拉姆齐选择原则对 ZF$\mathsf {ZF}$ 的影响

Pub Date : 2024-07-06 DOI:10.1002/malq.202300024
Lorenz Halbeisen, Riccardo Plati, Saharon Shelah
{"title":"拉姆齐选择原则对 ZF$\\mathsf {ZF}$ 的影响","authors":"Lorenz Halbeisen,&nbsp;Riccardo Plati,&nbsp;Saharon Shelah","doi":"10.1002/malq.202300024","DOIUrl":null,"url":null,"abstract":"<p>The Ramsey Choice principle for families of <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>-element sets, denoted <span></span><math>\n <semantics>\n <msub>\n <mo>RC</mo>\n <mi>n</mi>\n </msub>\n <annotation>$\\operatorname{RC}_{n}$</annotation>\n </semantics></math>, states that every infinite set <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> has an infinite subset <span></span><math>\n <semantics>\n <mrow>\n <mi>Y</mi>\n <mo>⊆</mo>\n <mi>X</mi>\n </mrow>\n <annotation>$Y\\subseteq X$</annotation>\n </semantics></math> with a choice function on <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mrow>\n <mo>[</mo>\n <mi>Y</mi>\n <mo>]</mo>\n </mrow>\n <mi>n</mi>\n </msup>\n <mo>:</mo>\n <mo>=</mo>\n <mrow>\n <mo>{</mo>\n <mi>z</mi>\n <mo>⊆</mo>\n <mi>Y</mi>\n <mo>:</mo>\n <mo>|</mo>\n <mi>z</mi>\n <mo>|</mo>\n <mo>=</mo>\n <mi>n</mi>\n <mo>}</mo>\n </mrow>\n </mrow>\n <annotation>$[Y]^n:= \\lbrace z\\subseteq Y: |z| = n\\rbrace$</annotation>\n </semantics></math>. We investigate for which positive integers <span></span><math>\n <semantics>\n <mi>m</mi>\n <annotation>$m$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math> the implication <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mo>RC</mo>\n <mi>m</mi>\n </msub>\n <mo>⇒</mo>\n <msub>\n <mo>RC</mo>\n <mi>n</mi>\n </msub>\n </mrow>\n <annotation>$\\operatorname{RC}_{m} \\implies \\operatorname{RC}_{n}$</annotation>\n </semantics></math> is provable in <span></span><math>\n <semantics>\n <mi>ZF</mi>\n <annotation>$\\mathsf {ZF}$</annotation>\n </semantics></math>. It will turn out that beside the trivial implications <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mo>RC</mo>\n <mi>m</mi>\n </msub>\n <mo>⇒</mo>\n <msub>\n <mo>RC</mo>\n <mi>m</mi>\n </msub>\n </mrow>\n <annotation>$\\operatorname{RC}_{m} \\implies \\operatorname{RC}_{m}$</annotation>\n </semantics></math>, under the assumption that every odd integer <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>&gt;</mo>\n <mn>5</mn>\n </mrow>\n <annotation>$n&amp;gt;5$</annotation>\n </semantics></math> is the sum of three primes (known as ternary Goldbach conjecture), the only non-trivial implication which is provable in <span></span><math>\n <semantics>\n <mi>ZF</mi>\n <annotation>$\\mathsf {ZF}$</annotation>\n </semantics></math> is <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mo>RC</mo>\n <mn>2</mn>\n </msub>\n <mo>⇒</mo>\n <msub>\n <mo>RC</mo>\n <mn>4</mn>\n </msub>\n </mrow>\n <annotation>$\\operatorname{RC}_{2} \\implies \\operatorname{RC}_{4}$</annotation>\n </semantics></math>.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300024","citationCount":"0","resultStr":"{\"title\":\"Implications of Ramsey Choice principles in \\n \\n ZF\\n $\\\\mathsf {ZF}$\",\"authors\":\"Lorenz Halbeisen,&nbsp;Riccardo Plati,&nbsp;Saharon Shelah\",\"doi\":\"10.1002/malq.202300024\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Ramsey Choice principle for families of <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math>-element sets, denoted <span></span><math>\\n <semantics>\\n <msub>\\n <mo>RC</mo>\\n <mi>n</mi>\\n </msub>\\n <annotation>$\\\\operatorname{RC}_{n}$</annotation>\\n </semantics></math>, states that every infinite set <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> has an infinite subset <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Y</mi>\\n <mo>⊆</mo>\\n <mi>X</mi>\\n </mrow>\\n <annotation>$Y\\\\subseteq X$</annotation>\\n </semantics></math> with a choice function on <span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mrow>\\n <mo>[</mo>\\n <mi>Y</mi>\\n <mo>]</mo>\\n </mrow>\\n <mi>n</mi>\\n </msup>\\n <mo>:</mo>\\n <mo>=</mo>\\n <mrow>\\n <mo>{</mo>\\n <mi>z</mi>\\n <mo>⊆</mo>\\n <mi>Y</mi>\\n <mo>:</mo>\\n <mo>|</mo>\\n <mi>z</mi>\\n <mo>|</mo>\\n <mo>=</mo>\\n <mi>n</mi>\\n <mo>}</mo>\\n </mrow>\\n </mrow>\\n <annotation>$[Y]^n:= \\\\lbrace z\\\\subseteq Y: |z| = n\\\\rbrace$</annotation>\\n </semantics></math>. We investigate for which positive integers <span></span><math>\\n <semantics>\\n <mi>m</mi>\\n <annotation>$m$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math> the implication <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mo>RC</mo>\\n <mi>m</mi>\\n </msub>\\n <mo>⇒</mo>\\n <msub>\\n <mo>RC</mo>\\n <mi>n</mi>\\n </msub>\\n </mrow>\\n <annotation>$\\\\operatorname{RC}_{m} \\\\implies \\\\operatorname{RC}_{n}$</annotation>\\n </semantics></math> is provable in <span></span><math>\\n <semantics>\\n <mi>ZF</mi>\\n <annotation>$\\\\mathsf {ZF}$</annotation>\\n </semantics></math>. It will turn out that beside the trivial implications <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mo>RC</mo>\\n <mi>m</mi>\\n </msub>\\n <mo>⇒</mo>\\n <msub>\\n <mo>RC</mo>\\n <mi>m</mi>\\n </msub>\\n </mrow>\\n <annotation>$\\\\operatorname{RC}_{m} \\\\implies \\\\operatorname{RC}_{m}$</annotation>\\n </semantics></math>, under the assumption that every odd integer <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>&gt;</mo>\\n <mn>5</mn>\\n </mrow>\\n <annotation>$n&amp;gt;5$</annotation>\\n </semantics></math> is the sum of three primes (known as ternary Goldbach conjecture), the only non-trivial implication which is provable in <span></span><math>\\n <semantics>\\n <mi>ZF</mi>\\n <annotation>$\\\\mathsf {ZF}$</annotation>\\n </semantics></math> is <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mo>RC</mo>\\n <mn>2</mn>\\n </msub>\\n <mo>⇒</mo>\\n <msub>\\n <mo>RC</mo>\\n <mn>4</mn>\\n </msub>\\n </mrow>\\n <annotation>$\\\\operatorname{RC}_{2} \\\\implies \\\\operatorname{RC}_{4}$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-07-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300024\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300024\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300024","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

元素集合族的拉姆齐选择原理(表示为 )指出,每个无限集合都有一个无限子集,其上有一个选择函数.我们将研究哪些正整数的蕴涵可以在 .中证明,除了三元蕴涵之外,在每个奇整数都是三个素数之和(称为三元哥德巴赫猜想)的假设下,唯一可以在 .中证明的非三元蕴涵是 .。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
分享
查看原文
Implications of Ramsey Choice principles in ZF $\mathsf {ZF}$

The Ramsey Choice principle for families of n $n$ -element sets, denoted RC n $\operatorname{RC}_{n}$ , states that every infinite set X $X$ has an infinite subset Y X $Y\subseteq X$ with a choice function on [ Y ] n : = { z Y : | z | = n } $[Y]^n:= \lbrace z\subseteq Y: |z| = n\rbrace$ . We investigate for which positive integers m $m$ and n $n$ the implication RC m RC n $\operatorname{RC}_{m} \implies \operatorname{RC}_{n}$ is provable in  ZF $\mathsf {ZF}$ . It will turn out that beside the trivial implications RC m RC m $\operatorname{RC}_{m} \implies \operatorname{RC}_{m}$ , under the assumption that every odd integer n > 5 $n&gt;5$ is the sum of three primes (known as ternary Goldbach conjecture), the only non-trivial implication which is provable in ZF $\mathsf {ZF}$ is RC 2 RC 4 $\operatorname{RC}_{2} \implies \operatorname{RC}_{4}$ .

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信