Section 1

Published in August 2017

Section 2

Published in December 2019

The functions collapsing α-weakly inaccessible cardinals were defined by me in August 2017 on base of Jäger’s and Hypcos’ functions. In the same month I also defined the fundamental sequences, which you can see above, and much later, in December 2019, created the ordinal notation, which is below.

Ordinal notation associated to the functions collapsing α-weakly inaccessible cardinals

We assume that <...> is a primitive recursive coding function on finite sequences of natural numbers.

1. Definition of set T

1. 0∈T

2. If α_1,…,α_n ∈ T then <1,α_1,…,α_n> ∈ T

3. If α, β ∈ T then <2,α,β> ∈ T

4. If α, β ∈ T then <3,α,β> ∈ T

2. Definition of V:On→T

1. V(0) = 0

2. V(α_1+…+ α_n) = <1,V(α_1),…,V( α_n)>

3. V(ψ_α(β)) =<2,V(α),V(β)>

4. V(I(α,β)) = <3,V(α),V(β)>

3. Definition of o:T→On

1. o(0) = 0

2. o(<1,α_1,…,α_n> ) = o(α_1)+…+o(α_n)

3. o(<2,α,β>) = ψ_{o(α)}(o(β))

4. o(<3,α,β>) = I(o(α), o(β))

4. Conventions

1. 1’ is an abbreviation of <2,<3,0,0>,0>

2. ω’ is an abbreviation of <2,<3,0,0>,<2,<3,0,0>,0>>

3. <α_1,…,α_m> = <β_1,…,β_n> ⇔ m = n and α_i = β_i for 1 ≤ i ≤ n

4. α ⊴ β ⇔ α ⊲ β or α = β

5. Definition of ⊗

α⊗η = 0 if η = 0

α⊗η = α if η = 1

α⊗η = <1,α,…,α> with η α's if η ≥ 2

Below we simultaneously define:

1. Sets of terms OT, P, R, L, S,

2. Function e(α) for α∈OT,

3. Coefficient sets G_π(α) for π∈R and α∈OT,

4. A linear ordering ⊲ on OT.

6. Definition of sets OT, P, R, L, S:

1. α=0 ⇒ α ∈ OT

2. α=1’ ⇒ α∈OT ∧ α∈P ∧ α∈S

3. α=<1,α_1,...,α_n> ∧ n≥2 ∧ α_1,...,α_n ∈ P ∧ α_n ⊴ … ⊴ α_1 ⊲ α ∧ α_n =1’ ⇒ α∈OT ∧ α∈S

4. α=<1,α_1,...,α_n> ∧ n≥2 ∧ α_1,...,α_n ∈ P ∧ α_n ⊴ … ⊴ α_1 ⊲ α ∧ 1’ ⊲ α_n ⇒ α∈OT ∧ α∈L

5. α=<3,β,γ> ∧ β,γ∈OT ∧ e(γ) ⊴ β ∧ γ∈L ⇒ α∈OT ∧ α∈P ∧ α∈L

6. α=<3,β,γ> ∧ β,γ∈OT ∧ e(γ) ⊴ β ∧ γ∉L⇒ α∈OT ∧ α∈P∧ α∈L ∧ α∈ R

7. α=<2,π,β> ∧ π,β∈OT ∧ π∈R ∧ G_π(β) ⊲ β ∧ 1’⊲ α ⇒ α∈OT ∧ α∈P ∧ α∈L

7. Definition of G_π(α) for π∈R and α∈OT

1. G_κ(0)=∅

2. G_κ (<1,α_1,…,α_n>) = G_κ(α_1) ∪… ∪ G_κ(α_n)

3. G_κ (<3,α,β>) = G_κ(α) ∪ G_κ(β)

4. G_κ (<2,α,β>) = ∅ if α ⊲ κ

5. G_κ (<2,α,β>) = {β} ∪ G_κ(α) ∪ G_κ(β) if κ ⊴ α

G_π(β) ⊲ β ⇔ o(β)∈C(o(β), ψ_{o(π)}(o(β)))

Note: G_π(β) ⊲ β abbreviates “γ⊲β for all γ∈G_π(β)”

8. Definition of function e(α) for α∈OT

1. e(0) = 0

2. e(<1,α_1,…,α_n>) = 0

3. e (<3,α,β>)= α

4. e (<2,π,β>)= e(π)

e(γ)⊴β ⇔ γ⊲<3,β,γ>

9. Definition of ⊲ for OT

All terms are supposed to be elements of OT.

a) 0 ⊲ α if 1’ ⊴ α.

b) <3,α,β> ⊲ <3,γ,δ> iff one of the following cases holds:

1. α ⊲ γ and β ⊲ <3,γ,δ>,

2. α = γ and β ⊲ δ,

3. γ ⊲ α and <3,α,β> ⊲ δ.

c) Let κ=<3,ρ,σ>. Then <3,β,γ> ⊲ <2,κ,α> iff one of the following cases holds:

1. β ⊲ ρ and γ ⊲ <2,κ,α>,

2. ρ ⊴ β and <3,β,γ> ⊲ κ.

otherwise <2,κ,α> ⊲ <3,β,γ>

d) <2,κ,α> ⊲ <2,π,β> iff one of the following cases holds:

1. κ ⊲ π and κ ⊲ <2,π,β>,

2. κ = π and α ⊲ β,

3. π ⊲ κ and <2,κ,α> ⊲ π.

e) <1,α_1,…,α_m> ⊲ <1,β_1,…,β_n> (2 ≤ m, 2 ≤ n) iff one of the following cases holds:

1. m < n and α_i = β_i for 1 ≤ i ≤ m,

2. there exists a k such that 1 ≤ k ≤ min{m,n} and α_k ⊲ β_k and α_i = β_i for 1 ≤ i < k.

f) Let β = <2,γ,δ> or β = <3,γ,δ>. Then

1. <1,α_1,…,α_k> ⊲ β if α_1 ⊲ β,

2. β ⊲ <1,α_1,…,α_k> if β ⊴ α_i for some 1 ≤ i ≤ k.

Note: α⊲β and α,β∈OT ⇔ o(α)<o(β).

10. Definition of fundamental sequences for α∈OT

1. If α =0 then R(α)=0 and α hasn’t a fundamental sequence.

2. If α=<2,<3,0,0>,0> then R(α)= α=1’ and α[0]=0.

3. If α=<3,β,γ> with R(γ)∈{0,1'} then R(α)=α and α[η]=η.

4. If α = <1,α_1,…,α_k> and k≥2 then R(α) = R(α_k) and

α[η] = <1,α_1,…,α_k[η]> if α_k[η] = <i,β,γ> where i∈{2,3},

α[η] = <1,α_1,…,α_{k-1},β_1,…,β_m> if α_k[η] = <1,β_1,…,β_m>,

α[η] = <1,α_1,…,α_{k-1}> if α_k[η] = 0 and k>2,

α[η] = α_1 if α_k[η] = 0 and k = 2.

5. If α = <2,<3,0,β>,0> and R(β)=1’ then R(α)=ω' and α[η] = <3,0,β>*⊗η

6. If α=<2,<3,0,β>,γ> and R(γ)=1’ and R(β)∈{0,1’} then R(α)=ω’ and α[η] = <2,<3,0,β>,γ[0]>⊗η.

7. If α=<2,<3,β,0>,0> and R(β)=1’ then R(α)=ω’ and α[0]=0 and α[η+1]=<3,β[0],α[η]>.

8. If α=<2,<3,β,γ>,0> and R(β)=R(γ)=1’ then R(α)=ω’ and α[0]=<1,<3,β,γ>*,1’> and α[η+1]=<3,β[0],α[η]>.

9. If α=<2,<3,β,γ>,δ> and R(β)=R(δ)=1’ and R(γ)∈{0,1’} then R(α)=ω’ and α[0]=<1,<2,<3,β,γ>,δ[0]>,1’> and α[η+1]=<3,β[0],α[η]>.

10. If α=<2,<3,β,0>,0> and R(β)∉{0,1’} then R(α)=R(β) and α[η]=<3,β[η],0>.

11. If α=<2,<3,β,γ>,0> and R(β)∉{0,1’}, R(γ)=1’ then R(α)=R(β) and α[η]=<3,β[η],<1,<3,β,γ>*,1’>>.

12. If α=<2,<3,β,γ>,δ> and R(β)∉{0,1’}, R(γ)∈{0,1’}, R(δ)=1’ then R(α)=R(β) and α[η]=<3,β[η],<1,<2,<3,β,γ>,δ[0]>,1’>>.

13. If α=<3,β,γ> and R(γ)∉{0,1’} then R(α)=R(γ) and α[η]=<3,β,γ[η]>.

14. If α=<2,π,β> and ω’⊴R(β)⊲π then R(α)=R(β) and α[η]=<2,π,β[η]>.

15. If α=<2,π,β> and π⊴R(β)=ρ then R(α)=ω’ and α[η]=<2,π,β[γ[η]]> with γ[0]=1’ and γ[η+1]=<2,ρ,β[γ[η]]>.

The function * used above is defined as follows:

1. <3,α,β>* = <3,α,β[0]> if e(β[0])⊴α

2. <3,α,β>* = β[0] if α⊲e(β[0])

11. Definition of FGH for α∈OT and n∈ℕ

1. If R(α)=0 then f(α,n)=n+1

2. If R(α)=1’ then f(α,n)= γ[n] where γ[0]=n and γ[m+1]= f(α[0],γ[m])

3. If R(α)=ω’ then f(α,n)= f(α[n],n)

If α = V(β) then for the function of fast-growing hierarchy f_β(n) = f(α,n)

12. Definition of F

F(n) = f(α[n],n) where α[n] = <2,<3,0,0>,β[n]> with β[0]=0 and β[m+1]=<3,β[m],0>

13. Definition of hyperoperations for α∈OT and n,k∈ℕ\{0}

1. If R(α)∈{0,1’} then h(n,α,1)=n

2. If R(α)=0 then h(n,α,k+1)=h(n,α,k)+n

3. If R(α)=1’ then h(n,α,k+1)=h(n,α[0],h(n,α,k))

4. If R(α)=ω’ then h(n,α,k)= h(n,α[k],n)

If α = V(β) then for extended arrows n↑^βk = h(n,α,k)

14. References

1. G. Jäger (1984). ρ-inaccessible ordinals, collapsing functions and a recursive notation system. Arch. Math. Logik Grundlagenforsch, Vol. 24, 49-62

2. M. Rathjen (1990). Proof-theoretic analysis of KPM. Arch. Math. Logic (1991) 30:377-403

3. Hyp cos/OCF vs Array Notation. Collapsing higher inaccessibility