https://en.wikipedia.org/wiki/Hereditarily_finite_set Hereditarily finite set In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set.
Representation This class of sets is naturally ranked by the number of bracket pairs necessary to represent the sets: ・{} (i.e. Φ , the Neumann ordinal "0"), ・{{}} (i.e. {Φ } or {0}, the Neumann ordinal "1"), ・{{{}}}, ・{{{{}}}} and then also {{},{{}}} (i.e. {0,1}, the Neumann ordinal "2"), ・{{{{{}}}}}, {{{},{{}}}} as well as {{},{{{
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？
