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DcN

2025-07-23T09:11:27Z

318`4😥：

DcN，是318`4创造的便于打字的记号，主要用于写FOS中作为序数的项，几乎不可能良定义，主要通过枚举列表来主观推算定义。DcN的极限是<math>\varphi(10,0)</math>，Y_cpper将其扩展到<math>\Gamma_0</math>，现在一般在<math>\varphi(\omega,0)</math>之后使用这个扩展。DcN的所有符号都集中在电脑键盘左上方的一小片区域，qwerty13456789，456789使用频率很低，Veblen记号需要随时要输入括号下标希腊字母，1-Y则有比DcN更长的常见表达式，所以DcN便于打字，是专为FOS中常见序数设计的记号；但因为可读性差，DcN一直没能流传开来。

因为完全无r的DcN存在歧义，比如wtw1wtww可以是<math>\omega^{\omega^2}+\omega^\omega</math>，也可以是<math>\omega^{\omega^{\omega^2}+\omega}</math>，因此在表达FOS的项时，建议使用含r的DcN，但是去掉所有位于表达式末尾的r，如<math>\omega^{\omega^{\omega^\omega}}</math>=wtwtwtwrrr记作wtwtwtw。

1-Y不必多说，是Yukito在2019年创造的序列记号，首次引入山脉图，在Y(1,3)之前有非常优秀的性质，因此也常用于表达FOS中作为项的序数。不过为简便，写1-Y表达式时通常省略Y( )外壳和逗号，超过10的数字用拉丁字母表示，超过36则无写法。

在研究FOS的定义、构造原理时，使用1-Y的情况更多，因为1-Y的山脉结构和FOS中项内部的山脉结构类似；而在研究FOS的分析、强度形成原理时，为简便起见，使用DcN更多。在阅读FOS的枚举列表之前，请务必将本文的枚举列表熟透于心中，否则用Veblen记号或OCF写FOS表达式费时费力。
{| class="wikitable"
|+
!序数（十进制+Cantor记号+Veblen记号）
!DcN
!1-Y
|-
|<math>0</math>
|0
|<math>\varnothing</math>
|-
|<math>1</math>
|1
|1
|-
|<math>2</math>
|11
|11
|-
|<math>3</math>
|111
|111
|-
|<math>\omega</math>
|w
|12
|-
|<math>\omega+1</math>
|1w
|121
|-
|<math>\omega+2</math>
|11w
|1211
|-
|<math>\omega2</math>
|w1
|1212
|-
|<math>\omega2+1</math>
|1w1
|12121
|-
|<math>\omega3</math>
|w11
|121212
|-
|<math>\omega3+1</math>
|1w11
|1212121
|-
|<math>\omega^2</math>
|ww
|122
|-
|<math>\omega^2+1</math>
|1ww
|1221
|-
|<math>\omega^2+\omega</math>
|w1ww
|12212
|-
|<math>\omega^2+\omega+1</math>
|1w1ww
|122121
|-
|<math>\omega^2+\omega2</math>
|w11ww
|1221212
|-
|<math>\omega^2+\omega2+1</math>
|1w11ww
|12212121
|-
|<math>\omega^2+\omega3</math>
|w111ww
|122121212
|-
|<math>\omega^22</math>
|ww1
|122122
|-
|<math>\omega^22+\omega</math>
|w1ww1
|12212212
|-
|<math>\omega^22+\omega2</math>
|w11ww1
|1221221212
|-
|<math>\omega^23</math>
|ww11
|122122122
|-
|<math>\omega^3</math>
|www
|1222
|-
|<math>\omega^3+\omega</math>
|w1www
|122212
|-
|<math>\omega^3+\omega^2</math>
|ww1www
|1222122
|-
|<math>\omega^3+\omega^2+1</math>
|1ww1www
|12221221
|-
|<math>\omega^3+\omega^2+\omega</math>
|w1ww1www
|122212212
|-
|<math>\omega^3+\omega^2+\omega+1</math>
|1w1ww1www
|1222122121
|-
|<math>\omega^3+\omega^2+\omega2</math>
|w11ww1www
|12221221212
|-
|<math>\omega^3+\omega^22+\omega</math>
|w1ww11www
|122212212212
|-
|<math>\omega^3+\omega^22+\omega2+1</math>
|1w11ww11www
|122212212212121
|-
|<math>\omega^32</math>
|www1
|12221222
|-
|<math>\omega^4</math>
|wwww
|12222
|-
|<math>\omega^5</math>
|wwwww
|122222
|-
|<math>\omega^\omega</math>
|wtw
|123
|-
|<math>\omega^\omega+1</math>
|1wtw
|1231
|-
|<math>\omega^\omega+\omega</math>
|w1wtw
|12312
|-
|<math>\omega^\omega+\omega2</math>
|w11wtw
|1231212
|-
|<math>\omega^\omega+\omega^2</math>
|ww1wtw
|123122
|-
|<math>\omega^\omega2</math>
|w1tw
|123123
|-
|<math>\omega^\omega2+\omega^2</math>
|ww1w1tw
|123123122
|-
|<math>\omega^\omega3</math>
|w11tw
|123123123
|-
|<math>\omega^{\omega+1}</math>
|wt1w
|1232
|-
|<math>\omega^{\omega+1}+\omega^2</math>
|ww1wt1w
|1232122
|-
|<math>\omega^{\omega+1}+\omega^\omega</math>
|wtwr1wt1w
|1232123
|-
|<math>\omega^{\omega+1}+\omega^\omega2</math>
|w1twr1wt1w
|1232123123
|-
|<math>\omega^{\omega+1}2+\omega^\omega</math>
|wtwr1w1t1w
|12321232123
|-
|<math>\omega^{\omega+1}2+\omega^\omega2</math>
|w1twr1w1t1w
|12321232123123
|-
|<math>\omega^{\omega+1}3</math>
|w11t1w
|123212321232
|-
|<math>\omega^{\omega+2}</math>
|wt11w
|12322
|-
|<math>\omega^{\omega+3}</math>
|wt111w
|123222
|-
|<math>\omega^{\omega2}</math>
|wtw1
|12323
|-
|<math>\omega^{\omega2+1}</math>
|wt1w1
|123232
|-
|<math>\omega^{\omega2+1}+\omega^{\omega2}</math>
|wtw1r1wt1w1
|12323212323
|-
|<math>\omega^{\omega2+1}+\omega^{\omega2}2</math>
|w1tw1r1wt1w1
|1232321232312323
|-
|<math>\omega^{\omega2+1}2+\omega^{\omega2}2</math>
|w1tw1r1w1t1w1
|1232321232321232312323
|-
|<math>\omega^{\omega2+1}2+\omega^{\omega2}2+\omega^{\omega+1}2+\omega^{\omega}2</math>
|w1twr1w1t1wr1w1tw1r1w1t1w1
|123232123232123231232312321232123123
|-
|<math>\omega^{\omega2+2}</math>
|wt11w1
|1232322
|-
|<math>\omega^{\omega2+2}+\omega^{\omega2+1}</math>
|wt1w1r1wt11w1
|1232322123232
|-
|<math>\omega^{\omega3}</math>
|wtw11
|1232323
|-
|<math>\omega^{\omega3+1}+\omega^{\omega3}</math>
|wtw11r1wt1w11
|123232321232323
|-
|<math>\omega^{\omega4}</math>
|wtw111
|123232323
|-
|<math>\omega^{\omega^2}</math>
|wtww
|1233
|-
|<math>\omega^{\omega^2+\omega+1}+\omega^{\omega^2+\omega}</math>
|wtw1wwr1wt1w1ww
|1233232123323
|-
|<math>\omega^{\omega^22}</math>
|wtww1
|1233233
|-
|<math>\omega^{\omega^3}</math>
|wtwww
|12333
|-
|<math>\omega^{\omega^\omega}</math>
|wtwtw
|1234
|-
|<math>\omega^{\omega^{\omega2+1}3}</math>
|wtw11t1w1
|1234343234343234343
|-
|<math>\omega^{\omega^{\omega2+2}}</math>
|wtwt11w1
|12343433
|-
|<math>\omega^{\omega^{\omega3}}</math>
|wtwtw11
|12343434
|-
|<math>\omega^{\omega^{\omega^2}}</math>
|wtwtww
|12344
|-
|<math>\omega^{\omega^{\omega^\omega}}</math>
|wtwtwtw
|12345
|-
|<math>\varepsilon_0</math>
|e
|124
|-
|<math>\varepsilon_0+\omega</math>
|w1e
|12412
|-
|<math>\varepsilon_0+\omega^\omega</math>
|wtwr1e
|124123
|-
|<math>\varepsilon_02</math>
|e1
|124124
|-
|<math>\varepsilon_02+\omega</math>
|w1e1
|12412412
|-
|<math>\varepsilon_0\omega=\omega^{\varepsilon_0+1}</math>
|wt1e
|1242
|-
|<math>\omega^{\varepsilon_0+1}+\varepsilon_0</math>
|e1wt1e
|1242124
|-
|<math>\omega^{\varepsilon_0+1}2</math>
|w1t1e
|12421242
|-
|<math>\omega^{\varepsilon_0+2}</math>
|wt11e
|12422
|-
|<math>\omega^{\varepsilon_0+\omega}</math>
|wtw1e
|12423
|-
|<math>\omega^{\varepsilon_0+\omega^\omega}</math>
|wtwtwr1e
|124234
|-
|<math>\omega^{\varepsilon_02}</math>
|wte1
|12424
|-
|<math>\omega^\omega^{\varepsilon_0+1}</math>
|wtwt1e
|1243
|-
|<math>\omega^\omega^{\varepsilon_02}</math>
|wtwte1
|12435
|-
|<math>\varepsilon_1</math>
|ee
|1244
|-
|<math>\omega^{\varepsilon_12}</math>
|wtee1
|1244244
|-
|<math>\varepsilon_2</math>
|eee
|12444
|-
|<math>\varepsilon_\omega</math>
|etw
|1245
|-
|<math>\varepsilon_\omega+\omega^\omega</math>
|wtwr1etw
|1245123
|-
|<math>\varepsilon_\omega+\varepsilon_0</math>
|e1etw
|1245124
|-
|<math>\varepsilon_\omega2</math>
|e1tw
|12451245
|-
|<math>\omega^{\varepsilon_\omega+1}</math>
|wt1etw
|12452
|-
|<math>\omega^{\varepsilon_\omega+\omega}</math>
|wtw1etw
|124523
|-
|<math>\omega^{\varepsilon_\omega2}</math>
|wte1tw
|1245245
|-
|<math>\varepsilon_{\omega+1}</math>
|et1w
|12454
|-
|<math>\varepsilon_{\omega2}</math>
|etw1
|124545
|-
|<math>\varepsilon_{\omega^\omega}</math>
|etwtw
|12456
|-
|<math>\varepsilon_{\varepsilon_0}</math>
|ete
|12457
|-
|<math>\varepsilon_{\varepsilon_0+1}</math>
|et1e
|12457
|-
|<math>\varepsilon_{\varepsilon_1}</math>
|etee
|12457
|-
|<math>\varepsilon_{\varepsilon_{\varepsilon_0}}</math>
|etete
|124578A
|-
|<math>\zeta_0</math>
|y
|1246
|-
|<math>\zeta_0+\varepsilon_0</math>
|e1y
|1246124
|-
|<math>\zeta_0+\varepsilon_{\varepsilon_0}</math>
|eter1y
|124612457
|-
|<math>\zeta_02</math>
|y1
|12461246
|-
|<math>\omega^{\zeta_0+1}</math>
|wt1y
|12462
|-
|<math>\omega^{\zeta_0+\varepsilon_\omega}</math>
|wtetwr1y
|1246245
|-
|<math>\omega^{\zeta_02}</math>
|wty1
|1246246
|-
|<math>\varepsilon_{\zeta_0+1}</math>
|ye=et1y
|12464
|-
|<math>\omega^{\varepsilon_{\zeta_0+1}+1}</math>
|wt1ye=wt1et1y
|124642
|-
|<math>\varepsilon_{\zeta_0+2}</math>
|yee=et11y
|124644
|-
|<math>\varepsilon_{\zeta_0+3}</math>
|yeee=et111y
|1246444
|-
|<math>\varepsilon_{\zeta_0+\omega}</math>
|etw1y
|124645
|-
|<math>\varepsilon_{\zeta_0+\omega^\omega}</math>
|etwtwr1y
|1246456
|-
|<math>\varepsilon_{\zeta_0+\varepsilon_0}</math>
|ete1y
|1246457
|-
|<math>\varepsilon_{\zeta_02}</math>
|ety1
|12464579
|-
|<math>\varepsilon_{\zeta_02+1}</math>
|et1y1
|124645794
|-
|<math>\varepsilon_{\zeta_03}</math>
|ety11
|124645794579
|-
|<math>\varepsilon_{\omega^{\zeta_0+1}}</math>
|etwt1y
|124645795
|-
|<math>\varepsilon_{\varepsilon_{\zeta_0+1}}</math>
|etet1y
|124645797
|-
|<math>\zeta_1</math>
|yy
|124646
|-
|<math>\varepsilon_{\zeta_1+1}</math>
|et1yy=yye
|1246464
|-
|<math>\zeta_2</math>
|yyy
|12464646
|-
|<math>\zeta_{\omega}</math>
|ytw
|12465
|-
|<math>\omega^{\zeta_{\omega}2}</math>
|wty1tw
|124652465
|-
|<math>\varepsilon_{\zeta_{\omega}+1}</math>
|et1ytw
|124654
|-
|<math>\varepsilon_{\zeta_{\omega}+\omega^{\varepsilon_0+1}}</math>
|etwt1er1ytw
|124654575
|-
|<math>\varepsilon_{\zeta_{\omega}+\varepsilon_\omega}</math>
|etetwr1ytw
|124654578
|-
|<math>\varepsilon_{\zeta_{\omega}+\zeta_0}</math>
|ety1ytw
|124654579
|-
|<math>\varepsilon_{\varepsilon_{\zeta_{\omega}2}}</math>
|etety1tw
|124654579878ACB
|-
|<math>\zeta_{\omega+1}</math>
|yt1w
|1246546
|-
|<math>\zeta_{\omega2}</math>
|ytw1
|124655
|-
|<math>\zeta_{\varepsilon_0}</math>
|yte
|124657
|-
|<math>\zeta_{\zeta_0}</math>
|yty
|1246579
|-
|<math>\eta_0</math>
|3
|12466
|-
|<math>\omega^{\eta_0+1}</math>
|wt13
|124662
|-
|<math>\omega^{\eta_0+\varepsilon_0}</math>
|wte13
|1246624
|-
|<math>\omega^{\eta_0+\zeta_0}</math>
|wty13
|12466246
|-
|<math>\omega^{\eta_02}</math>
|wt31
|124662466
|-
|<math>\varepsilon_{\eta_0+1}</math>
|3e=et13
|124664
|-
|<math>\varepsilon_{\eta_0+\omega}</math>
|etw13
|1246645
|-
|<math>\varepsilon_{\eta_0+\varepsilon_0}</math>
|ete13
|12466457
|-
|<math>\varepsilon_{\eta_0+\zeta_0}</math>
|ety13
|124664579
|-
|<math>\varepsilon_{\eta_02}</math>
|et31
|1246645799
|-
|<math>\zeta_{\eta_0+1}</math>
|3y=yt13
|1246646
|-
|<math>\zeta_{\eta_0+\omega}</math>
|ytw13
|12466465
|-
|<math>\zeta_{\eta_0+\varepsilon_0}</math>
|yte13
|124664657
|-
|<math>\zeta_{\eta_0+\zeta_0}</math>
|yty13
|1246646579
|-
|<math>\zeta_{\eta_02}</math>
|yt31
|12466465799
|-
|<math>\eta_1</math>
|33
|12466466
|-
|<math>\eta_\omega</math>
|3tw
|124665
|-
|<math>\eta_{\eta_0}</math>
|3t3
|124665799
|-
|<math>\varphi(4,0)</math>
|4
|124666
|-
|<math>\omega^{\varphi(4,0)2}</math>
|wt41
|1246662466
|-
|<math>\varepsilon_{\varphi(4,0)+1}</math>
|et41
|12466645799
|-
|<math>\zeta_{\varphi(4,0)+1}</math>
|yt41
|124666465799
|-
|<math>\eta_{\varphi(4,0)+1}</math>
|3t41
|1246664665799
|-
|<math>\varphi(4,1)</math>
|44
|1246664666
|-
|<math>\varphi(5,0)</math>
|5
|1246666
|-
|<math>\varphi(6,0)</math>
|6
|12466666
|-
|<math>\varphi(7,0)</math>
|7
|124666666
|-
|<math>\varphi(8,0)</math>
|8
|1246666666
|-
|<math>\varphi(4,\varphi(8,0)+1)+\eta_0</math>
|3184=314t18
|1246666666466657999999912466
|-
|<math>\varphi(9,0)</math>
|9
|12466666666
|-
|<math>\varphi(9,\varphi(9,0))</math>
|9t9
|124666666665799999999
|-
|<math>\varphi(10,0)</math>
|<math>\mathrm{limit}</math>
|124666666666
|+
!序数（十进制+Cantor记号+Veblen记号）
!扩展DcN
!1-Y
|-
|<math>\varphi(10,0)</math>
|q1111111111r
|124666666666
|-
|<math>\varphi(11,0)</math>
|q11111111111r
|1246666666666
|-
|<math>\varphi(\omega,0)</math>
|qwr
|12467
|-
|<math>\varepsilon_{\varphi(\omega,0)+1}</math>
|et1qwr=qwre

=q1rt1qwr=qwrq1r
|124674
|-
|<math>\varphi(\omega,1)</math>
|qwrqwr
|12467467
|-
|<math>\varphi(\omega,\omega)</math>
|qwrtw
|124675
|-
|<math>\varphi(\omega+1,0)</math>
|q1wr
|124676
|-
|<math>\varphi(\omega2,0)</math>
|qw1r
|1246767
|-
|<math>\varphi(\varepsilon_0,0)</math>
|qer
|124679
|-
|<math>\varphi(\varepsilon_0+\omega,\zeta_0)</math>
|qw1erty（qwerty）
|12467967579
|-
|<math>\varphi(\zeta_0,0)</math>
|qyr
|124679B
|-
|<math>\varphi(\varphi(\omega,0),0)</math>
|qqwrr
|124679BC
|-
|<math>\Gamma_0</math>
|扩展极限
|12468
|-
|}
{| class="wikitable"
|+
!序数（十进制+BOCF）
!1-Y
|-
|<math>\psi(\Omega^{\Omega})</math>
|12468
|-
|<math>\psi(\Omega^{\Omega}+1)</math>
|124682
|-
|<math>\psi(\Omega^{\Omega}+\psi(\Omega^{\Omega}))</math>
|124682468
|-
|<math>\psi(\Omega^{\Omega}+\Omega)</math>
|124684
|-
|<math>\psi(\Omega^{\Omega}+\Omega^\omega)</math>
|12468467
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\omega)})</math>
|124684679AB
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\Omega)})</math>
|124684679AC
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\Omega)+1})</math>
|124684679AC6
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\Omega+1)})</math>
|124684679AC7
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\Omega+\Omega)})</math>
|124684679AC9
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\Omega+\Omega^\omega)})</math>
|124684679AC9AB
|-
|<math>\psi(\Omega^{\Omega}2)</math>
|12468468
|-
|<math>\psi(\Omega^{\Omega+1})</math>
|124686
|-
|<math>\psi(\Omega^{\Omega+\psi(\Omega^{\Omega})})</math>
|12468679BD
|-
|<math>\psi(\Omega^{\Omega2})</math>
|1246868
|-
|<math>\psi(\Omega^{\Omega\psi(\Omega^{\Omega})})</math>
|1246879BD
|-
|<math>\psi(\Omega^{\Omega^2})</math>
|124688
|-
|<math>\psi(\Omega^{\Omega^\omega})</math>
|124689
|-
|<math>\psi(\Omega^{\Omega^{\psi(\Omega^{\Omega^\omega})}})</math>
|124689BDFG
|-
|<math>\psi(\Omega^{\Omega^\Omega})</math>
|12468A
|-
|<math>\psi(\Omega_2)</math>
|1247
|-
|<math>\psi(\Omega_2+\Omega^\Omega)</math>
|1247468
|-
|<math>\psi(\Omega_2+\psi_1(\Omega_2))</math>
|124747
|-
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\Omega))</math>
|12476
|-
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\Omega^\Omega))</math>
|124768
|-
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))</math>
|124769
|-
|<math>\psi(\Omega_22)</math>
|12477
|-
|<math>\psi(\Omega_2\omega)</math>
|12478
|-
|<math>\psi(\Omega_2\Omega)</math>
|12479
|-
|<math>\psi(\Omega_2^2)</math>
|1247A
|-
|<math>\psi(\Omega_2^{\Omega_2})</math>
|1247AD
|-
|<math>\psi(\Omega_3)</math>
|1247B
|-
|<math>\psi(\Omega_4)</math>
|1247BG
|-
|<math>\psi(\Omega_\omega)</math>
|1248
|-
|}
再往后涉及项嵌套的提升效应，目前对此研究不清晰，故枚举列表暂时到此为止。

DcN

2025-07-23T09:09:31Z

318`4😥：

DcN，是318`4创造的便于打字的记号，主要用于写FOS中作为序数的项，几乎不可能良定义，主要通过枚举列表来主观推算定义。DcN的极限是<math>\varphi(10,0)</math>，Y_cpper将其扩展到<math>\Gamma_0</math>，现在一般在<math>\varphi(\omega,0)</math>之后使用这个扩展。DcN的所有符号都集中在电脑键盘左上方的一小片区域，qwerty13456789，456789使用频率很低，Veblen记号需要随时要输入括号下标希腊字母，1-Y则有比DcN更长的常见表达式，所以DcN便于打字，是专为FOS中常见序数设计的记号；但因为可读性差，DcN一直没能流传开来。

因为完全无r的DcN存在歧义，比如wtw1wtww可以是<math>\omega^{\omega^2}+\omega^\omega</math>，也可以是<math>\omega^{\omega^{\omega^2}+\omega}</math>，因此在表达FOS的项时，建议使用含r的DcN，但是去掉所有位于表达式末尾的r，如\omega^{\omega^{\omega^\omega}=wtwtwtwrrr记作wtwtwtw。

1-Y不必多说，是Yukito在2019年创造的序列记号，首次引入山脉图，在Y(1,3)之前有非常优秀的性质，因此也常用于表达FOS中作为项的序数。不过为简便，写1-Y表达式时通常省略Y( )外壳和逗号，超过10的数字用拉丁字母表示，超过36则无写法。

在研究FOS的定义、构造原理时，使用1-Y的情况更多，因为1-Y的山脉结构和FOS中项内部的山脉结构类似；而在研究FOS的分析、强度形成原理时，为简便起见，使用DcN更多。在阅读FOS的枚举列表之前，请务必将本文的枚举列表熟透于心中，否则用Veblen记号或OCF写FOS表达式费时费力。
{| class="wikitable"
|+
!序数（十进制+Cantor记号+Veblen记号）
!DcN
!1-Y
|-
|<math>0</math>
|0
|<math>\varnothing</math>
|-
|<math>1</math>
|1
|1
|-
|<math>2</math>
|11
|11
|-
|<math>3</math>
|111
|111
|-
|<math>\omega</math>
|w
|12
|-
|<math>\omega+1</math>
|1w
|121
|-
|<math>\omega+2</math>
|11w
|1211
|-
|<math>\omega2</math>
|w1
|1212
|-
|<math>\omega2+1</math>
|1w1
|12121
|-
|<math>\omega3</math>
|w11
|121212
|-
|<math>\omega3+1</math>
|1w11
|1212121
|-
|<math>\omega^2</math>
|ww
|122
|-
|<math>\omega^2+1</math>
|1ww
|1221
|-
|<math>\omega^2+\omega</math>
|w1ww
|12212
|-
|<math>\omega^2+\omega+1</math>
|1w1ww
|122121
|-
|<math>\omega^2+\omega2</math>
|w11ww
|1221212
|-
|<math>\omega^2+\omega2+1</math>
|1w11ww
|12212121
|-
|<math>\omega^2+\omega3</math>
|w111ww
|122121212
|-
|<math>\omega^22</math>
|ww1
|122122
|-
|<math>\omega^22+\omega</math>
|w1ww1
|12212212
|-
|<math>\omega^22+\omega2</math>
|w11ww1
|1221221212
|-
|<math>\omega^23</math>
|ww11
|122122122
|-
|<math>\omega^3</math>
|www
|1222
|-
|<math>\omega^3+\omega</math>
|w1www
|122212
|-
|<math>\omega^3+\omega^2</math>
|ww1www
|1222122
|-
|<math>\omega^3+\omega^2+1</math>
|1ww1www
|12221221
|-
|<math>\omega^3+\omega^2+\omega</math>
|w1ww1www
|122212212
|-
|<math>\omega^3+\omega^2+\omega+1</math>
|1w1ww1www
|1222122121
|-
|<math>\omega^3+\omega^2+\omega2</math>
|w11ww1www
|12221221212
|-
|<math>\omega^3+\omega^22+\omega</math>
|w1ww11www
|122212212212
|-
|<math>\omega^3+\omega^22+\omega2+1</math>
|1w11ww11www
|122212212212121
|-
|<math>\omega^32</math>
|www1
|12221222
|-
|<math>\omega^4</math>
|wwww
|12222
|-
|<math>\omega^5</math>
|wwwww
|122222
|-
|<math>\omega^\omega</math>
|wtw
|123
|-
|<math>\omega^\omega+1</math>
|1wtw
|1231
|-
|<math>\omega^\omega+\omega</math>
|w1wtw
|12312
|-
|<math>\omega^\omega+\omega2</math>
|w11wtw
|1231212
|-
|<math>\omega^\omega+\omega^2</math>
|ww1wtw
|123122
|-
|<math>\omega^\omega2</math>
|w1tw
|123123
|-
|<math>\omega^\omega2+\omega^2</math>
|ww1w1tw
|123123122
|-
|<math>\omega^\omega3</math>
|w11tw
|123123123
|-
|<math>\omega^{\omega+1}</math>
|wt1w
|1232
|-
|<math>\omega^{\omega+1}+\omega^2</math>
|ww1wt1w
|1232122
|-
|<math>\omega^{\omega+1}+\omega^\omega</math>
|wtwr1wt1w
|1232123
|-
|<math>\omega^{\omega+1}+\omega^\omega2</math>
|w1twr1wt1w
|1232123123
|-
|<math>\omega^{\omega+1}2+\omega^\omega</math>
|wtwr1w1t1w
|12321232123
|-
|<math>\omega^{\omega+1}2+\omega^\omega2</math>
|w1twr1w1t1w
|12321232123123
|-
|<math>\omega^{\omega+1}3</math>
|w11t1w
|123212321232
|-
|<math>\omega^{\omega+2}</math>
|wt11w
|12322
|-
|<math>\omega^{\omega+3}</math>
|wt111w
|123222
|-
|<math>\omega^{\omega2}</math>
|wtw1
|12323
|-
|<math>\omega^{\omega2+1}</math>
|wt1w1
|123232
|-
|<math>\omega^{\omega2+1}+\omega^{\omega2}</math>
|wtw1r1wt1w1
|12323212323
|-
|<math>\omega^{\omega2+1}+\omega^{\omega2}2</math>
|w1tw1r1wt1w1
|1232321232312323
|-
|<math>\omega^{\omega2+1}2+\omega^{\omega2}2</math>
|w1tw1r1w1t1w1
|1232321232321232312323
|-
|<math>\omega^{\omega2+1}2+\omega^{\omega2}2+\omega^{\omega+1}2+\omega^{\omega}2</math>
|w1twr1w1t1wr1w1tw1r1w1t1w1
|123232123232123231232312321232123123
|-
|<math>\omega^{\omega2+2}</math>
|wt11w1
|1232322
|-
|<math>\omega^{\omega2+2}+\omega^{\omega2+1}</math>
|wt1w1r1wt11w1
|1232322123232
|-
|<math>\omega^{\omega3}</math>
|wtw11
|1232323
|-
|<math>\omega^{\omega3+1}+\omega^{\omega3}</math>
|wtw11r1wt1w11
|123232321232323
|-
|<math>\omega^{\omega4}</math>
|wtw111
|123232323
|-
|<math>\omega^{\omega^2}</math>
|wtww
|1233
|-
|<math>\omega^{\omega^2+\omega+1}+\omega^{\omega^2+\omega}</math>
|wtw1wwr1wt1w1ww
|1233232123323
|-
|<math>\omega^{\omega^22}</math>
|wtww1
|1233233
|-
|<math>\omega^{\omega^3}</math>
|wtwww
|12333
|-
|<math>\omega^{\omega^\omega}</math>
|wtwtw
|1234
|-
|<math>\omega^{\omega^{\omega2+1}3}</math>
|wtw11t1w1
|1234343234343234343
|-
|<math>\omega^{\omega^{\omega2+2}}</math>
|wtwt11w1
|12343433
|-
|<math>\omega^{\omega^{\omega3}}</math>
|wtwtw11
|12343434
|-
|<math>\omega^{\omega^{\omega^2}}</math>
|wtwtww
|12344
|-
|<math>\omega^{\omega^{\omega^\omega}}</math>
|wtwtwtw
|12345
|-
|<math>\varepsilon_0</math>
|e
|124
|-
|<math>\varepsilon_0+\omega</math>
|w1e
|12412
|-
|<math>\varepsilon_0+\omega^\omega</math>
|wtwr1e
|124123
|-
|<math>\varepsilon_02</math>
|e1
|124124
|-
|<math>\varepsilon_02+\omega</math>
|w1e1
|12412412
|-
|<math>\varepsilon_0\omega=\omega^{\varepsilon_0+1}</math>
|wt1e
|1242
|-
|<math>\omega^{\varepsilon_0+1}+\varepsilon_0</math>
|e1wt1e
|1242124
|-
|<math>\omega^{\varepsilon_0+1}2</math>
|w1t1e
|12421242
|-
|<math>\omega^{\varepsilon_0+2}</math>
|wt11e
|12422
|-
|<math>\omega^{\varepsilon_0+\omega}</math>
|wtw1e
|12423
|-
|<math>\omega^{\varepsilon_0+\omega^\omega}</math>
|wtwtwr1e
|124234
|-
|<math>\omega^{\varepsilon_02}</math>
|wte1
|12424
|-
|<math>\omega^\omega^{\varepsilon_0+1}</math>
|wtwt1e
|1243
|-
|<math>\omega^\omega^{\varepsilon_02}</math>
|wtwte1
|12435
|-
|<math>\varepsilon_1</math>
|ee
|1244
|-
|<math>\omega^{\varepsilon_12}</math>
|wtee1
|1244244
|-
|<math>\varepsilon_2</math>
|eee
|12444
|-
|<math>\varepsilon_\omega</math>
|etw
|1245
|-
|<math>\varepsilon_\omega+\omega^\omega</math>
|wtwr1etw
|1245123
|-
|<math>\varepsilon_\omega+\varepsilon_0</math>
|e1etw
|1245124
|-
|<math>\varepsilon_\omega2</math>
|e1tw
|12451245
|-
|<math>\omega^{\varepsilon_\omega+1}</math>
|wt1etw
|12452
|-
|<math>\omega^{\varepsilon_\omega+\omega}</math>
|wtw1etw
|124523
|-
|<math>\omega^{\varepsilon_\omega2}</math>
|wte1tw
|1245245
|-
|<math>\varepsilon_{\omega+1}</math>
|et1w
|12454
|-
|<math>\varepsilon_{\omega2}</math>
|etw1
|124545
|-
|<math>\varepsilon_{\omega^\omega}</math>
|etwtw
|12456
|-
|<math>\varepsilon_{\varepsilon_0}</math>
|ete
|12457
|-
|<math>\varepsilon_{\varepsilon_0+1}</math>
|et1e
|12457
|-
|<math>\varepsilon_{\varepsilon_1}</math>
|etee
|12457
|-
|<math>\varepsilon_{\varepsilon_{\varepsilon_0}}</math>
|etete
|124578A
|-
|<math>\zeta_0</math>
|y
|1246
|-
|<math>\zeta_0+\varepsilon_0</math>
|e1y
|1246124
|-
|<math>\zeta_0+\varepsilon_{\varepsilon_0}</math>
|eter1y
|124612457
|-
|<math>\zeta_02</math>
|y1
|12461246
|-
|<math>\omega^{\zeta_0+1}</math>
|wt1y
|12462
|-
|<math>\omega^{\zeta_0+\varepsilon_\omega}</math>
|wtetwr1y
|1246245
|-
|<math>\omega^{\zeta_02}</math>
|wty1
|1246246
|-
|<math>\varepsilon_{\zeta_0+1}</math>
|ye=et1y
|12464
|-
|<math>\omega^{\varepsilon_{\zeta_0+1}+1}</math>
|wt1ye=wt1et1y
|124642
|-
|<math>\varepsilon_{\zeta_0+2}</math>
|yee=et11y
|124644
|-
|<math>\varepsilon_{\zeta_0+3}</math>
|yeee=et111y
|1246444
|-
|<math>\varepsilon_{\zeta_0+\omega}</math>
|etw1y
|124645
|-
|<math>\varepsilon_{\zeta_0+\omega^\omega}</math>
|etwtwr1y
|1246456
|-
|<math>\varepsilon_{\zeta_0+\varepsilon_0}</math>
|ete1y
|1246457
|-
|<math>\varepsilon_{\zeta_02}</math>
|ety1
|12464579
|-
|<math>\varepsilon_{\zeta_02+1}</math>
|et1y1
|124645794
|-
|<math>\varepsilon_{\zeta_03}</math>
|ety11
|124645794579
|-
|<math>\varepsilon_{\omega^{\zeta_0+1}}</math>
|etwt1y
|124645795
|-
|<math>\varepsilon_{\varepsilon_{\zeta_0+1}}</math>
|etet1y
|124645797
|-
|<math>\zeta_1</math>
|yy
|124646
|-
|<math>\varepsilon_{\zeta_1+1}</math>
|et1yy=yye
|1246464
|-
|<math>\zeta_2</math>
|yyy
|12464646
|-
|<math>\zeta_{\omega}</math>
|ytw
|12465
|-
|<math>\omega^{\zeta_{\omega}2}</math>
|wty1tw
|124652465
|-
|<math>\varepsilon_{\zeta_{\omega}+1}</math>
|et1ytw
|124654
|-
|<math>\varepsilon_{\zeta_{\omega}+\omega^{\varepsilon_0+1}}</math>
|etwt1er1ytw
|124654575
|-
|<math>\varepsilon_{\zeta_{\omega}+\varepsilon_\omega}</math>
|etetwr1ytw
|124654578
|-
|<math>\varepsilon_{\zeta_{\omega}+\zeta_0}</math>
|ety1ytw
|124654579
|-
|<math>\varepsilon_{\varepsilon_{\zeta_{\omega}2}}</math>
|etety1tw
|124654579878ACB
|-
|<math>\zeta_{\omega+1}</math>
|yt1w
|1246546
|-
|<math>\zeta_{\omega2}</math>
|ytw1
|124655
|-
|<math>\zeta_{\varepsilon_0}</math>
|yte
|124657
|-
|<math>\zeta_{\zeta_0}</math>
|yty
|1246579
|-
|<math>\eta_0</math>
|3
|12466
|-
|<math>\omega^{\eta_0+1}</math>
|wt13
|124662
|-
|<math>\omega^{\eta_0+\varepsilon_0}</math>
|wte13
|1246624
|-
|<math>\omega^{\eta_0+\zeta_0}</math>
|wty13
|12466246
|-
|<math>\omega^{\eta_02}</math>
|wt31
|124662466
|-
|<math>\varepsilon_{\eta_0+1}</math>
|3e=et13
|124664
|-
|<math>\varepsilon_{\eta_0+\omega}</math>
|etw13
|1246645
|-
|<math>\varepsilon_{\eta_0+\varepsilon_0}</math>
|ete13
|12466457
|-
|<math>\varepsilon_{\eta_0+\zeta_0}</math>
|ety13
|124664579
|-
|<math>\varepsilon_{\eta_02}</math>
|et31
|1246645799
|-
|<math>\zeta_{\eta_0+1}</math>
|3y=yt13
|1246646
|-
|<math>\zeta_{\eta_0+\omega}</math>
|ytw13
|12466465
|-
|<math>\zeta_{\eta_0+\varepsilon_0}</math>
|yte13
|124664657
|-
|<math>\zeta_{\eta_0+\zeta_0}</math>
|yty13
|1246646579
|-
|<math>\zeta_{\eta_02}</math>
|yt31
|12466465799
|-
|<math>\eta_1</math>
|33
|12466466
|-
|<math>\eta_\omega</math>
|3tw
|124665
|-
|<math>\eta_{\eta_0}</math>
|3t3
|124665799
|-
|<math>\varphi(4,0)</math>
|4
|124666
|-
|<math>\omega^{\varphi(4,0)2}</math>
|wt41
|1246662466
|-
|<math>\varepsilon_{\varphi(4,0)+1}</math>
|et41
|12466645799
|-
|<math>\zeta_{\varphi(4,0)+1}</math>
|yt41
|124666465799
|-
|<math>\eta_{\varphi(4,0)+1}</math>
|3t41
|1246664665799
|-
|<math>\varphi(4,1)</math>
|44
|1246664666
|-
|<math>\varphi(5,0)</math>
|5
|1246666
|-
|<math>\varphi(6,0)</math>
|6
|12466666
|-
|<math>\varphi(7,0)</math>
|7
|124666666
|-
|<math>\varphi(8,0)</math>
|8
|1246666666
|-
|<math>\varphi(4,\varphi(8,0)+1)+\eta_0</math>
|3184=314t18
|1246666666466657999999912466
|-
|<math>\varphi(9,0)</math>
|9
|12466666666
|-
|<math>\varphi(9,\varphi(9,0))</math>
|9t9
|124666666665799999999
|-
|<math>\varphi(10,0)</math>
|<math>\mathrm{limit}</math>
|124666666666
|+
!序数（十进制+Cantor记号+Veblen记号）
!扩展DcN
!1-Y
|-
|<math>\varphi(10,0)</math>
|q1111111111r
|124666666666
|-
|<math>\varphi(11,0)</math>
|q11111111111r
|1246666666666
|-
|<math>\varphi(\omega,0)</math>
|qwr
|12467
|-
|<math>\varepsilon_{\varphi(\omega,0)+1}</math>
|et1qwr=qwre

=q1rt1qwr=qwrq1r
|124674
|-
|<math>\varphi(\omega,1)</math>
|qwrqwr
|12467467
|-
|<math>\varphi(\omega,\omega)</math>
|qwrtw
|124675
|-
|<math>\varphi(\omega+1,0)</math>
|q1wr
|124676
|-
|<math>\varphi(\omega2,0)</math>
|qw1r
|1246767
|-
|<math>\varphi(\varepsilon_0,0)</math>
|qer
|124679
|-
|<math>\varphi(\varepsilon_0+\omega,\zeta_0)</math>
|qw1erty（qwerty）
|12467967579
|-
|<math>\varphi(\zeta_0,0)</math>
|qyr
|124679B
|-
|<math>\varphi(\varphi(\omega,0),0)</math>
|qqwrr
|124679BC
|-
|<math>\Gamma_0</math>
|扩展极限
|12468
|-
|}
{| class="wikitable"
|+
!序数（十进制+BOCF）
!1-Y
|-
|<math>\psi(\Omega^{\Omega})</math>
|12468
|-
|<math>\psi(\Omega^{\Omega}+1)</math>
|124682
|-
|<math>\psi(\Omega^{\Omega}+\psi(\Omega^{\Omega}))</math>
|124682468
|-
|<math>\psi(\Omega^{\Omega}+\Omega)</math>
|124684
|-
|<math>\psi(\Omega^{\Omega}+\Omega^\omega)</math>
|12468467
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\omega)})</math>
|124684679AB
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\Omega)})</math>
|124684679AC
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\Omega)+1})</math>
|124684679AC6
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\Omega+1)})</math>
|124684679AC7
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\Omega+\Omega)})</math>
|124684679AC9
|-
|<math>\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^\Omega+\Omega^\omega)})</math>
|124684679AC9AB
|-
|<math>\psi(\Omega^{\Omega}2)</math>
|12468468
|-
|<math>\psi(\Omega^{\Omega+1})</math>
|124686
|-
|<math>\psi(\Omega^{\Omega+\psi(\Omega^{\Omega})})</math>
|12468679BD
|-
|<math>\psi(\Omega^{\Omega2})</math>
|1246868
|-
|<math>\psi(\Omega^{\Omega\psi(\Omega^{\Omega})})</math>
|1246879BD
|-
|<math>\psi(\Omega^{\Omega^2})</math>
|124688
|-
|<math>\psi(\Omega^{\Omega^\omega})</math>
|124689
|-
|<math>\psi(\Omega^{\Omega^{\psi(\Omega^{\Omega^\omega})}})</math>
|124689BDFG
|-
|<math>\psi(\Omega^{\Omega^\Omega})</math>
|12468A
|-
|<math>\psi(\Omega_2)</math>
|1247
|-
|<math>\psi(\Omega_2+\Omega^\Omega)</math>
|1247468
|-
|<math>\psi(\Omega_2+\psi_1(\Omega_2))</math>
|124747
|-
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\Omega))</math>
|12476
|-
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\Omega^\Omega))</math>
|124768
|-
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))</math>
|124769
|-
|<math>\psi(\Omega_22)</math>
|12477
|-
|<math>\psi(\Omega_2\omega)</math>
|12478
|-
|<math>\psi(\Omega_2\Omega)</math>
|12479
|-
|<math>\psi(\Omega_2^2)</math>
|1247A
|-
|<math>\psi(\Omega_2^{\Omega_2})</math>
|1247AD
|-
|<math>\psi(\Omega_3)</math>
|1247B
|-
|<math>\psi(\Omega_4)</math>
|1247BG
|-
|<math>\psi(\Omega_\omega)</math>
|1248
|-
|}
再往后涉及项嵌套的提升效应，目前对此研究不清晰，故枚举列表暂时到此为止。

DcN

2025-07-21T13:28:15Z

318`4😥：

DcN

2025-07-19T10:19:22Z

318`4😥：

DcN

2025-07-19T09:38:42Z

318`4😥：

DcN

2025-07-19T09:30:12Z

318`4😥：test

DcN，是318`4创造的便于打字的记号，主要用于写FOS中作为序数的项，几乎不可能良定义，主要通过枚举列表来主观推算定义。DcN的极限是φ(10,0)<math>\varphi(10,0)</math>，Y_cpper将其扩展到Г₀，现在一般在φ(ω,0)之后使用这个扩展。DcN的所有符号都集中在电脑键盘左上方的一小片区域，qwerty13456789，456789使用频率很低，Veblen记号需要随时要输入括号下标希腊字母，1-Y则有比DcN更长的常见表达式，所以DcN便于打字，是专为FOS中常见序数设计的记号；但因为可读性差，DcN一直没能流传开来。

1-Y不必多说，是Yukito在2019年创造的序列记号，首次引入山脉图，在Y(1,3)之前有非常优秀的性质，因此也常用于表达FOS中作为项的序数。不过为简便，写1-Y表达式时通常省略Y( )外壳和逗号，超过10的数字用拉丁字母表示，超过36则无写法。

在研究FOS的定义、构造原理时，使用1-Y的情况更多，因为1-Y的山脉结构和FOS中项内部的山脉结构类似；而在研究FOS的分析、强度形成原理时，为简便起见，使用DcN更多。在阅读FOS的枚举列表之前，请务必将本文的枚举列表熟透于心中，否则用Veblen记号或OCF写FOS表达式费时费力。
{| class="wikitable"
|+
!序数（十进制+Cantor记号+Veblen记号+BOCF）
!DcN
!1-Y
|-
|0
|0
|[空序列]
|-
|1
|1
|1
|-
|2
|11
|11
|-
|3
|111
|111
|-
|1111<math display="block">\omega</math>
|w
|12
|-
|
|
|
|-
|
|
|
|-
|
|
|
|-
|
|
|
|-
|
|
|
|-
|
|
|
|-
|
|
|
|-
|
|
|
|-
|
|
|
|-
|
|
|
|-
|
|
|
|-
|
|
|
|}