DcN

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 $\phi (10,0)$，Y_cpper 将其扩展到 ${\mathrm{\Gamma }}_0$，现在一般在 $\phi (\omega ,0)$ 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。

因为完全无r的 DcN 存在歧义，比如 wtw1wtww 可以是 ${\omega }^{{\omega }^2}+{\omega }^{\omega }$，也可以是 ${\omega }^{{\omega }^{{\omega }^2}+\omega }$，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。

序数（十进制 + Cantor 记号 + Veblen 记号）	DcN	1-Y
$0$	0	$\varnothing$
$1$	1	1
$2$	11	11
$3$	111	111
$\omega$	w	12
$\omega +1$	1w	121
$\omega +2$	11w	1211
$\omega 2$	w1	1212
$\omega 2+1$	1w1	12121
$\omega 3$	w11	121212
$\omega 3+1$	1w11	1212121
${\omega }^2$	ww	122
${\omega }^2+1$	1ww	1221
${\omega }^2+\omega$	w1ww	12212
${\omega }^2+\omega +1$	1w1ww	122121
${\omega }^2+\omega 2$	w11ww	1221212
${\omega }^2+\omega 2+1$	1w11ww	12212121
${\omega }^2+\omega 3$	w111ww	122121212
${\omega }^{2}2$	ww1	122122
${\omega }^{2}2+\omega$	w1ww1	12212212
${\omega }^{2}2+\omega 2$	w11ww1	1221221212
${\omega }^{2}3$	ww11	122122122
${\omega }^3$	www	1222
${\omega }^3+\omega$	w1www	122212
${\omega }^3+{\omega }^2$	ww1www	1222122
${\omega }^3+{\omega }^2+1$	1ww1www	12221221
${\omega }^3+{\omega }^2+\omega$	w1ww1www	122212212
${\omega }^3+{\omega }^2+\omega +1$	1w1ww1www	1222122121
${\omega }^3+{\omega }^2+\omega 2$	w11ww1www	12221221212
${\omega }^3+{\omega }^{2}2+\omega$	w1ww11www	122212212212
${\omega }^3+{\omega }^{2}2+\omega 2+1$	1w11ww11www	122212212212121
${\omega }^{3}2$	www1	12221222
${\omega }^4$	wwww	12222
${\omega }^5$	wwwww	122222
${\omega }^{\omega }$	wtw	123
${\omega }^{\omega }+1$	1wtw	1231
${\omega }^{\omega }+\omega$	w1wtw	12312
${\omega }^{\omega }+\omega 2$	w11wtw	1231212
${\omega }^{\omega }+{\omega }^2$	ww1wtw	123122
${\omega }^{\omega }2$	w1tw	123123
${\omega }^{\omega }2+{\omega }^2$	ww1w1tw	123123122
${\omega }^{\omega }3$	w11tw	123123123
${\omega }^{\omega +1}$	wt1w	1232
${\omega }^{\omega +1}+{\omega }^2$	ww1wt1w	1232122
${\omega }^{\omega +1}+{\omega }^{\omega }$	wtwr1wt1w	1232123
${\omega }^{\omega +1}+{\omega }^{\omega }2$	w1twr1wt1w	1232123123
${\omega }^{\omega +1}2+{\omega }^{\omega }$	wtwr1w1t1w	12321232123
${\omega }^{\omega +1}2+{\omega }^{\omega }2$	w1twr1w1t1w	12321232123123
${\omega }^{\omega +1}3$	w11t1w	123212321232
${\omega }^{\omega +2}$	wt11w	12322
${\omega }^{\omega +3}$	wt111w	123222
${\omega }^{\omega 2}$	wtw1	12323
${\omega }^{\omega 2+1}$	wt1w1	123232
${\omega }^{\omega 2+1}+{\omega }^{\omega 2}$	wtw1r1wt1w1	12323212323
${\omega }^{\omega 2+1}+{\omega }^{\omega 2}2$	w1tw1r1wt1w1	1232321232312323
${\omega }^{\omega 2+1}2+{\omega }^{\omega 2}2$	w1tw1r1w1t1w1	1232321232321232312323
${\omega }^{\omega 2+1}2+{\omega }^{\omega 2}2+{\omega }^{\omega +1}2+{\omega }^{\omega }2$	w1twr1w1t1wr1w1tw1r1w1t1w1	123232123232123231232312321232123123
${\omega }^{\omega 2+2}$	wt11w1	1232322
${\omega }^{\omega 2+2}+{\omega }^{\omega 2+1}$	wt1w1r1wt11w1	1232322123232
${\omega }^{\omega 3}$	wtw11	1232323
${\omega }^{\omega 3+1}+{\omega }^{\omega 3}$	wtw11r1wt1w11	123232321232323
${\omega }^{\omega 4}$	wtw111	123232323
${\omega }^{{\omega }^2}$	wtww	1233
${\omega }^{{\omega }^2+\omega +1}+{\omega }^{{\omega }^2+\omega }$	wtw1wwr1wt1w1ww	1233232123323
${\omega }^{{\omega }^{2}2}$	wtww1	1233233
${\omega }^{{\omega }^3}$	wtwww	12333
${\omega }^{{\omega }^{\omega }}$	wtwtw	1234
${\omega }^{{\omega }^{\omega 2+1}3}$	wtw11t1w1	1234343234343234343
${\omega }^{{\omega }^{\omega 2+2}}$	wtwt11w1	12343433
${\omega }^{{\omega }^{\omega 3}}$	wtwtw11	12343434
${\omega }^{{\omega }^{{\omega }^2}}$	wtwtww	12344
${\omega }^{{\omega }^{{\omega }^{\omega }}}$	wtwtwtw	12345
${\epsilon }_0$	e	124
${\epsilon }_0+\omega$	w1e	12412
${\epsilon }_0+{\omega }^{\omega }$	wtwr1e	124123
${\epsilon }_{0}2$	e1	124124
${\epsilon }_{0}2+\omega$	w1e1	12412412
${\epsilon }_0\omega ={\omega }^{{\epsilon }_0+1}$	wt1e	1242
${\omega }^{{\epsilon }_0+1}+{\epsilon }_0$	e1wt1e	1242124
${\omega }^{{\epsilon }_0+1}2$	w1t1e	12421242
${\omega }^{{\epsilon }_0+2}$	wt11e	12422
${\omega }^{{\epsilon }_0+\omega }$	wtw1e	12423
${\omega }^{{\epsilon }_0+{\omega }^{\omega }}$	wtwtwr1e	124234
${\omega }^{{\epsilon }_{0}2}$	wte1	12424
${\omega }^{\omega }{}^{{\epsilon }_0+1}$	wtwt1e	1243
${\omega }^{\omega }{}^{{\epsilon }_{0}2}$	wtwte1	12435
${\epsilon }_1$	ee	1244
${\omega }^{{\epsilon }_{1}2}$	wtee1	1244244
${\epsilon }_2$	eee	12444
${\epsilon }_{\omega }$	etw	1245
${\epsilon }_{\omega }+{\omega }^{\omega }$	wtwr1etw	1245123
${\epsilon }_{\omega }+{\epsilon }_0$	e1etw	1245124
${\epsilon }_{\omega }2$	e1tw	12451245
${\omega }^{{\epsilon }_{\omega }+1}$	wt1etw	12452
${\omega }^{{\epsilon }_{\omega }+\omega }$	wtw1etw	124523
${\omega }^{{\epsilon }_{\omega }2}$	wte1tw	1245245
${\epsilon }_{\omega +1}$	et1w	12454
${\epsilon }_{\omega 2}$	etw1	124545
${\epsilon }_{{\omega }^{\omega }}$	etwtw	12456
${\epsilon }_{{\epsilon }_0}$	ete	12457
${\epsilon }_{{\epsilon }_0+1}$	et1e	12457
${\epsilon }_{{\epsilon }_1}$	etee	12457
${\epsilon }_{{\epsilon }_{{\epsilon }_0}}$	etete	124578A
${\zeta }_0$	y	1246
${\zeta }_0+{\epsilon }_0$	e1y	1246124
${\zeta }_0+{\epsilon }_{{\epsilon }_0}$	eter1y	124612457
${\zeta }_{0}2$	y1	12461246
${\omega }^{{\zeta }_0+1}$	wt1y	12462
${\omega }^{{\zeta }_0+{\epsilon }_{\omega }}$	wtetwr1y	1246245
${\omega }^{{\zeta }_{0}2}$	wty1	1246246
${\epsilon }_{{\zeta }_0+1}$	ye=et1y	12464
${\omega }^{{\epsilon }_{{\zeta }_0+1}+1}$	wt1ye=wt1et1y	124642
${\epsilon }_{{\zeta }_0+2}$	yee=et11y	124644
${\epsilon }_{{\zeta }_0+3}$	yeee=et111y	1246444
${\epsilon }_{{\zeta }_0+\omega }$	etw1y	124645
${\epsilon }_{{\zeta }_0+{\omega }^{\omega }}$	etwtwr1y	1246456
${\epsilon }_{{\zeta }_0+{\epsilon }_0}$	ete1y	1246457
${\epsilon }_{{\zeta }_{0}2}$	ety1	12464579
${\epsilon }_{{\zeta }_{0}2+1}$	et1y1	124645794
${\epsilon }_{{\zeta }_{0}3}$	ety11	124645794579
${\epsilon }_{{\omega }^{{\zeta }_0+1}}$	etwt1y	124645795
${\epsilon }_{{\epsilon }_{{\zeta }_0+1}}$	etet1y	124645797
${\zeta }_1$	yy	124646
${\epsilon }_{{\zeta }_1+1}$	et1yy=yye	1246464
${\zeta }_2$	yyy	12464646
${\zeta }_{\omega }$	ytw	12465
${\omega }^{{\zeta }_{\omega }2}$	wty1tw	124652465
${\epsilon }_{{\zeta }_{\omega }+1}$	et1ytw	124654
${\epsilon }_{{\zeta }_{\omega }+{\omega }^{{\epsilon }_0+1}}$	etwt1er1ytw	124654575
${\epsilon }_{{\zeta }_{\omega }+{\epsilon }_{\omega }}$	etetwr1ytw	124654578
${\epsilon }_{{\zeta }_{\omega }+{\zeta }_0}$	ety1ytw	124654579
${\epsilon }_{{\epsilon }_{{\zeta }_{\omega }2}}$	etety1tw	124654579878ACB
${\zeta }_{\omega +1}$	yt1w	1246546
${\zeta }_{\omega 2}$	ytw1	124655
${\zeta }_{{\epsilon }_0}$	yte	124657
${\zeta }_{{\zeta }_0}$	yty	1246579
${\eta }_0$	3	12466
${\omega }^{{\eta }_0+1}$	wt13	124662
${\omega }^{{\eta }_0+{\epsilon }_0}$	wte13	1246624
${\omega }^{{\eta }_0+{\zeta }_0}$	wty13	12466246
${\omega }^{{\eta }_{0}2}$	wt31	124662466
${\epsilon }_{{\eta }_0+1}$	3e=et13	124664
${\epsilon }_{{\eta }_0+\omega }$	etw13	1246645
${\epsilon }_{{\eta }_0+{\epsilon }_0}$	ete13	12466457
${\epsilon }_{{\eta }_0+{\zeta }_0}$	ety13	124664579
${\epsilon }_{{\eta }_{0}2}$	et31	1246645799
${\zeta }_{{\eta }_0+1}$	3y=yt13	1246646
${\zeta }_{{\eta }_0+\omega }$	ytw13	12466465
${\zeta }_{{\eta }_0+{\epsilon }_0}$	yte13	124664657
${\zeta }_{{\eta }_0+{\zeta }_0}$	yty13	1246646579
${\zeta }_{{\eta }_{0}2}$	yt31	12466465799
${\eta }_1$	33	12466466
${\eta }_{\omega }$	3tw	124665
${\eta }_{{\eta }_0}$	3t3	124665799
$\phi (4,0)$	4	124666
${\omega }^{\phi (4,0)2}$	wt41	1246662466
${\epsilon }_{\phi (4,0)+1}$	et41	12466645799
${\zeta }_{\phi (4,0)+1}$	yt41	124666465799
${\eta }_{\phi (4,0)+1}$	3t41	1246664665799
$\phi (4,1)$	44	1246664666
$\phi (5,0)$	5	1246666
$\phi (6,0)$	6	12466666
$\phi (7,0)$	7	124666666
$\phi (8,0)$	8	1246666666
$\phi (4,\phi (8,0)+1)+{\eta }_0$	3184=314t18	1246666666466657999999912466
$\phi (9,0)$	9	12466666666
$\phi (9,\phi (9,0))$	9t9	124666666665799999999
$\phi (10,0)$	limit	124666666666
序数（十进制 + Cantor 记号 + Veblen 记号）	扩展DcN	1-Y
$\phi (10,0)$	q1111111111r	124666666666
$\phi (11,0)$	q11111111111r	1246666666666
$\phi (\omega ,0)$	qwr	12467
${\epsilon }_{\phi (\omega ,0)+1}$	et1qwr=qwre =q1rt1qwr=qwrq1r	124674
$\phi (\omega ,1)$	qwrqwr	12467467
$\phi (\omega ,\omega )$	qwrtw	124675
$\phi (\omega +1,0)$	q1wr	124676
$\phi (\omega 2,0)$	qw1r	1246767
$\phi ({\epsilon }_0,0)$	qer	124679
$\phi ({\epsilon }_0+\omega ,{\zeta }_0)$	qw1erty（qwerty）	12467967579
$\phi ({\zeta }_0,0)$	qyr	124679B
$\phi (\phi (\omega ,0),0)$	qqwrr	124679BC
${\mathrm{\Gamma }}_0$	limit	12468

序数（十进制+BOCF）	1-Y
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})$	12468
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+1)$	124682
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}))$	124682468
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+\mathrm{\Omega })$	124684
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\omega })$	12468467
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })})$	124684679AB
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})})$	124684679AC
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})+1})$	124684679AC6
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+1)})$	124684679AC7
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+\mathrm{\Omega })})$	124684679AC9
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\omega })})$	124684679AC9AB
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}2)$	12468468
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1})$	124686
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})})$	12468679BD
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }2})$	1246868
$\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})})$	1246879BD
$\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2})$	124688
$\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})$	124689
$\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})}})$	124689BDFG
$\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})$	12468A
$\psi ({\mathrm{\Omega }}_2)$	1247
$\psi ({\mathrm{\Omega }}_2+{\mathrm{\Omega }}^{\mathrm{\Omega }})$	1247468
$\psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2))$	124747
$\psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+\mathrm{\Omega }))$	12476
$\psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\mathrm{\Omega }}^{\mathrm{\Omega }}))$	124768
$\psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2)))$	124769
$\psi ({\mathrm{\Omega }}_{2}2)$	12477
$\psi ({\mathrm{\Omega }}_2\omega )$	12478
$\psi ({\mathrm{\Omega }}_2\mathrm{\Omega })$	12479
$\psi ({\mathrm{\Omega }}_2^2)$	1247A
$\psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2})$	1247AD
$\psi ({\mathrm{\Omega }}_3)$	1247B
$\psi ({\mathrm{\Omega }}_4)$	1247BG
$\psi ({\mathrm{\Omega }}_{\omega })$	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。