591 名前：}}}}, ・... sets represented with 6}6 bracket pairs, e.g. {{{{{{}}}}}}, ・... sets represented with 7}7 bracket pairs, e.g. {{{{{{{}}}}}}}, ・... sets represented with 8}8 bracket pairs, e.g. {{{{{{{{}}}}}}}} or {{},{{}},{{},{{}}}} (i.e. {0,1,2}, the Neumann ordinal "3") ・... etc. In this way, the number of sets with n bracket pairs is[1] 1,1,1,2,3,6,12,25,52,113,247,548,1226,2770,6299,14426,・・・
Axiomatizations Theories of finite sets
ZF
See also Hereditary set Hereditarily countable set Hereditary property Rooted trees Constructive set theory Finite set []
Addition The first transfinite ordinal is ω, the set of all natural numbers. For example, the ordinal ω + ω is obtained by two copies of the natural numbers ordered in the usual fashion and the second copy completely to the right of the first. Writing 0' < 1' < 2' < ... for the second copy, ω + ω looks like 0 < 1 < 2 < 3 < ... < 0' < 1' < 2' < ... This is different from ω because in ω only 0 does not have a direct predecessor while in ω + ω the two elements 0 and 0' do not have direct predecessors.
Multiplication Here is ω・2: 00 < 10 < 20 < 30 < ... < 01 < 11 < 21 < 31 < ..., which has the same order type as ω + ω.
Exponentiation For instance, ω^2 = ω・ω using the operation of ordinal multiplication. Note that ω・ω can be defined using the set of functions from 2 = {0,1} to ω = {0,1,2,...}, ordered lexicographically with the least significant position first: (0,0) < (1,0) < (2,0) < (3,0) < ... < (0,1) < (1,1) < (2,1) < (3,1) < ... < (0,2) < (1,2) < (2,2) < ... Here for brevity, we have replaced the function {(0,k), (1,m)} by the ordered pair (k, m). (引用終り) 以上
596１３２人目の素数さん2022/01/28(金) 10:20:39.06ID:XHv+DeMU 594 >This is different from ω because in ω only 0 does not have a direct predecessor while in ω + ω the two elements 0 and 0' do not have direct predecessors. しっかり書いてありますね 。0'の前者は無いと。英語読めますか？
