つの理論に対する分析を行い,
存在条件をある程度特徴付けることに成功した.
今回木更津工業高等専門学校の倉橋太志講師との共同研究により ,
この Bennet の結果は任意有限個の理論に拡張可能であることが判明した.
本講演ではその拡張した結果を紹介する.
