Analyses of binary bisector notation with BMS and Y-sequence

[1] {0}

• Ordinal: 0

• BMS: (0)

• Y: (1)

[2] {1}

• Ordinal: 1

• BMS: (0)(0)

• Y: (1,1)

[3] {2}

• Ordinal: 2

• BMS: (0)(0)(0)

• Y: (1,1,1)

[4] {3}

• Ordinal: 3

• BMS: (0)(0)(0)(0)

• Y: (1,1,1,1)

[5] {4}

• Ordinal: 4

• BMS: (0)(0)(0)(0)(0)

• Y: (1,1,1,1,1)

[6] {{1}}

• Ordinal: ω

• BMS: (0)(1)

• Y: (1,2)

[7] {{1} + 1}

• Ordinal: ω + 1

• BMS: (0)(1)(0)

• Y: (1,2,1)

[8] {{1} + 2}

• Ordinal: ω + 2

• BMS: (0)(1)(0)(0)

• Y: (1,2,1,1)

[9] {{1}{1}}

• Ordinal: ω·2

• BMS: (0)(1)(0)(1)

• Y: (1,2,1,2)

[10] {{1}{1} + 1}

• Ordinal: ω·2 + 1

• BMS: (0)(1)(0)(1)(0)

• Y: (1,2,1,2,1)

[11] {{1}{1}{1}}

• Ordinal: ω·3

• BMS: (0)(1)(0)(1)(0)(1)

• Y: (1,2,1,2,1,2)

[12] {{1}{1}{1}{1}}

• Ordinal: ω·4

• BMS: (0)(1)(0)(1)(0)(1)(0)(1)

• Y: (1,2,1,2,1,2,1,2)

[13] {{2}}

• Ordinal: ω^2

• BMS: (0)(1)(1)

• Y: (1,2,2)

[14] {{2} + 1}

• Ordinal: ω^2 + 1

• BMS: (0)(1)(1)(0)

• Y: (1,2,2,1)

[15] {{2}{1}}

• Ordinal: ω^2 + ω

• BMS: (0)(1)(1)(0)(1)

• Y: (1,2,2,1,2)

[16] {{2}{2}}

• Ordinal: ω^2·2

• BMS: (0)(1)(1)(0)(1)(1)

• Y: (1,2,2,1,2,2)

[17] {{2}{2}{2}}

• Ordinal: ω^2·3

• BMS: (0)(1)(1)(0)(1)(1)(0)(1)(1)

• Y: (1,2,2,1,2,2,1,2,2)

[18] {{3}}

• Ordinal: ω^3

• BMS: (0)(1)(1)(1)

• Y: (1,2,2,2)

[19] {{3}{3}}

• Ordinal: ω^3·2

• BMS: (0)(1)(1)(1)(0)(1)(1)(1)

• Y: (1,2,2,2,1,2,2,2)

[20] {{4}}

• Ordinal: ω^4

• BMS: (0)(1)(1)(1)

• Y: (1,2,2,2,2)

[21] {{5}}

• Ordinal: ω^5

• BMS: (0)(1)(1)(1)(1)

• Y: (1,2,2,2,2,2)

[22] {{{1}}}

• Ordinal: ω^ω

• BMS: (0)(1)(2)

• Y: (1,2,3)

[23] {{{1}} + 1}

• Ordinal: ω^ω + 1

• BMS: (0)(1)(2)(0)

• Y: (1,2,3,1)

[24] {{{1}}{1}}

• Ordinal: ω^ω + ω

• BMS: (0)(1)(2)

• Y: (1,2,3,1,2)

[25] {{{1}}{{1}}}

• Ordinal: ω^ω·2

• BMS: (0)(1)(2)

• Y: (1,2,3,1,2,3)

[26] {{{1} + 1}}

• Ordinal: ω^(ω + 1)

• BMS: (0)(1)(2)(1)

• Y: (1,2,3,2)

[27] {{{1} + 2}}

• Ordinal: ω^(ω + 2)

• BMS: (0)(1)(2)(1)(1)

• Y: (1,2,3,2,2)

[28] {{{1}{1}}}

• Ordinal: ω^(ω·2)

• BMS: (0)(1)(2)(1)(2)

• Y: (1,2,3,2,3)

[29] {{{1}{1}{1}}}

• Ordinal: ω^(ω·3)

• BMS: (0)(1)(2)(1)(2)(1)(2)

• Y: (1,2,3,2,3,2,3)

[30] {{{2}}}

• Ordinal: ω^ω^2

• BMS: (0)(1)(2)(2)

• Y: (1,2,3,3)

[31] {{{2}{2}}}

• Ordinal: ω^(ω^2·2)

• BMS: (0)(1)(2)(2)(1)(2)(2)

• Y: (1,2,3,3,2,3,3)

[32] {{{3}}}

• Ordinal: ω^ω^3

• BMS: (0)(1)(2)(2)(2)

• Y: (1,2,3,3,3)

[33] {{{4}}}

• Ordinal: ω^ω^4

• BMS: (0)(1)(2)(2)(2)(2)

• Y: (1,2,3,3,3,3)

[34] {{{{1}}}}

• Ordinal: ω^ω^ω

• BMS: (0)(1)(2)(3)

• Y: (1,2,3,4)

[35] {{{{1} + 1}}}

• Ordinal: ω^ω^(ω + 1)

• BMS: (0)(1)(2)(3)(2)

• Y: (1,2,3,4,3)

[36] {{{{1}{1}}}}

• Ordinal: ω^ω^(ω·2)

• BMS: (0)(1)(2)(3)(2)(3)

• Y: (1,2,3,4,3,4)

[37] {{{{2}}}}

• Ordinal: ω^ω^ω^2

• BMS: (0)(1)(2)(3)(3)

• Y: (1,2,3,4,4)

[38] {{{{3}}}}

• Ordinal: ω^ω^ω^3

• BMS: (0)(1)(2)(3)(3)(3)

• Y: (1,2,3,4,4,4)

[39] {{{{{1}}}}}

• Ordinal: ω^ω^ω^ω

• BMS: (0)(1)(2)(3)(4)

• Y: (1,2,3,4,5)

[40] {{{{{2}}}}}

• Ordinal: ω^ω^ω^ω^2

• BMS: (0)(1)(2)(3)(4)(4)

• Y: (1,2,3,4,5,5)

[41] {{{{{{1}}}}}}

• Ordinal: ω^ω^ω^ω^ω

• BMS: (0)(1)(2)(3)(4)(5)

• Y: (1,2,3,4,5,6)

[42] {{{{{{{1}}}}}}}

• Ordinal: ω^ω^ω^ω^ω^ω

• BMS: (0)(1)(2)(3)(4)(5)(6)

• Y: (1,2,3,4,5,6,7)

[43] {{{{{{{{1}}}}}}}}

• Ordinal: ω^ω^ω^ω^ω^ω^ω

• BMS: (0)(1)(2)(3)(4)(5)(6)(7)

• Y: (1,2,3,4,5,6,7,8)

[44] {{0, 1}}

• Ordinal: ε0

• Buchholz: ψ0(Ω)

• Madore: ψ(0)

• BMS: (0,0)(1,1)

• Y: (1,2,4)

[45] {{0, 1} + 1}

• Ordinal: ε0 + 1

• Buchholz: ψ0(Ω) + 1

• Madore: ψ(0) + 1

• BMS: (0,0)(1,1)(0,0)

• Y: (1,2,4,1)

[46] {{0, 1}{1}}

• Ordinal: ε0 + ω

• Buchholz: ψ0(Ω) + ω

• Madore: ψ(0) + ω

• BMS: (0,0)(1,1)(0,0)(1,0)

• Y: (1,2,4,1,2)

[47] {{0, 1}{{0, 1}}}

• Ordinal: ε0·2

• Buchholz: ψ0(Ω)·2

• Madore: ψ(0)·2

• BMS: (0,0)(1,1)(0,0)(1,1)

• Y: (1,2,4,1,2,4)

[48] {{0, 1}{{0, 1}}{{0, 1}}}

• Ordinal: ε0·3

• Buchholz: ψ0(Ω)·3

• Madore: ψ(0)·3

• BMS: (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)

• Y: (1,2,4,1,2,4,1,2,4)

[49] {{0, 1}{{0, 1} + 1}}

• Ordinal: ε0·ω

• Buchholz: ψ0(Ω)·ω

• Madore: ψ(0)·ω

• BMS: (0,0)(1,1)(1,0)

• Y: (1,2,4,2)

[50] {{0, 1}{{0, 1}{1}}}

• Ordinal: ε0·ω^ω

• Buchholz: ψ0(Ω)·ω^ω

• Madore: ψ(0)·ω^ω

• BMS: (0,0)(1,1)(1,0)(2,0)

• Y: (1,2,4,2,3)

[51] {{0, 1}{{0, 1}{{0, 1}}}}

• Ordinal: ε0^2

• Buchholz: ψ0(Ω)^2

• Madore: ψ(0)^2

• BMS: (0,0)(1,1)(1,0)(2,1)

• Y: (1,2,4,2,4)

[52] {{0, 1}{{0, 1}{{0, 1} + 1}}}

• Ordinal: ε0^ω

• Buchholz: ψ0(Ω)^ω

• Madore: ψ(0)^ω

• BMS: (0,0)(1,1)(1,0)(2,1)(2,0)

• Y: (1,2,4,3)

[53] {{0, 1}{{0, 1}{{0, 1}{{0, 1}}}}}

• Ordinal: ε0^ε0

• Buchholz: ψ0(Ω)^ψ0(Ω)

• Madore: ψ(0)^ψ(0)

• BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)

• Y: (1,2,4,3,5)

[54] {{0, 1}{{0, 1}{{0, 1}{{0, 1} + 1}}}}

• Ordinal: ε0^ε0^ω

• Buchholz: ψ0(Ω)^ψ0(Ω)^ω

• Madore: ψ(0)^ψ(0)^ω

• BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)

• Y: (1,2,4,3,5,4)

[55] {{0, 1}{{0, 1}{{0, 1}{{0, 1}{{0, 1}}}}}}

• Ordinal: ε0^ε0^ε0

• Buchholz: ψ0(Ω)^ψ0(Ω)^ψ0(Ω)

• Madore: ψ(0)^ψ(0)^ψ(0)

• BMS: (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)

• Y: (1,2,4,3,5,4,6)

[56] {{0, 1}{0, 1}}

• Ordinal: ε1

• Buchholz: ψ0(Ω·2)

• Madore: ψ(1)

• BMS: (0,0)(1,1)(1,1)

• Y: (1,2,4,4)

[57] {{0, 1}{0, 1}{{0, 1}{0, 1}}}

• Ordinal: ε1·2

• Buchholz: ψ0(Ω·2)·2

• Madore: ψ(1)·2

• BMS: (0,0)(1,1)(1,1)(0,0)(1,1)(1,1)

• Y: (1,2,4,4,1,2,4,4)

[58] {{0, 1}{0, 1}{{0, 1}{0, 1}{{0, 1}{0, 1}}}}

• Ordinal: ε1^2

• Buchholz: ψ0(Ω·2)^2

• Madore: ψ(1)^2

• BMS: (0,0)(1,1)(1,1)(1,0)(2,1)(2,1)

• Y: (1,2,4,4,2,4,4)

[59] {{0, 1}{0, 1}{0, 1}}

• Ordinal: ε2

• Buchholz: ψ0(Ω·3)

• Madore: ψ(2)

• BMS: (0,0)(1,1)(1,1)(1,1)

• Y: (1,2,4,4,4)

[60] {{0, 1}{0, 1}{0, 1}{0,1}}

• Ordinal: ε3

• Buchholz: ψ0(Ω·4)

• Madore: ψ(3)

• BMS: (0,0)(1,1)(1,1)(1,1)(1,1)

• Y: (1,2,4,4,4,4)

[61] {{1, 1}}

• Ordinal: ε(ω)

• Buchholz: ψ0(Ω·ω)

• Madore: ψ(ω)

• BMS: (0,0)(1,1)(2,0)

• Y: (1,2,4,5)

[62] {{1, 1}{{1, 1}}}

• Ordinal: ε(ω)·2

• Buchholz: ψ0(Ω·ω)·2

• Madore: ψ(ω)·2

• BMS: (0,0)(1,1)(2,0)(0,0)(1,1)(2,0)

• Y: (1,2,4,5,1,2,4,5)

[63] {{1, 1}{{1, 1}{{1, 1}}}}

• Ordinal: ε(ω)^2

• Buchholz: ψ0(Ω·ω)^2

• Madore: ψ(ω)^2

• BMS: (0,0)(1,1)(2,0)(1,0)(2,1)(3,0)

• Y: (1,2,4,5,2,4,5)

[64] {{1, 1}{0, 1}}

• Ordinal: ε(ω + 1)

• Buchholz: ψ0(Ω·(ω + 1))

• Madore: ψ(ω + 1)

• BMS: (0,0)(1,1)(2,0)(1,1)

• Y: (1,2,4,5,4)

[65] {{1, 1}{0, 1}{0, 1}}

• Ordinal: ε(ω + 2)

• Buchholz: ψ0(Ω·(ω + 2))

• Madore: ψ(ω + 2)

• BMS: (0,0)(1,1)(2,0)(1,1)(1,1)

• Y: (1,2,4,5,4,4)

[66] {{1, 1}{1, 1}}

• Ordinal: ε(ω·2)

• Buchholz: ψ0(Ω·ω·2)

• Madore: ψ(ω·2)

• BMS: (0,0)(1,1)(2,0)(1,1)(2,0)

• Y: (1,2,4,5,4,5)

[67] {{1, 1}{1, 1}{1, 1}}

• Ordinal: ε(ω·3)

• Buchholz: ψ0(Ω·ω·3)

• Madore: ψ(ω·3)

• BMS: (0,0)(1,1)(2,0)(1,1)(2,0)(1,1)(2,0)

• Y: (1,2,4,5,4,5,4,5)

[68] {{2, 1}}

• Ordinal: ε(ω^2)

• Buchholz: ψ0(Ω·ω^2)

• Madore: ψ(ω^2)

• BMS: (0,0)(1,1)(2,0)(2,0)

• Y: (1,2,4,5,5)

[69] {{3, 1}}

• Ordinal: ε(ω^3)

• Buchholz: ψ0(Ω·ω^3)

• Madore: ψ(ω^3)

• BMS: (0,0)(1,1)(2,0)(2,0)(2,0)

• Y: (1,2,4,5,5,5)

[70] {{{1}, 1}}

• Ordinal: ε(ω^ω)

• Buchholz: ψ0(Ω·ω^ω)

• Madore: ψ(ω^ω)

• BMS: (0,0)(1,1)(2,0)(3,0)

• Y: (1,2,4,5,6)

[71] {{{{1}}, 1}}

• Ordinal: ε(ω^ω^ω)

• Buchholz: ψ0(Ω·ω^ω^ω)

• Madore: ψ(ω^ω^ω)

• BMS: (0,0)(1,1)(2,0)(3,0)(4,0)

• Y: (1,2,4,5,6,7)

[72] {{{{0, 1}}, 1}}

• Ordinal: ε(ε0)

• Buchholz: ψ0(Ω·ψ0(Ω))

• Madore: ψ(ψ(0))

• BMS: (0,0)(1,1)(2,0)(3,1)

• Y: (1,2,4,5,7)

[73] {{{{0, 1}}, 1}{0, 1}}

• Ordinal: ε(ε0 + 1)

• Buchholz: ψ0(Ω·(ψ0(Ω) + 1))

• Madore: ψ(ψ(0) + 1)

• BMS: (0,0)(1,1)(2,0)(3,1)(1,1)

• Y: (1,2,4,5,7,4)

[74] {{{{0, 1}}, 1}{1, 1}}

• Ordinal: ε(ε0 + ω)

• Buchholz: ψ0(Ω·(ψ0(Ω) + ω))

• Madore: ψ(ψ(0) + ω)

• BMS: (0,0)(1,1)(2,0)(3,1)(1,1)(2,0)

• Y: (1,2,4,5,7,4,5)

[75] {{{{0, 1}}, 1}{{{0, 1}}, 1}}

• Ordinal: ε(ε0·2)

• Buchholz: ψ0(Ω·ψ0(Ω)·2)

• Madore: ψ(ψ(0)·2)

• BMS: (0,0)(1,1)(2,0)(3,1)(1,1)(2,0)(3,1)

• Y: (1,2,4,5,7,4,5,7)

[76] {{{{0, 1}} + 1, 1}}

• Ordinal: ε(ε0·ω)

• Buchholz: ψ0(Ω·ψ0(Ω)·ω)

• Madore: ψ(ψ(0)·ω)

• BMS: (0,0)(1,1)(2,0)(3,1)(2,0)

• Y: (1,2,4,5,7,5)

[77] {{{{0, 1}}{{0, 1}}, 1}}

• Ordinal: ε(ε0^2)

• Buchholz: ψ0(Ω·ψ0(Ω)^2)

• Madore: ψ(ψ(0)^2)

• BMS: (0,0)(1,1)(2,0)(3,1)(2,0)(3,1)

• Y: (1,2,4,5,7,5,7)

[78] {{{{0, 1} + 1}, 1}}

• Ordinal: ε(ε0^ω)

• Buchholz: ψ0(Ω·ψ0(Ω)^ω)

• Madore: ψ(ψ(0)^ω)

• BMS: (0,0)(1,1)(2,0)(3,1)(3,0)

• Y: (1,2,4,5,7,6)

[79] {{{{0, 1}{{0, 1}}}, 1}}

• Ordinal: ε(ε0^ε0)

• Buchholz: ψ0(Ω·ψ0(Ω)^ψ0(Ω))

• Madore: ψ(ψ(0)^ψ(0))

• BMS: (0,0)(1,1)(2,0)(3,1)(3,0)(4,1)

• Y: (1,2,4,5,7,6,8)

[80] {{{{0, 1}{0, 1}}, 1}}

• Ordinal: ε(ε1)

• Buchholz: ψ0(Ω·ψ0(Ω·2))

• Madore: ψ(ψ(1))

• BMS: (0,0)(1,1)(2,0)(3,1)(3,1)

• Y: (1,2,4,5,7,7)

[81] {{{{0, 1}{0, 1}{0, 1}}, 1}}

• Ordinal: ε(ε2)

• Buchholz: ψ0(Ω·ψ0(Ω·3))

• Madore: ψ(ψ(2))

• BMS: (0,0)(1,1)(2,0)(3,1)(3,1)(3,1)

• Y: (1,2,4,5,7,7,7)

[82] {{{{1, 1}}, 1}}

• Ordinal: ε(ε(ω))

• Buchholz: ψ0(Ω·ψ0(Ω·ω))

• Madore: ψ(ψ(ω))

• BMS: (0,0)(1,1)(2,0)(3,1)(4,0)

• Y: (1,2,4,5,7,8)

[83] {{{{{1}, 1}}, 1}}

• Ordinal: ε(ε(ω^ω))

• Buchholz: ψ0(Ω·ψ0(Ω·ω^ω))

• Madore: ψ(ψ(ω^ω))

• BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,0)

• Y: (1,2,4,5,7,8,9)

[84] {{{{{{0, 1}}, 1}}, 1}}

• Ordinal: ε(ε(ε0))

• Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω)))

• Madore: ψ(ψ(ψ(0)))

• BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)

• Y: (1,2,4,5,7,8,10)

[85] {{{{{{1, 1}}, 1}}, 1}}

• Ordinal: ε(ε(ε(ω)))

• Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ω)))

• Madore: ψ(ψ(ψ(ω)))

• BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)

• Y: (1,2,4,5,7,8,10,11)

[86] {{{{{{{{0, 1}}, 1}}, 1}}, 1}}

• Ordinal: ε(ε(ε(ε0)))

• Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω))))

• Madore: ψ(ψ(ψ(ψ(0))))

• BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,1)

• Y: (1,2,4,5,7,8,10,11,13)

[87] {{{{{{{{{{0, 1}}, 1}}, 1}}, 1}}, 1}}

• Ordinal: ε(ε(ε(ε(ε0))))

• Buchholz: ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω·ψ0(Ω)))))

• Madore: ψ(ψ(ψ(ψ(ψ(0)))))

• BMS: (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,1)(8,0)(9,1)

• Y: (1,2,4,5,7,8,10,11,13,14,16)

[88] {{{0, 1}, 1}}

• Ordinal: ζ0

• Buchholz: ψ0(Ω^2)

• Madore: ψ(Ω)

• BMS: (0,0)(1,1)(2,1)

• Y: (1,2,4,6)

[] {{0, 1}}

• Ordinal: 

• Buchholz: ψ0(Ω)

• Madore: ψ(0)

• BMS: (0,0)(1,1)

• Y: (1,2,4)